Approximate Labeling via the Primal-Dual Schema

Transcription

1 Aroximate Labeling via the Primal-Dual Shema Nikos Komodakis and Georgios Tziritas Tehnial Reort CSD-TR February 1, 2005

2 Aroximate Labeling via the Primal-Dual Shema Nikos Komodakis and Georgios Tziritas Comuter Siene Deartment, University of Crete s: {komod, Tehnial Reort CSD-TR February 1, 2005 Abstrat A linear rogramming based framework is resented whih is aable of roviding ombinatorial-based aroximation algorithms to a ertain lass of NP-omlete lassifiation roblems. The resulting algorithms utilize tools from the duality theory of linear rogramming and have guaranteed otimality roerties. Finally, it is shown that state-of-the-art lassifiation tehniues an be derived merely as a seial ase of the onsidered framework.

3 Contents 1 Introdution 1 2 The rimal-dual shema Metri Labeling as a linear rogram Relaxed omlementary slakness onditions An intuitive view of the dual variables and some extra terminology Alying the rimal-dual shema to Metri Labeling The PD1 algorithm An intuitive understanding of the algorithm Construting the aaitated grah G x,y Udate of the rimal and dual variables PD2 algorithm Algorithm overview Udate of the rimal and dual variables PD3 algorithms: extending PD2 to the semimetri ase Algorithms PD3 a and PD3 b Algorithm PD Algorithmi roerties of the resented rimal-dual algorithms 23

4 1 Introdution The Metri Labeling Problem, introdued by Kleinberg and Tardos [1], an ature a broad range of lassifiation roblems that arise in early vision (e.g. image restoration, stereo mathing, image segmentation et.). Aording to this roblem, the task is to lassify a set V of n objets by assigning to eah objet a label from a given set L of labels. Eah labeling, i.e. a funtion f : V L, is assoiated with a ertain ost whih has 2 omonents. On one hand, for eah V there is a label ost,a 0 for assigning label a = f() to. On the other hand, for eah air of objets, there is a so-alled searation ost for assigning labels a = f(), b = f() to them. This searation ost is eual to w d ab where the uantities w are the edge weights of a grah G = (V, E) and reresent the strength of the relationshi between, while d ab is a distane funtion between labels whih is assumed to be a metri 1. Thus, the total ost euals C(f) = V,f() + (,) E w d f()f() and the goal is to find a labeling f with the minimum ost. For a onnetion between Metri Labeling and Markov Random Fields the reader is referred to [1]. Aording to one lass of aroximation algorithms [1, 2, 3] the Metri Labeling roblem is formulated as the Linear Programming relaxation of an integer rogram. This LP relaxation is then solved and a randomized rounding tehniue is being used to extrat a near the otimum integer solution. These algorithms have good theoretial roerties but sine they reuire the solution of a linear rogram whih, in the ase of early vision roblems, an grow very large, this makes them imratial to use. On the other hand, a variety of ombinatorial-based aroximation algorithms [4, 5, 6, 7, 8] have been develoed. These state-of-the-art tehniues are very effiient and have been alied with great suess to many roblems in omuter vision [9, 10, 11, 12, 13]. However, they have been interreted only as greedy loal searh tehniues u to now. The major ontributions of this aer are: A linear rogramming based framework that makes use of the rimal-dual shema in order to rovide effiient (i.e. ombinatorial-based) aroximation algorithms to the Metri Labeling roblem, thus bridging the ga between the two lasses of aroximation algorithms mentioned above. The derived algorithms have guaranteed otimality roerties even in the more general ase where d ab is merely a semimetri 2. These roerties assert the existene of worst-ase subotimality bounds meaning that the minimum generated by any of the onsidered algorithms is always within a known fator of the global otimum. Grah-ut tehniues introdued in [4] an be derived as a seial ase of our framework whih is thus shedding further light on the essene of those algorithms. In artiular, it is the first time that these (state of the art) algorithms are being interreted not merely as greedy loal searh tehniues but in terms of riniles drawn from the theory of linear rogramming. In addition to the theoretial (worst-ase) subotimality bounds, the onsidered algorithms also rovide er-instane subotimality bounds for the generated solutions. This way one may better inset how suessful the onvergene of the algorithm has been for eah seifi instane. In ratie these er-instane bounds always rove to be muh tighter than the theoretial (worst-ase) ones, thus showing that the generated minimum is very lose to the global otimum eah time. Sine grah-ut tehniues an be inluded as a seial ase, this last fat exlains in another way the great suess that these tehniues exhibit in ratie. 2 The rimal-dual shema Consider the following rimal-dual air of linear rograms: PRIMAL: min T x DUAL: max b T y s.t. Ax = b, x 0 s.t. A T y where A = [a ij ] is an m n matrix and b, are vetors of size m, n resetively. We would like to find an otimal integral solution to the rimal rogram. But sine this is in general an NP-omlete roblem we need to settle with estimating aroximate solutions. A rimal-dual f-aroximation algorithm ahieves that by use of the following rinile: 1 i.e. d ab = 0 a = b, d ab = d ba 0, d ab d a + d b 2 i.e. d ab = 0 a = b, d ab = d ba 0 1

5 Primal-Dual Prinile. If x and y are integral-rimal and dual feasible solutions satisfying: then x is an f-aroximation to the otimal integral solution x i.e. T x f T x T x f b T y (1) The above rinile, whih is a a onseuene of the Weak Duality Theorem, lies at the heart of any rimal-dual tehniue. In fat, the various rimal-dual methods mostly differ in the way that they manage to estimate a air (x, y) satisfying the fundamental ineuality (1). One very ommon way for that (but not the only one) is by relaxing the so-alled rimal omlementary slakness onditions [14]: Theorem (Relaxed Comlementary Slakness). If the air (x, y) of integral-rimal and dual feasible solutions satisfies the so-alled relaxed rimal omlementary slakness onditions: m x j > 0 a ij y i j /f j i=1 then (x, y) also satisfies the Primal-Dual Prinile with f = max j f j and therefore x is an f-aroximation to the otimal integral solution. Based on the above theorem, during a rimal-dual f-aroximation algorithm the following iterative shema is usually being used: Primal-Dual Shema. Generate a seuene of airs of integral-rimal, dual solutions {x k, y k } t k=1 until the elements x = xt and y = y t of the last air of the seuene are both feasible and satisfy the relaxed rimal omlementary slakness onditions. It should be noted that the exat slakness onditions (i.e. f j = 1) are always satisfied by the rimal-dual air of otimal frational solutions. 2.1 Metri Labeling as a linear rogram Here we will onsider the following integer rogramming formulation of the Metri Labeling roblem, introdued in [2]: min,a x,a + d ab x,ab (2) V,a L (,) E w a,b L s.t. x,a = 1 V (3) a x,ab = x,b b L, (, ) E (4) a x,ab = x,a a L, (, ) E (5) b x,a, x,ab {0, 1} V, (, ) E, a, b L The {0, 1}-variable x,a indiates that vertex is assigned label a while the {0, 1}-variable x,ab indiates that vertex is labeled a and vertex is labeled b. The variables x,ab, x,ba therefore indiate exatly the same thing and should oinide. So, in order to redue the number of variables in the rimal roblem, we adot the onvention that for any neighbors, exatly one of (, ), (, ) E. This in turn imlies that exatly one of the variables x,ab, x,ba is being used for eah air of neighbors,. The notation will hereafter denote the fat that, are neighboring verties and will mean either (, ) E or (, ) E. The first onstraints (3) simly exress the fat that eah vertex must reeive a label while onstraints (4), (5) maintain onsisteny between variables x,a, x,b and x,ab in the sense that if x,a = 1, x,b = 1 they fore x,ab = 1 as well. By relaxing the {0, 1} onstraints to x,a 0, x,ab 0 we get a linear rogram. The dual of that linear rogram has the following form: max y s.t. y,a + : y,a y,a + y,b w d ab V, a L a, b L, (, ) E 2

6 To eah vertex, there orresonds one dual variable y. Also, to eah air of neighboring verties, (and any label a), there orresond 2 dual variables y,a and y,a. Note that this is in ontrast to the rimal roblem where, for eah air of neighboring verties, (and any labels a, b), only one of the 2 variables x ab, x,ba is being used deending on whether (, ) E or (, ) E. All the dual variables {y,a } a L,: 3 will be alled balane variables hereafter while also for eah air y,a, y,a of these balane variables we will say that y,a is the onjugate balane variable of y,a (and vie versa) or euivalently that y,a, y,a are onjugate variables. By defining the auxiliary variables ht y,a as: the dual roblem is trivially transformed into: ht y,a,a + : y,a (6) max y s.t. y ht y,a V, a L (7) y,a + y,b w d ab a, b L, (, ) E (8) The dual variables ht y,a will be alled height variables hereafter. The reason for this as well as for introduing these redundant variables will beome lear in the setions that are following. For defining a dual solution, only the balane variables y,a as well as the y variables need to be seified. The auxiliary height variables ht y,a are then omuted by (6). 2.2 Relaxed omlementary slakness onditions The relaxed rimal omlementary slakness onditions, related to the seifi air of rimal-dual linear rograms of the revious setion, are: x,a > 0 y,a /f 1 + : y,a (9) x,ab > 0 y,a + y,b w d ab /f 2 (10) During the rimal-dual shema we will be onsidering only feasible {0, 1}-rimal solutions. It is not diffiult to see that suh solutions an be omletely seified (i.e. all rimal variables x,a, x,ab an be estimated) one we know what label has been assigned to eah vertex. For this reason, a rimal solution x will hereafter refer to a set of labels {x } V where x denotes the label assigned to vertex. Under this notation, x,a > 0 (i.e. x,a = 1 sine we are dealing only with {0, 1} solutions) is euivalent to x = a and so the relaxed rimal omlementary slakness onditions (9) are trivially redued to: y,x /f 1 + : y,x (11) In a similar fashion, x,ab > 0 is euivalent to x = a and x = b and the omlementary slakness onditions (10) are trivially redued to: x x y,x + y,x w d x x /f 2 (12) x = x = a y,a + y,a = 0 (13) where we onsider the ases a b and a = b searately. During any of the algorithms that will follow our objetive will be to find feasible solutions x, y satisfying the above omlementary slakness onditions (11), (12) and (13). The last slakness onditions (13) simly exress the fat that onjugate balane variables should be oosite to eah other. For this reason we set by definition: y,a y,a (, ) E, a L (14) Therefore slakness ondition (13) will always be true hereafter and so we will have to take are for fulfilling only onditions (11) and (12). 3

7 w y ht, a y y b ht, ht, α=x b y, y, α=x b y ht, y ht, a y b ht, referene lane Fig. 1: A visualization of the dual for a very simle instane of the Metri Labeling Problem where the grah G onsists of just 2 neighboring verties, and the set of labels L is eual to {a, b, }. To eah of the verties, there orresonds a searate set of labels {a, b, } (eah label is reresented by a irle) and all of these labels are loated at ertain heights relative to a ommon referene lane. The values of these heights are set eual to the dual variables ht and therefore deend on the balane variables. Label at is ulled u due to the inrease of the balane variable y, and so the orresonding label at neighboring vertex is ulled down due to the derease of the onjugate variable y,. The labels whih are urrently assigned to verties, are drawn with a thiker line. 2.3 An intuitive view of the dual variables and some extra terminology A way of viewing/visualizing the dual variables, that will rove useful when designing our aroximation algorithms later, is the following: for eah vertex, we onsider a searate oy of the omlete set of labels L. One then may assume that all of these labels are objets whih are loated at ertain heights relative to a ommon referene lane (see Fig. 1). The height of label a at vertex is given by the dual variable ht y,a. Exressions like label a at is below/above label b imly ht y,a ht y,b. The role of the balane variables is to ontribute to the inrease or derease of a vertex s height. In artiular, due to (6), the height of a label a at an be altered only if at least one of the balane variables {y,a } : is altered as well. In addition, due to the fat that onjugate balane variables are oosite to eah other (see (14)), hanges in the height of label a at also affet the height of that label at a neighboring vertex. In Fig. 1, for examle, eah time we inrease the height of label at, say by inreasing balane variable y,, the height of at the neighboring vertex is dereased by the same amount due to the derease of the onjugate variable y,. Before roeeding let us also define some terminology that will be used freuently throughout this doument. Let x, y be any air of integral-rimal, dual solutions. We will all label x the assigned label to or euivalently the ative label at. We will also refer to the height of an assigned label to vertex (i.e. ht y,x ) as merely the height of. Based on this definition, a funtion (denoted as AP F x,y hereafter) whih will lay an imortant role in all of the onsidered rimal-dual algorithms is the sum of the heights of all verties i.e. AP F x,y = hty,x. If x, y satisfy the exat (i.e. f 1 = 1, f 2 = 1) slakness onditions (11),(12), then it is easy to rove that APF oinides with the value of the rimal objetive funtion while if the relaxed slakness onditions hold then it is also easy to rove that APF stays lose to the atual value of the rimal objetive funtion. For this reason APF will be alled the Aroximate Primal Funtion hereafter. Another signifiant onet is that of an ative balane variable. We define as an ative balane variable at a vertex any balane variable belonging to the following set {y,x } : (i.e. any balane variable of the form y,x where is any neighboring vertex of ). Based on the ative balane variable onet, we may also introdue another very imortant uantity whih is alled the load between two neighbors, (load x,y ) and is eual to the sum of the 2 ative balane variables y,x, y,x i.e. load x,y = y,x + y,x. If relaxed slakness onditions (12) are to be satisfied, then it is easy to see that the load between, an be thought of as a virtual searation ost whih is always a rough aroximation of the atual searation ost w d xx between,. This an be verified as follows: if the searation ost between, is zero (i.e. x = x ) then so is load x,y due to (14). While if the searation ost is not zero (i.e. x x ) then it holds that w d xx /f 2 load x,y w d xx where the 1 st ineuality is due to slakness onditions (12) and the 2 nd one is due to the dual onstraints (8). Another useful thing to note is that there exists a diret relationshi between the value of the APF funtion and the loads. In artiular, it holds that: AP F x,y =,x + load x,y (15) (,) E 3 Aording to our notation the set {y,a } a L,: euals the set {y,a, y,a } a L,:(,) E 4

8 1: k 1; x k INIT PRIMALS( ); y k INIT DUALS( ); 2: LabelChange 0 3: for eah label in L do 4: ȳ k PREEDIT DUALS(, x k, y k ); 5: [x k+1, ȳ k+1 ] UPDATE DUALS PRIMALS(, x k, ȳ k ); 6: y k+1 POSTEDIT DUALS(, x k+1, ȳ k+1 ); 7: if x k+1 x k then LabelChange 1 8: k++; 9: end for 10: if LabelChange = 1 (i.e. at least one vertex has hanged its label) then goto 2; 11: if algorithm PD1 then y fit DUAL FIT(y k ); Fig. 2: Pseudoode showing the basi struture of the algorithms PD1, PD2 and PD3. One an very easily verify the above euation as follows: AP F x,y = ht y,x = (,x + ) y,x = : =,x + load x,y (,) E,x + ) (y,x + y,x (,) E 2.4 Alying the rimal-dual shema to Metri Labeling The majority of the aroximation algorithms that will be resented here ahieve an aroximation fator of f a = 2 dmax d min with d min min a b d ab and d max max a b d ab. The distintion between the onsidered algorithms will be lying in the exat values they assign to the onstants f 1, f 2 that are used in the relaxed omlementary slakness onditions (11),(12). Another imortant differene will be that some of them are aliable even in the more general ase of d ab being a semimetri 4. The basi struture of any of the onsidered algorithms an be seen in Fig. 2. The initial rimal-dual solutions are generated inside INIT PRIMALS and INIT DUALS resetively. During eah inner iteration (lines 4-8 in Fig. 2) a label is seleted and a new rimal-dual air of solutions (x k+1, y k+1 ) is generated by udating the urrent air (x k, y k ). It should be noted that among all balane variables of y k (i.e. {y,a} k a L,: ) only the balane variables of the labels (i.e. {y,} k,: ) are modified. We all this a -iteration of the algorithm. L suh iterations (one -iteration for eah label in the set L) make u an outer iteration (lines 2-9 in Fig. 2) and if no vertex hanges its label during the urrent outer iteration the algorithm then terminates. The role of the routines whih are being exeuted during an inner -iteration is as follows: the role of PREEDIT DUALS is to edit solution y k into solution ȳ k that is going to be used as an inut to the UPDATE DUALS PRIMALS routine. That routine is resonsible for the main udate of the rimal and dual variables and to this end it generates the air of solutions (x k+1, ȳ k+1 ). Finally, POSTEDIT DUALS alies further modifiations to ȳ k+1 thus roduing the next dual solution y k+1. This solution y k+1 along with x k+1 onstitute the next rimal-dual air of solutions. The algorithms to be onsidered are named PD1, PD2, PD3 and the DUAL FIT routine, whih is being used only in the last two of them, serves only the urose of alying a saling oeration to the last dual solution (as we shall later see). Sine we will be dealing only with aroximation algorithms, we may hereafter assume w.l.o.g. that all oeffiients,a, w, d ab of the Metri Labeling integer rogram (2) are nonnegative integers. If this was not true, then we ould aroximate (to any reision) those oeffiients by rational numbers thus generating an instane of the Metri Labeling roblem whih in turn an be trivially transformed into an integer rogram (2) with integral oeffiients. We ould then aly all of our aroximation algorithms to this last integer rogram. 4 The linear rogramming formulation of Metri Labeling is still valid 5

9 3 The PD1 algorithm During this setion we will assume that d ab is a semimetri. In the artiular ase of the PD1 algorithm our goal will be to find feasible solutions x, y satisfying slakness onditions (11), (12) with f 1 = 1 and f 2 = f a. By relaing f 1 = 1 in (11) that ondition beomes y ht y,x. Sine it also holds that y min a ht y,a (by the dual onstraints (7)), it is easy to see that (11) redues to the following 2 euations: y = min ht y,a a (16) ht y,x = min ht y,a a (17) In addition, by making use of the definition of the load as load x,y = y,x + y,x and by relaing f 2 = f a in (12) that ondition beomes euivalent to: x x load x,y w d xx /f a (18) Therefore the objetive of PD1 is to find feasible x, y satisfying onditions (16)-(18). PD1 uses the following strategy to ahieve its goal: during its exeution it generates a series of rimal-dual airs of solutions, one rimal-dual air er iteration. At eah iteration it makes sure that onditions (16) and (18) are automatially satisfied by the urrent rimal-dual air. In addition, it makes sure that the urrent dual solution is feasible (rimal solutions are always integral-feasible by onstrution). To this end it enfores that the urrent dual solution always satisfies the following onstraints: y,a w d min /2 a L, (19) To see that (19) ensures feasibility, it is enough to observe that due to this onstraint the following ineuality an be derived: y,a + y,b 2w d min /2 = w d min w d ab and so the dual onstraints (8) hold true. This imlies that solution y is indeed feasible sine the other dual onstraints (7) already hold true due to ondition (16). All that remains then for PD1 to ahieve its goal is just to ensure that after a finite number of iterations slakness onditions (17) are satisfied as well. This last objetive (i.e. driving the rimal-dual airs towards satisfying (17)) will be the final and key issue for the suess of the onsidered algorithm. To this end, PD1 will be trying to ensure that the number of verties for whih euation (17) holds true inreases after eah one of its iterations. 3.1 An intuitive understanding of the algorithm Let us now give some feel for how PD1 is really trying to ahieve that last objetive. Before roeeding, it should be noted that enforing ondition (16) is always a trivial thing to do (we simly need to set eah dual variable y eual to min a ht y,a), so we do not really have to worry about that ondition throughout PD1. Let x, y be the urrent air of integral-rimal and dual feasible solutions satisfying all reuired onditions (16)-(19) exet for (17). That ondition simly reuires that the label x assigned to any vertex must be lower than all other labels at that vertex. So let be a vertex for whih this ondition fails i.e. one of its labels, say, is loated below the ative label x at that vertex. To restore (17) we need to raise label u to x by inreasing one of the balane variables {y, } :. But as already mentioned, eah time we inrease y, its onjugate variable y, dereases and so the height of at the neighboring vertex dereases as well. This may have as a result that label at gets below the ative label x. Therefore we must be areful whih balane variables we hoose to inrease and by how muh or otherwise we may break ondition (17) for some neighboring vertex. This means that the inrease/derease of the balane variables must roeed in an otimal way so that ondition (17) is restored for as many verties as ossible. Based on this observation, during any iteration of the algorithm the udate of the rimal and dual variables roughly roeeds as follows: Dual variables udate: given the urrent rimal solution (i.e. the urrent label assignment), we kee the heights of all ative labels fixed and then for eah vertex we try so that all of its labels are raised above the vertex s ative label. Of ourse, we only need to do that for the labels that are below the ative label. To this end we need to udate the dual balane variables in an otimal way. As we shall see any udate of the balane variables an be simulated by ushing flow through an aroriately onstruted aaitated grah while the otimal udate an be ahieved by ushing the maximum flow through that grah. Of ourse, are must also be taken so that onstraints (19) do not beome violated during this udate of the dual variables or else the resulting dual solution might not be feasible. Constraints (19) imose uer bounds on the values of the balane variables, restriting this way the maximum allowed inrease that we may aly to the height of a label. 6

10 t (sink) y x ht, y ht, w w r r a=x y ht, f -f y ht r, a a f - -f ht y ht y, x =x r, x r a=x r a = wdmin 2 y, f y y f a t = ht, ht, x a r = 0 f f f r a = wdmin 2 y, a s = ht y y, x ht, s (soure) f r f r a r = 0 a sr =1 r (a) (b) Fig. 3: (a) An arrangement of labels (reresented by irles) for a simle instane of the Metri Labeling roblem onsisting of 3 verties,, r and 2 edges, r with weights w, w r. The label set is L = {a, }. The irles with the thiker line reresent the ative labels. Also, the red arrows indiate how the labels will move in resond to the udate of the dual variables while the irles with the dashed line show the final osition of those labels after the udate. (b) The orresonding aaitated grah G x,y is shown. A maximum flow algorithm is alied to this grah for udating the dual variables. Interior edges are drawn with a solid line while exterior edges are drawn with a dashed line. Caaities of both interior and exterior edges are also shown. Primal variables udate: after the otimal rearrangement of the labels heights, there might still be some verties whose ative labels are not the ones with the lowest height (among all labels at the vertex), violating this way ondition (17). We selet a suitable subset of these verties and assign to them new labels whih are at lower heights than the revious ative labels so that the resulting rimal solution is taken loser to satisfying (17) too. The reason we may not be able to do that for all the verties is that we must still take are that the other slakness onditions (18) are maintained as well. Nevertheless, the number of verties violating (17) dereases er iteration and so by kee reeating this udate of the rimal and dual variables it an be shown that in the end the ative label of eah vertex will have the lowest height at that vertex and the last rimal-dual air will therefore satisfy all reuired onditions (16)-(19). 3.2 Construting the aaitated grah G x,y The rearrangement of the label heights takes lae in grous. Given as inut a urrent rimal-dual air of solutions (x, y) and a label, UPDATE DUALS PRIMALS(, x, y) rearranges only the heights of the labels. To this end it hanges solution y into solution y by hanging only the balane variables {y, },: (i.e. the balane variables of all labels) into {y,},:. The goal of this udate will be to have the resulting heights ht y, rearranged in an otimal way so that as many of the labels as ossible end u being above the urrently ative labels. In addition, we need to make sure that the new dual solution y does not break onditions (19). In the simle ase resented in Fig. 3(a), for examle, we would like to have label at move at least as high as label a at (the ative label of ) without, at the same time, label at gets lower than label a at (the ative label of ). Label at r does not need to move at all sine it is already the ative label of vertex r and has the lowest height in there. It turns out that in the general ase the udate of both the balane variables and the labels heights an be simulated by ushing flow through an aroriately onstruted direted grah G x,y = (V x,y, E x,y, C x,y ) with aaities C x,y. In fat, as we shall see later, the otimal udate orresonds to ushing the maximum amount of flow through that grah. Suh a aaitated grah, assoiated to the simle roblem of Fig. 3(a), is resented in Fig. 3(b). Let us now exlain how suh a grah an be onstruted as well as how ushing flow through that grah relates to the udate of the dual variables. The nodes V x,y of G x,y onsist of all the nodes of grah G (these are the internal nodes) lus two seial external nodes the soure s and the sink t. The nodes of G x,y are onneted by two tyes of edges: interior edges (drawn with solid lines in Fig. 3(b)) and exterior edges (drawn with dashed lines in Fig. 3(b)). Interior edges: For eah edge (, ) G, we insert 2 direted interior edges and in grah G x,y. The amount of flow f leaving through reresents the inrease of the balane variable y, while the amount of flow f entering through interior edge reresents the derease of the same variable y,. The total hange of y, will therefore be: y, = y, + f f (20) The total hange in y, is defined symmetrially sine any flow oming out of through will enter (and vie versa). It 7

11 is then obvious that y, = y, and so onjugate balane variables remain oosite to eah other as they should. Based on (20), it is also easy to see that the aaity a of an interior edge reresents the maximum allowed inrease of the y, variable (attained if f = a, f = 0) and therefore the uantity y, + a reresents the maximum value of the new balane variable y,. Similar onlusions an be drawn regarding the aaity a of the reverse edge. Based on these observations the aaities a, a are assigned as follows: if the urrent label of (or ) is already then we want to kee the height of at (or ) fixed during the urrent iteration and so the aaity for all interior edges in or out of (or ) must be zero. Therefore: x = or x = a = a = 0 (21) Otherwise (i.e. x, x ), we set the aaity of the edges, so that the values of the new balane variables y,, y, an never exeed w d min /2 and feasibility onditions (19) are therefore maintained for the new dual solution y. For this reason we set: y, + a = w d min /2 = y, + a (22) Exterior edges: Eah internal node will be onneted to either the soure node s or the sink node t through an exterior edge. Whether we hoose the soure or the sink deends on the relative heights at vertex of the labels and x (the ative label of ). There are 3 ossible ases: Case 1: If is below x (i.e. ht y, < ht y,x ) then (as exlained in setion 3.1) we would like to raise label by exatly as muh as needed so that it reahes label x. This is the ase, for examle, with vertex in Fig. 3(a) where we would like that label at reahes label a at. To this end, we onnet the soure node s to node through a direted edge s. The flow f assing through that edge has the following interretation: it reresents the total inrease in the height of label (taking into aount the ontribution from the hange of all balane variables): ht y, = ht y, + f (23) To verify this, it is enough to ombine the flow onservation onstraint at node whih an be easily seen to redue to: f = ( ) f f (24) : and the fat that f f reresents the total hange of the balane variable y, i.e. f f = y, y, (see (20)). Indeed, it then follows that: ht y, + f = (, + ) y, + f : = (, + : = (, + : y, ) + : y, ) + : ( f f ) ( y, y, ) =, + : y, = ht y, Based on this observation, the aaity a s of the edge s reresents the maximum allowed raise in the height of. Therefore, sine we need to raise only as high as the urrent label of but not higher than that, we simly set this aaity as follows: a s = ht y,x ht y, (25) The aaity of edge s in Fig. 3(b) is defined this way. Case 2: If is not below x (i.e. ht y, ht y,x ) and is also not the ative label of (i.e. x ) then we an afford a derease in the height of as long as remains above x. In suh a ase, we onnet the node to the sink node t through a direted edge t. This time the flow assing through edge t will reflet the total derease in the height of (taking again into aount the ontribution from the hange of all balane variables): ht y, = ht y, f (26) The aaity of this edge therefore reresents the maximum derease in the height of label and sine this label must remain above x, aaity a t is defined, in a similar fashion to (25), as: a t = ht y, ht y,x (27) 8

12 In Fig. 3(a), for examle, label at vertex needs to remain above label a at and so in Fig. 3(b) the aaity of edge t is defined by alying (27) to vertex. Case 3: Finally, if is the ative label of (i.e. = x ) then we want to kee the height of fixed at the urrent iteration. As in ase 1 above, we again onnet the soure node s to node through direted edge s. This time, however, no flow asses through the interior edges inident to (i.e. f = f = 0 for any neighboring vertex ) sine these edges have zero aaity now (see (21)). So f = 0 (due to (24)) and no flow asses through edge s as well whih in turn imlies that the height of label will not hange (see (23)), as was intended. By onvention we set the aaity a s of edge s eual to one: a s = 1 (28) Vertex r in Fig. 3(a) belongs to this ase and this is the reason why we set a sr = 1 in the grah of Fig. 3(b). 3.3 Udate of the rimal and dual variables We are now ready to desribe what ations are erformed by eah of the main routines of PD1 during a -iteration. PREEDIT DUALS(, x k, y k ): For all of the onsidered algorithms the role of this routine will be to edit urrent solution y k into solution ȳ k that will be used (along with x k ) as inut for the onstrution of the aaitated grah G xk,ȳ k of setion 3.2. In the seifi ase of PD1, no udate takes lae inside PREEDIT DUALS and so ȳ k = y k. UPDATE DUALS PRIMALS(, x k, ȳ k ): After the onstrution of the grah G xk,ȳ k (as exlained in setion 3.2), a maximum flow algorithm [15] is alied to it and the resulting flows on interior edges are used in udating the dual balane variables. More seifially, only the balane variables of the labels are udated as follows (see (20)): ȳ k+1, = ȳ k, + f f (29) Therefore the heights of all labels will also hange as (see (23), (26)): { htȳk+1, = htȳk f if is onneted to node s, + f if is onneted to node t (30) In the toy examle of Fig. 4(a) you an see an initial arrangement of labels heights based on the values that the dual variables take at the start of a -iteration while in Fig. 4(b) you an see the resulting rearrangement of the labels heights due to the udate of the dual variables after alying the maximum-flow algorithm to the assoiated grah G xk,ȳ k of Fig. 4(d). The orresonding flows are also shown in that figure. Based on the resulting heights, we now need to udate the rimal variables i.e. assign new labels to the verties of G. Sine only the heights of labels have been altered, the latter amounts to deiding whether eah vertex kees its urrent label or is assigned the label. On one hand, this must be done so that the labels of any vertex are taken loser to satisfying (17) (i.e. the ative label of should also be the lowest one at ). This ratially means that if label at has not managed to get above the ative label of, then we need to assign as the new label of that vertex. For examle, in the ase of the udated heights of Fig. 4(b) we would like vertex to be assigned label while the rest of the verties, r should maintain their urrent labels. On the other hand, we must also take are maintaining ondition (18). It turns out that both of the above riteria an be fulfilled by onsidering the flows through both interior and exterior edges of G xk,ȳ k and making use of the following rule: REASSIGN RULE. Label will be the new label of (i.e. x k+1 (otherwise kees its urrent label i.e. x k+1 = x k ). = ) unsaturated 5 ath between the soure and node In Fig. 4() you an see the new assignment of labels to verties that has resulted after alying the above rule to the grah of 4(d). Vertex gets indeed a new label beause edge s is unsaturated while, r maintain their ative labels sine no unsaturated ath to them exists in G xk,ȳ k. As mentioned above, the new assignment obviously oinides with what we would like to ahieve initially. Before roeeding let us state some very useful roerties resulting out of the hoie of the reassign rule (these roerties will hold for all of the onsidered algorithms): 5 A ath is unsaturated if flow < aaity for all forward ars and flow > 0 for all bakward ars 9

13 w =200 w r =1000 r a=x a=x a=x r w =200 w r =1000 r a=x 120 f 70 -f a=x f r 70 -f r a=x r (a) w =200 w r =1000 r a =x a=x a=x r (b) t a t =50 a rt =50 f f r =50 =50 a =100 a r =500 f =100 f r =50 r f =0 f r =0 a =100 a r =500 f =100 a s =125 s () (d) Fig. 4: (a) The initial arrangement of the labels heights at the start of the urrent -iteration for a toy examle of the Metri Labeling roblem. All of the verties,, r are urrently assigned label a (as indiated by the irles with the thiker line). (b) The red and blue arrows show how the labels will move due to the udate of the balane variables after alying a maximum flow algorithm to the grah in (d). Label movements due to hanges in balane variables that are onjugate to eah other are drawn with the same line style and olor. Furthermore, the dashed irles indiate the final ositions of the labels. () The new ative labels (thik irles) that have been seleted based on the reassign rule are shown here. Only vertex had to hange its label into sine the exterior edge s of the grah in (d) is unsaturated. (d) The assoiated aaitated grah (assuming that initially all balane variables are zero) and the resulting flows after alying a maximum flow algorithm. The flows f, f, f r at the exterior edges are eual to the total hange in the height of the labels at,, r resetively. In this examle the Potts metri has been used for the distane d ab (i.e. if a b d ab = 1). Proerties 3.1. Let, be two neighboring verties i.e.. Then during a -iteration: (a) a ȳ,a k+1 = ȳ,a, k htȳk+1,a = htȳk,a (b) x k = x k+1 =, ( ȳ k+1,, ȳ k+1, ) = (ȳk,, ȳ k, () htȳk+1 htȳk,x k+1,x k (d) htȳk+1,x k+1 htȳk+1, ), ht ȳ k+1,x k+1 = htȳk,x k (e) if is assigned label but kees its urrent label (i.e. x k+1 = and x k+1 = x k ), then ȳ, k+1 = ȳ, k + a i.e. the balane variable ȳ, k+1 attains its maximum value (f) (APF monotoniity) AP F xk+1,ȳ k+1 AP F xk,ȳ k Furthermore, if at least one hange of label has taken lae during the urrent -iteration then AP F xk+1,ȳ k+1 < AP F xk,ȳ k Proof: (a) This roerty follows diretly from the fat that only the balane variables of the labels are udated during a - iteration, by definition. (b) Due to x k = and (21), the aaities of all interior edges, with adjaent to (i.e. ) will be zero and so no flow an ass through them i.e.: f = f = 0 : (31) 10

14 If we then aly the flow onservation at node (24), we an see that the flow through edge s will be zero as well (i.e. f = 0) whih in turn imlies that the edge s is unsaturated (sine a s = 1 by (28)). Therefore by the reassign rule it will also be x k+1 =. Finally, the euality ( ) ȳ, k+1, ȳ, k+1 (ȳk =,, ȳ,) k follows diretly from alying (31) to = and then using (29) while the other euality htȳk+1 = htȳk follows from f,x k = 0 and (30).,x k+1 () if x k+1 then it will neessarily hold x k+1 kees its urrent label x k. Therefore it will also be x k. So by setting x k+1 roerty (a) and easily onlude that htȳk+1,x k+1 Therefore we may hereafter assume that x k+1 easily verify the roerty as follows: = x k sine by the reassign rule a vertex is either assigned label or = x k = a we may now aly = htȳk whih means that the roerty holds in this ase.,x k =. In that ase, if is onneted to the soure node s then we may htȳk+1 = htȳk+1,x k+1, = htȳk, + f by (30) htȳk, + a s = htȳk, + ( htȳk,x k = htȳk,x k ) htȳk, by (25) Let us now onsider the ase where is onneted to the sink t: sine we assume x k+1 = the reassign rule imlies that there must be an unsaturated ath, say s, from s to. But it must then hold f = a t or else there will also be an unsaturated ath s t between the soure and the sink whih is imossible due to the max-flow min-ut theorem [15]. Combining this fat (i.e. f = a t ) with (30) and the definition of a t in (27) it then follows that: htȳk+1 = htȳk+1,x k+1, = htȳk, f = htȳk, a t = htȳk, ( htȳk, htȳk,x k ) = ht ȳ k (d) If x k+1 = the roerty obviously holds. So we may assume that x k+1. In that ase, it will also be x k as well (due to roerty (b)). If is onneted to the soure node s then the ar s must be saturated i.e. f = a s or else it would hold x k+1 = aording to the reassign rule. Using this fat as well as (30) and the definition of a s in (25) the roerty then follows: htȳk+1, = htȳk, + f = htȳk, + a s = htȳk, + ( htȳk,x k where the last euality is true due to the fat x k and roerty (a). On the other hand, if is onneted to the sink t then: htȳk+1,,x k ) htȳk, = ht ȳ k = htȳk,x k,x k+1 = htȳk, f by (30) htȳk, a t = htȳk, ( htȳk, htȳk,x k = htȳk = htȳk,x k,x k+1 where again the last euality is true due to the fat x k and roerty (a). ) by (27) 11

15 (e) If x k = then a = 0 (due to (21)) while also ȳ, k+1 = ȳ, k (by roerty (b)) and so the roerty obviously holds. Therefore we may assume that x k whih imlies that x k+1 as well (sine x k+1 = x k ). Sine has been assigned the label there must exist an unsaturated ath s from s to. But then the forward ar as well as the bakward ar of the ath s must be saturated i.e.: f = a and f = 0 (32) or else that ath would also be unsaturated (whih would in turn imly that x k+1 above). Due to (32) and (29) the roerty then follows. = ontrary to our assumtion (f) The first ineuality follows diretly from () and the definition of the APF funtion. Furthermore, if at least one hange of label has taken lae then aording to the reassign rule there must be at least one unsaturated ar, say s, between the soure and some node. This imlies that f < a s and so by also using (30) and the definition of a s in (25) it is then trivial to show that htȳk+1 < htȳk Due to this fat and by alying roerty () to all other verties the,x. k desired strit ineuality follows.,x k+1 Proerty 3.1(a) simly exresses the fat that only dual variables related to the -labels may hange during a -iteration while roerty 3.1(b) simly says that if label is the ative label during a -iteration then its height is ket fixed during that iteration. Note also that due to roerty 3.1() the height of a vertex (i.e. the height of its ative label) dereases after eah iteration of the algorithm. Furthermore, due to roerty 3.1(d), the new ative label at the end of a -iteration (i.e. the label assigned to a vertex by x k+1 ) is always loated below that vertex s label (as was intended). Based on these observations the new label assigned to a vertex by x k+1 is always taken loser to having the minimum height at that vertex after the end of eah iteration. We may therefore hoe that by kee reeating this roedure we will finally make sure that (17) is satisfied after a suffiient number of iterations has elased. Besides that, however, we also need to ensure that x k+1, y k+1 satisfy onditions (18). To this end, the routine POSTE- DIT DUALS(, x k+1, ȳ k+1 ) still needs to aly an additional orretion to the resulting dual solution ȳ k+1. In artiular, the role of that routine will be to hange ȳ k+1 into y k+1 so that all of the ative balane variables of solution y k+1 beome nonnegative (this should ome as no surrise sine onditions (18) involve only sums of ative balane variables). It turns out that in the ase of algorithm PD1, only neighbors, with x k+1 = x k+1 ȳ k+1,x k+1 = may have one of the ative balane variables = 0. No, ȳ k+1 being negative during a -iteration. In that ase,x k+1 POSTEDIT DUALS simly sets y, k+1 = y, k+1 other differenes between ȳ k+1, y k+1 exist. It is then obvious that for any neighboring verties, the following euation holds: y k+1 + y k+1 = ȳ k+1,x k+1,x k+1,x k+1 solution ȳ k+1 to y k+1 (i.e. load xk+1,y k+1 + ȳ k+1,x k+1. This means that all the loads are reserved during the transition from = load xk+1,ȳ k+1 ) whih in turn imlies (due to (15)) that POSTEDIT DUALS does not alter the value of the AP F funtion as well i.e. AP F xk+1,y k+1 = AP F xk+1,ȳ k+1 This, seemingly minor, modifiation of the dual solution by POSTEDIT DUALS lays, nevertheless, a very ruial role in the suess of the PD1 algorithm. In artiular, the onsidered modifiation along with roerty 3.1(e) will be absolutely neessary for ensuring that onditions (18) are satisfied by all dual solutions generated throughout PD1. This will beome lear during the roof of the theorem 3.2 ahead, whih is the main result of this setion and roves that PD1 always leads to an f a -aroximate solution. The seudoode for the PD1 algorithm is shown in Fig. 5. Theorem 3.2. The final rimal and dual solutions generated by PD1 satisfy all onditions (16) - (19). Therefore, (as exlained in setion 3) these solutions are feasible and satisfy the relaxed omlementary slakness onditions with f 1 = 1, f 2 = f a. Proof: Due to the integrality assumtion of the uantities,a, w, d ab, both the initial dual solution as well as the aaities of the grah G xk,ȳ k are always of the form n 0 2 with n 0 N. It an then be easily verified that any balane variable, and therefore the APF funtion too, an take values only of that form. So after every -iteration any derease of APF will always have magnitude 1/2. Based on this observation and the fat that, as mentioned above, POSTEDIT DUALS does not alter the value of the APF funtion, the algorithm termination (i.e. no hange of label taking lae for L onseutive inner iterations) is guaranteed by the APF monotoniity roerty 3.1(f). 12

16 INIT PRIMALS: initialize x k by a random label assignment INIT DUALS y k = 0 for eah air (, ) E with x k x k do y k,x = k yk,x = wdmin/2 = k yk,x = k yk,x k y k = min a ht yk,a V {It imoses onditions (16)} PREEDIT DUALS(, x k, y k ): ȳ k = y k UPDATE DUALS PRIMALS(, x k, ȳ k ) x k+1 = x k, ȳ k+1 = ȳ k Aly max-flow to G xk,ȳ k and omute flows f, f ȳ, k+1 = ȳ, k + f f, : V x k+1 = unsaturated ath s in G xk,ȳ k POSTEDIT DUALS(, x k+1, ȳ k+1 ) y k+1 = ȳ k+1 for eah air (, ) E with x k+1 = x k+1 = do if ȳ, k+1 < 0 or ȳ, k+1 < 0 then y, k+1 = y, k+1 = 0 y k+1 = min a ht yk+1,a V {It imoses onditions (16)} Fig. 5: Pseudoode for the PD1 algorithm. Feasibility onditions (16) are enfored by the definition of the PD1 algorithm (see Fig. 5). In addition, due to the seifi assignment of aaities to interior edges (see (22)), the balane variables of edges, are not allowed to grow larger than w d min /2 and so onstraints (19) are also enfored. Furthermore, we an rove by indution that solutions x k, y k (for any k) satisfy slakness onditions (18) and have all of their ative balane variables nonnegative i.e. y k,x k 0 (33) These onditions are obviously true at initialization (by the definition of INIT DUALS), so let us assume that they hold for x k, y k and let the urrent iteration be a -iteration. We will then show that these onditions hold for x k+1, y k+1 as well. To this end, we will onsider 3 ases: Case 1: let us first onsider the ase where x k+1 = x k+1 =. Then load xk+1,y k+1 = 0 (due to (14)) and so (18) obviously holds while (33) is guaranteed to be restored by the definition of POSTEDIT DUALS. Case 2: Next, let us examine the ase where neither nor is assigned a new label (i.e. x k+1 = x k, x k+1 = x k ). We an then show that: y k+1 = y k,x k+1,x k y k+1 = y k,x k+1,x k and so both onditions (18), (33) follow diretly from the indution hyothesis. Indeed, by alying either roerty 3.1(a) or 3.1(b), deending on whether x k+1 = x k = a or x k+1 = x k =, we onlude that ȳ k+1 = ȳ k,x k+1,x In addition,. k ȳ k,x = y k k,x 0 with the euality being true due to the definition of the k PREEDIT DUALS funtion and the ineuality following by the indution hyothesis. Combining the above relations we get: while with similar reasoning we an also show that: Therefore, both ȳ k+1,x k+1 (34) ȳ k+1 = y k,x k+1,x 0 (35) k ȳ k+1 = y k,x k+1,x 0 (36) k, ȳ k+1 are nonnegative and so their values will not be altered by,x k+1 POSTEDIT DUALS: y k+1,x k+1 = ȳ k+1,x k+1 y k+1,x k+1 = ȳ k+1,x k+1 (37) 13

17 The above euation, in onjuntion with (35), (36), imlies that (34) holds true, as laimed. Case 3: Finally, let us onsider the only remaining ase aording to whih only one of, (say ) is assigned a new label i.e. x k+1 = x k while the other one (say ) kees its urrent label i.e. x k+1 = x k = a with a. In this ase due to roerty 3.1(e) and (22) it follows that ȳ, k+1 = ȳ, k + a = w d min /2. In addition, it holds that ȳ,a k+1 = ȳ,a k = y,a k 0 where the 1 st euality is true due to a and roerty 3.1(a), the 2 nd euality is true due to the definition of PREEDIT DUALS and the ineuality follows from the indution hyothesis. Sine a (or euivalently x k+1 ), POSTEDIT DUALS (by definition) will alter none of the ative balane variables ȳ, k+1, ȳ,a k+1 and so y, k+1 = ȳ, k+1, y,a k+1 = ȳ,a. k+1 By ombining all of the above eualities it is now trivial to verify that (18), (33) hold for x k+1, y k+1 as well. Finally, to onlude the roof of this theorem we need to show that the last rimal-dual air of solutions satisfies ondition (17). Aording to the termination riterion of the PD1 algorithm, during its last L inner iterations there should be no label hange. Let be any label and onsider the -iteration out of these last L iterations. During that iteration it will hold that: htȳk+1 htȳk+1,x k+1, (38) where the above ineuality is true due to roerty 3.1(d). In addition, sine no hange of label takes lae, we an aly the same reasoning as in ase 2 above and show again that (37) holds for any neighboring verties,. This imlies that all ative balane variables are ket onstant during the transition from ȳ k+1 into y k+1 whih in turn imlies that y k+1 = ȳ k+1 sine, by definition, POSTEDIT DUALS annot touh any non-ative balane variables. Therefore, the heights of labels do not hange during the transition from ȳ k+1 into y k+1 and so, based on the revious ineuality (38), it will also hold that: ht yk+1 ht yk+1,x k+1, (39) Furthermore, the value of ht yk+1 is not altered during any of the next iterations. This is true beause kees its urrent,x k+1 label (by the termination riterion) and so we may again show that (34) holds for all of the remaining iterations. Similarly, the value of ht yk+1, will not hange hereafter sine by assumtion this is the last -iteration i.e. the last time the balane variables of the labels are udated. Therefore, ineuality (39) will be maintained until the end of the algorithm. Sine the same reasoning an be alied to any label, ondition (17) will finally hold true at the end of the last iteration. 4 PD2 algorithm The PD2 aroximation algorithm an be alied only in the ase of d ab being a metri (the neessity of this assumtion will beome lear later in the setion). In fat, PD2 reresents not just one algorithm but instead a family of algorithms PD2 µ 1 whih is arameterized by a variable µ [ f a 1]. The distintion between the various PD2 µ algorithms for different values of µ lies in the exat form of slakness onditions that they are trying to ahieve. Seifially, the rimal-dual solutions generated by PD2 µ will satisfy slakness onditions (11), (12) with f 1 = µf a and f 2 = f a. The reason for reuiring µ 1 f a is that it an never be f 1 < Algorithm overview A main differene between algorithms PD1 and PD2 µ is that the former is always generating a feasible dual solution at any of its inner iterations while the latter will allow an intermediate dual solution of beoming infeasible. However, the PD2 µ algorithm ensures that the (robably) infeasible dual solutions generated are not too far away from feasibility. This notion of not too far away ratially means that if the infeasible dual solutions are divided by a suitable fator, they will beome feasible again. This method (i.e. turning an infeasible dual solution into a feasible one by division) is also known as dual-fitting [14] in the linear rogramming literature. More seifially, we will rove that the algorithm is generating a series of intermediate airs onsisting of rimal-dual solutions with the following roerties: all of them satisfy slakness ondition (12) as an euality with f 2 = 1 µ : x x load x,y = µw d x x (40) 14

18 In addition, the last intermediate air satisfies the exat (i.e. f 1 = 1) slakness ondition (11) whih, as exlained in setion 3, redues to: y = min ht y,a a (41) ht y,x = min ht y,a a (42) However, the dual solution of this last air is robably infeasible sine although it satisfies dual onstraints (7) (due to (41)), it an be guaranteed to satisfy only: y,a + y,b 2µw d max a, b L, (, ) E (43) in lae of the dual onstraints (8). Nevertheless the above onditions ensures that the last dual solution is not too far away from feasibility. This ratially means that by relaing this last solution, say y, with y fit = y µf a, it is then easy to show that the resulting dual solution y fit satisfies the dual onstraints (8) as well (and is therefore feasible): y fit,a + y fit,b = y,a + y,b µf a 2µw d max µf a = 2µw d max µ2d max /d min = w d min w d ab Furthermore, by making use of (40)-(42) it again takes only elementary algebra to show that the feasible rimal-dual air (x, y fit ) (where x is the last rimal solution) satisfies the relaxed slakness onditions (11), (12) with f 1 = µf a and f 2 = f a and thus omrises an f a -aroximate solution. Indeed, this an be verified for the ase of onditions (11) by making use of the next eualities: y fit = y = hty,x =,x + : y,x µf a µf a while for the ase of onditions (12) one may roeed as follows: =,x µf a + =,x µf a + µf a y,x µf : a : y fit,x y,x fit + y,x fit = y,x + y,x = loadx,y = µw d x x = w d x x µf a µf a µf a The generation of y fit (given y) is exatly what the DUAL FIT routine does. The goal of the P D2 µ algorithm will therefore be to extrat a rimal-dual air (x, y) satisfying onditions (40)-(43). It should be noted that (as in the PD1 ase) imosing ondition (41) is always a trivial thing to do. We now roeed to desribe the bulk of the algorithm i.e. eah of the main routines aearing during a -iteration. 4.2 Udate of the rimal and dual variables The udate of the rimal and dual variables inside UPDATE DUALS PRIMALS(, x k, ȳ k ) takes lae in exatly the same way as in the PD1 algorithm (see (29), (30) and the reassign rule ). In addition, the onstrution of the grah G xk,ȳ k (as desribed in setion 3.2) an be reliated here as well exet for the assignment of aaities to a ertain subset of interior edges. This subset will onsist of all interior edges, (orresonding to edge (, ) of the original grah G) whose endoints, have labels at the start of the urrent -iteration i.e. x k = a and x k = b. Then, in lae of (22), we instead define: f a a = µw (d a + d b d ab ) (44) a = 0 (45) 15

19 In addition, in this ase (i.e. when x k = a, x k = b ) PREEDIT DUALS hanges y k, (and therefore its onjugate y k,) into ȳ k, (and ȳ k,) so as to ensure: ȳ k,a + ȳ k, = µw d a (46) This is done without modifying y k,a i.e. ȳ k,a = y k,a. Regarding the rest of the balane variables, PREEDIT DUALS alies no hanges to them, during the transition from y k to ȳ k, just like in the PD1 ase. No further differenes exist between the routines PREEDIT DUALS, UPDATE DUALS PRIMALS of the PD1 algorithm and the orresonding routines in PD2 µ. Based on the above observations it is easy to see that PREEDIT DUALS does not alter any of the ative balane variables (indeed, aording to its definition, PREEDIT DUALS modifies a balane variable only if it is of the form y k, or y k, with x k and x k ). Therefore the values of all the loads are not altered during the transition from yk to ȳ k : load xk,ȳ k = load xk,y k (47) In addition, the above euation (47) (along with (15)) imlies that the APF funtion is also not modified: AP F xk,ȳ k = AP F xk,y k (48) Furthermore, (44) exlains why d ab needs to be a metri (or else a would be negative). The rationale behind the aaity assignments in (44), (45) and the seifi definition of PREEDIT DUALS is to ensure that in the ase whih only one of, is assigned the label (by the new rimal solution x k+1 ), then ondition (40) still remains valid. The basi tool to be used for the roof of this assertion will be roerty 3.1(e). All of the above an be seen in the following key lemma. Lemma 4.1. During a -iteration, let, be two neighbors (i.e. ) with x k = a, x k = b and assume that x k, ȳ k satisfy ondition (40) i.e. load xk,ȳ k = µw d x k x k. (a) if a, b then ȳ, k+1 µw d b ȳ,b k and ȳk+1, µw d a ȳ,a k (b) x k+1, ȳ k+1 satisfy ondition (40) as well i.e. load xk+1,ȳ k+1 = µw d x k+1 x k+1 Proof: (a) Sine both of a, b are then (44), (45), (46) hold. In addition, by the lemma hyothesis: load xk,ȳ k = ȳ k,a + ȳ k,b = µw d ab (49) By roerty 3.1(e) the maximum value of ȳ, k+1 will be: ȳ,+a k = ȳ,+µw k (d a +d b d ab ) = µw d b ȳ,b k where the first euality is due to (44) and the last euality follows by substituting d ab, d a from (49), (46). Likewise the maximum value of ȳ, k+1 will be: ȳ, k + a = ȳ, k = µw d a ȳ,a k where the first euality is due to (45) and the last euality follows from (46). (b) If x k+1 = x k+1 then load xk+1,ȳ k+1 = 0 and art (b) of the lemma obviously holds. Therefore we may hereafter assume that x k+1 x k+1. This assumtion has as a result that not both of a, b an be eual to or else it would hold x k+1 = x k+1 = due to roerty 3.1(b). On the other hand, if either one of a, b is eual to (say x k = a =, x k = b ), this imlies that ȳ k+1,b = ȳk,b (due to b and roerty 3.1(a)) as well as xk+1 = and ȳ, k+1 = ȳ, k (due to x k = and roerty 3.1(b)). Then neessarily xk+1 = x k = b (sine we assume x k+1 x k+1 = and by definition of x k+1 any vertex is either assigned label or else kees its urrent label x k ). By ombining all of the above eualities it follows that load xk+1,ȳ k+1 = ȳ, k+1 + ȳ k+1,b = ȳk, + ȳ,b k =,ȳ k loadxk = µw d b (where the last euality is true due to the lemma hyothesis) and art (b) of the lemma therefore holds in this ase. We still need to onsider only the ase where both of a, b are different than (i.e. a, b ). Sine we assume x k+1 x k+1 only one of, may be assigned label by x k+1. If label is assigned to but kees its urrent label b (i.e. x k+1 =, x k+1 = b) then by roerty 3.1(e) ȳ, k+1 attains its maximum value and so by art (a) ȳ, k+1 = µw d b ȳ,b k. In addition ȳk+1,b = ȳk,b (due to b and roerty 3.1(a)) and so,ȳ k+1 loadxk+1 = ȳ, k+1 + ȳ k+1 ( ),b = µw d b ȳ,b k + ȳ k,b = µw d b. Likewise we an show that if label is assigned to (by x k+1 ) but kees its urrent label a then load xk+1,ȳ k+1 = µw d a. 16

20 As will beome lear during the roof of theorem 4.3 ahead, the above lemma lays a very signifiant role for the suess of the PD2 µ algorithm. The seond art of that lemma an lead, as we shall see, diretly to the roof that onditions (40) remain valid throughout PD2 µ. While the first art of the above lemma an be used in showing that the balane variables do not atually inrease too muh er iteration. This way we will be able later to show that the balane variables are always bounded above by the uantity µw d max, thus roving that onditions (43) are satisfied throughout PD2 µ as well. However, for roving this last assertion the first art of lemma 4.1 does not suffie. We still need to aly an additional modifiation to the dual solution ȳ k+1. This will be arried out by the POSTEDIT DUALS routine, whih also lays a very ruial role for the suess of the PD2 µ algorithm. More seifially, as in the PD1 ase, the role of that routine will be to hange solution ȳ k+1 into y k+1 so as to ensure that the ative balane variables of the resulting solution y k+1 are nonnegative. The differene with the PD1 algorithm is that a greater number of ative balane variables may turn out to be negative now. In addition, are must be taken (by POSTEDIT DUALS) so that the value of the loads and the APF funtion are not altered during this transition from ȳ k+1 to y k+1. To this end, POSTEDIT DUALS is alying an oerator RECTIFY(, ) to any air (, ) E. This oerator is defined as follows: let x k+1 = a, x k+1 = b and let us also assume that at least one of the ative balane variables ȳ,a, k+1 ȳ k+1,b is negative, say ȳ k+1,b < 0. Then if a = b the oerator RECTIFY(, ) simly sets yk+1,a = y,a k+1 = 0 while if a b it sets y,a k+1 = ȳ,a k+1 + ȳ k+1,b, yk+1,b = 0.6 During the transition from ȳ k+1 to y k+1 no other balane variables are modified by the RECTIFY oerator. The resulting dual solution y k+1 then satisfies the following roerties for any air of neighboring verties, : Proerties 4.2. (a) load xk+1,y k+1 = load xk+1,ȳ k+1 (b) The rimal-dual solutions x k+1, y k+1 satisfy onditions (40) i.e. load xk+1,y k+1 = µw d x k+1 x k+1 () AP F xk+1,y k+1 = AP F xk+1,ȳ k+1 (d) y k+1,a ȳ k+1,a, y k+1,a ȳ k+1,a a L (e) y k+1,x k+1 Proof: 0, y k+1 0,x k+1 (a) This euality an be easily verified diretly from the definition of the oerator RECTIFY. (b) This roerty follows easily by indution from lemma 4.1(b). Indeed, assuming that the air x k, y k satisfies onditions (40) then the same thing alies to air x k, ȳ k as well due to (47). Therefore the hyothesis of lemma 4.1 holds and by use of 4.1(b) in that lemma the air x k+1, ȳ k+1 also satisfies (40). By then alying roerty (a) the same onlusion an be drawn regarding the air x k+1, y k+1 and this omletes the indution. () It follows by ombining roerty (a) above and euation (15). (d) This an be trivially verified based on the definition of oerator RECTIFY(, ). (e) Combining roerties (a) and (b) we onlude that load xk+1,ȳ k+1 roerty based on the definition of the RECTIFY oerator. 0. Using this fat it is then trivial to verify the It should be noted that roerty 4.2(e) above ensures that all ative balane variables remain nonnegative throughout P D2 µ (as was intended) i.e. for any air of neighboring verties, it holds that: y k,x k 0 k (50) The seudoode for PD2 µ is shown in Fig. 6. We are now ready to rove the main theorem of this setion. Theorem 4.3. The final rimal-dual solutions generated by PD2 µ are feasible and satisfy the relaxed omlementary slakness onditions with f 1 = µf a and f 2 = f a. 6 Of ourse we also set their onjugate balane variables as: y k+1,a = y k+1,a, y k+1,b = yk+1,b. 17

21 INIT DUALS y k = 0 for eah air (, ) E with labels x k x k do y k,x = k yk,x = µw d k x k x k/2 = yk,x = k yk,x k y k = min a ht yk,a V {It imoses onditions (41)} PREEDIT DUALS(, x k, y k ) ȳ k = y k for eah (, ) E with x k = a, x k = b do ȳ k, = µw d a ȳ k,a; ȳ k, = ȳ k, POSTEDIT DUALS(, x k+1, ȳ k+1 ) for eah air (, ) E do RECTIFY(, ) y k+1 = min a ht yk+1,a V {It imoses onditions (41)} DUAL FIT( y k ): y fit = yk µf a Fig. 6: Pseudoode for the PD2 µ algorithm. Only those routines differing from their ounterarts in algorithm PD1 are shown. Proof: Due to the integrality assumtion of the uantities,a, w, d ab, both the initial dual solution as well as the aaities of the grah G xk,ȳ k are always of the form n 0 2 with n 0 N. It is then easy to verify that the APF funtion an take values only of the form n0 2 with n 0 N, so any derease of APF will neessarily be of magnitude 1 2. The algorithm termination is then guaranteed by the APF monotoniity roerty 3.1(f) and the observation that neither PREEDIT DUALS (due to (48)) nor POSTEDIT DUALS (due to roerty 4.2()) alters the value of APF. Also, onditions (41) are enfored by the definition of the P D2 µ algorithm (see Fig. 6) while onditions (40) follows diretly from roerty 4.2(b). In order to rove that onditions (43) hold as well it is enough to show by indution that y,, k y, k µw d max L. This is obviously true at initialization so let s assume it holds for y,, k y,. k We will show that during a -iteration this holds for y, k+1, y, k+1 as well. Due to roerty 4.2(d) it is enough to show ȳ, k+1, ȳ, k+1 µw d max. If either one of x k = a, x k = b euals then by roerty 3.1(b) ȳ, k+1 = ȳ,, k ȳ, k+1 = ȳ,. k Also, ȳ, k = y,, k ȳ, k = y, k (sine PREEDIT DUALS may alter y, k only if x k and x k ). The assertion then follows from the indution hyothesis. If both of a, b are then by lemma 4.1(a) ȳ, k+1 µw d b ȳ,b k while also ȳk,b = yk,b (sine b and PREEDIT DUALS, by definition, may alter only balane variables of the form y, k during a -iteration). But y,b k 0 sine x k = b i.e. this is an ative balane variable (see (50)). Therefore ȳ, k+1 µw d b µw d max. Likewise, using again lemma 4.1(a), we an rove that ȳ, k+1 µw d a µw d max and the assertion follows. Finally, we may show that the last rimal-dual air of solutions (say x, y) also satisfies onditions (42), by following the same reasoning that has been used in the roof of theorem 3.2 to show the satisfiability of the euivalent onditions (17). Therefore all onditions (40)-(43) hold true and so (as exlained in setion 4.1) the air (x, y fit ) generated by DUAL FIT will be feasible and will also satisfy all reuired slakness onditions, thus onluding the roof of the theorem. All PD2 µ algorithms with µ < 1 (as well as PD1) are non-greedy algorithms. That means that neither the rimal objetive funtion (nor the dual objetive funtion) are neessarily dereasing (inreasing) during eah iteration. Instead, it is the value of the APF funtion whih is onstantly dereasing but sine APF is always ket lose to the true rimal funtion the derease in APF is finally refleted to the values of the rimal funtion as well. However, a notable thing haens when µ = 1. In that ase, due to (40), the load between any neighboring verties, reresents exatly their searation ost (i.e. load xk,y k = w d x k x ) and so it an be roved that APF oinides with the rimal funtion (see lemma 4.4). In addition it k an be shown that the resulting PD2 µ=1 algorithm is atually euivalent to the -exansion algorithm introdued by Boykov et.al. in [4] (whih has been interreted only as a greedy loal searh tehniue u to now). The roof of this fat omes in theorem 4.6 ahead. Before roeeding, we reall that a label assignment x is alled a -exansion of (another label assignment) x k, if it holds that either x = x k or x = for any V (i.e. only label may be assigned as a new label by x ). 18

22 Lemma 4.4. Let x be a label assignment and y a dual solution (not neessarily feasible) satisfying the following onditions: load x,y w d xx (, ) E (51) Let us also denote by P RIMAL x the value of the rimal objetive funtion at x. Under these assumtions it is always true that AP F x,y P RIMAL x while if, in addition, onditions (51) hold as eualities then AP F x,y = P RIMAL x. Proof: Euation (15) and the assumtions of the lemma imly: AP F x,y =,x + load x,y (,) E,x + w d xx = P RIMAL x (,) E Lemma 4.5. Let x k, y k be a rimal dual air of solutions at the start of a -iteration of the PD2 µ=1 algorithm. Let x be any label assignment due to a -exansion of x k. (a) AP F xk+1,ȳ k+1 AP F x,ȳ k+1 (b) AP F x,ȳ k+1 P RIMAL x Proof: (a) Sine x is a label assignment due to a -exansion, this means that x may either kee the urrent label x k of a vertex or assign label to it. If x = x k (i.e. x kees the urrent label of ) then by roerty 3.1() htȳk+1 htȳk = htȳk,x k+1,x k,x = htȳk+1,x other hand if x = then by roerty 3.1(d) htȳk+1 therefore AP F xk+1,ȳ k+1 AP F x,ȳ k+1. where the last euality is true due to x and roerty 3.1(a). On the,x k+1 htȳk+1, = htȳk+1,x. So in any ase htȳk+1,x k+1 htȳk+1,x and (b) Let us first reall that the following euation holds: load xk,ȳ k = load xk,y k = w d x k x k (52) where the 1 st euality is due to the reservation of the load by PREEDIT DUALS (see (47)) while the 2 nd one follows from the fat that all x k, y k generated by PD2 µ=1 satisfy slakness onditions (40). Aording to lemma 4.4, to rove the assertion it is enough to show that x, ȳ k+1 satisfy (51). If x = x this is obviously true sine in that ase load x,ȳ k+1 = 0 due to (14). If x = x k, x = x k then by alying either roerty = ȳ k,x Similarly we. k 3.1(a) or 3.1(b) (deending on whether x k = a or x k = ) we an onlude that ȳ k+1,x k an show that ȳ k+1 = ȳ k,x k,x and so: k load xk,ȳ k+1 = load xk,ȳ k (53) The roerty then follows sine: load x,ȳ k+1 = load xk,ȳ k+1 = load xk,ȳ k = w d x k x = w d k x x where the the first and last eualities are true due to our assumtion that x = x k, x = x k while the 2 nd and 3 rd eualities are true due to euations (53) and (52) resetively. Also, if x, x then neessarily x = x k, x = x k (sine x is a -exansion) and so we fall bak into the revious ase. Therefore we still need to onsider only the ase where x x, (x, x ) (x k, x k ) and one of x, x is eual to. Assume that x = and let us also set x k = a, x k = b. Based on all of the above assumtions and the fat that x is a -exansion of x k, one may then easily rove that: x = b, a, b. This together with (52) imlies that the hyothesis of lemma 4.1(a) holds and so ȳ k+1, w d b ȳ k,b while also ȳk+1,b = ȳk,b (due to b and roerty 3.1(a)). Therefore load x,ȳ k+1 = ȳ k+1, + ȳ k+1,b ( w d b ȳ k,b) + ȳ k,b = w d b and the lemma follows. 19

23 Theorem 4.6. The label assignment x k+1 seleted during a -iteration of the PD2 µ=1 algorithm, has the minimum rimal ost among all label assignments that an result after a -exansion of x k. Proof: Assignment x k+1 is indeed a -exansion sine no label may be relaed during a -iteration (see roerty 3.1(b)). In addition load xk+1,ȳ k+1 = load xk+1,y k+1 = w d x k+1 x k+1 due to roerties 4.2(a) and 4.2(b). So, solutions x k+1, ȳ k+1 satisfy onditions (51) of lemma 4.4 as eualities and therefore P RIMAL xk+1 = AP F xk+1,ȳ k+1 by that lemma. Let now x be any other label assignment due to a -exansion of x k. Combining the above euality with the (a) and (b) ineualities of lemma 4.5 we get: P RIMAL xk+1 = AP F xk+1,ȳ k+1 AP F x,ȳ k+1 P RIMAL x and the theorem therefore follows. 5 PD3 algorithms: extending PD2 to the semimetri ase By modifiations to the PD2 µ algorithm, three different variations (PD3 a, PD3 b, PD3 ) of that algorithm may result whih are aliable even in the ase of d ab being a semimetri. Only two of these variations lead to aroximations with guaranteed otimality roerties. The roof of this fat makes again use of the dual fitting tehniue and follows along the same lines as those of roving the PD2 µ otimality roerties. For this reason that roof will be omitted. For simliity we will onsider only the µ = 1 ase i.e. only the variations of the PD2 µ=1 algorithm. An additional reason for that is beause in that ase, the resulting algorithms have nie, intuitive interretations in the rimal domain. Before roeeding, we are realling a fat that roves to be very useful for roviding these intuitive interretations as well as for exlaining the rationale lying behind the algorithms definition: if exat slakness onditions hold and the aroximation fator is therefore eual to 1 (i.e. the urrent solution is otimal) then the load between any two neighbors should reresent exatly their searation ost. The main diffiulty of extending PD2 µ=1 to the ase of a semimetri relates to how the aaities of ertain interior edges of G xk,ȳ k are defined during a -iteration. In artiular, these are all edges whose aaity is defined by euation (44). (This should ome as no surrise sine (44) has been the only lae where the metri hyothesis has been used.) Euivalently, these are all interior edges, whose endoints, are urrently assigned labels (i.e. x k = a, x k = b ) and in addition the following ineuality holds: d ab > d a + d b (54) Hereafter we will all any suh air a onfliting air while the orresonding trilet of labels (a, b, ) will be alled a onfliting label-trilet. The only differenes between a PD3 algorithm and PD2 µ=1 originate from the way the former deals with any onfliting airs that might our. Also, as we shall see, these differenes will onern only the definition of aaity of the above mentioned edges and/or ertain modifiations to the behavior of routines PREEDIT DUALS and POSTEDIT DUALS. The above observations imly that if the distane d ab is a metri then the PD3 algorithms always oinide with PD2 µ= Algorithms PD3 a and PD3 b The 2 algorithms of the urrent setion differ from the PD2 µ=1 algorithm only when a onfliting air is met during their exeution. In all other ases, i.e. at non-onfliting airs, these algorithms omletely oinide with PD2 µ=1, meaning that their routines PREEDIT DUALS, UPDATE DUALS PRIMALS and POSTEDIT DUALS work in exatly the same way with the orresonding routines from PD2 µ=1. Uon meeting a onfliting air, (with x k = a, x k = b ) during a -iteration, both algorithms roeed then as follows in order to define the aaities of the edges, : They first make use of a rule RESOLVE([a, b], ) in order to do what we all resolving the onfliting air. RE- SOLVE([a, b], ) (whih an be freely seified by the user on a er-aliation basis) takes as inut a onfliting labeltrilet (a, b, ) (with d ab > d a + d b ) and selets one of the 2 airs (a, ), (, b) while exluding the other one. The hysial meaning of the resolve rule will beome lear later in the setion. Then the value for one of the 2 aaities a, a is defined based on the outut of this RESOLVE routine. In artiular, if RESOLVE([a, b], ) selets (a, ), then a = 0 and PREEDIT DUALS sets ȳ, k so that ȳ,a k + ȳ, k = w d a. While if (, b) is seleted, then a = 0 and PREEDIT DUALS assigns a value to ȳ, k so that ȳ, k + ȳ,b k = w d b. In both ases, no other hange of balane variables takes lae during PREEDIT DUALS i.e. ȳ,a k = y,a, k ȳ,b k = yk,b. It should be noted that in the original PD2 µ=1 algorithm it has been always the first ase that was taking lae (see (45), (46)). Let us now assume w.l.o.g. that RESOLVE([a, b], ) has seleted air (a, ). Then the only thing that we still need to define, 20

24 before being able to aly UPDATE DUALS PRIMALS, is the aaity a. Two otions will be onsidered, giving rise to 2 different algorithms: PD3 a algorithm: We hoose to set a = 0 as well 7. In this ase, if the air of labels a, (the air seleted by RESOLVE) is assigned to, (i.e. x k+1 = a, x k+1 = ), then (as in lemma 4.1(b)) we may easily show that load xk+1,ȳ k+1 = w d a. In addition, due to roerty 4.2(a), it is always the ase that load xk+1,y k+1 = load xk+1,ȳ k+1. Therefore, at the start of the next iteration the load between verties, (i.e. load xk+1,y k+1 ) will reresent exatly the atual searation ost of, (i.e. w d a ) as it should in the otimal ase. However, if the air of labels, b (the air exluded by RESOLVE) is assigned to, (i.e. x k+1 =, x k+1 = b) then due to our hoie of setting a = 0 it is again easy to show that load xk+1,ȳ k+1 = w (d ab d a ) whih is > w d b sine, is a onfliting air (see (54)). So in this ase, the load overestimates the atual searation ost. In order that this overestimation is not maintained during the next iteration of the algorithm, POSTEDIT DUALS hanges the ative balane variables ȳ, k+1, ȳ k+1,b 8 into y, k+1, y k+1,b so that the value of the sum yk+1, +y k+1,b dereases to w d b and the euality between load xk+1,y k+1 and searation ost is therefore restored. The rest of the POSTEDIT DUALS routine is exatly the same as in the PD2 µ=1 algorithm. For an intuitive understanding of what is really haening, one an think of the situation as follows: sine d ab > d a + d b no matter how the aaities a, a are defined there will always exist one air among (a, ), (, b) whih, if assigned to, by x k+1, will lead to an overestimation of the searation ost in the sense that (for the next rimaldual air x k+1, y k+1 ) the load between, will be greater than the atual searation ost of these verties. RESOLVE selets whih one of the two label airs will ause an overestimated ost while at the same time the assignment of zero aaity to both edges, by the algorithm ensures that this overestimation error will be as small as ossible. In addition, POSTEDIT DUALS is trying so that this overestimation is aneled before the beginning of the algorithm s next iteration. One may also view this ost overestimation as an euivalent overestimation of the distane between labels. In the above ase, for examle, we saw that if labels, b are assigned to, by x k+1 then instead of the atual searation ost w d b the resulting overestimated ost has been w db with d b = d ab d a. This is euivalent to saying that the algorithm has assigned the distane d b > d b to labels, b instead of their atual distane d b. In all other ases (i.e. when (a, b) or (a, ) are assigned to, by x k+1 ) no ost overestimation takes lae and so the distanes assigned to the orresonding labels by the algorithm are eual to the atual distanes i.e. d ab = d ab, d a = d a. Sine d a + d b = d ab one ould then argue that the P D3 a algorithm hose to overestimate the distane between labels, b in order to restore the triangle ineuality for the urrent label-trilet (a, b, ). Put otherwise, it is as if a dynami aroximation of the d semimetri by a varying metri d is taking lae. The distane d b assigned to any air of labels (, b) by this metri is not ket fixed throughout PD3 a. Instead, the PD3 a algorithm onstantly adats d aording to the onfliting label trilets ourring during its exeution, always trying to restore the triangle ineuality for the urrent onfliting label trilet while introduing the least amount of overestimation error to the d semimetri at the same time. Furthermore, an advantage of this metri aroximation to d is that it an be exliitly ontrolled through the RESOLVE sheme. That sheme is seified by the user and an therefore be hosen on a er-aliation basis. It an be shown (using similar reasoning with that in theorem 4.3) that the intermediate rimal-dual solutions generated by both algorithms PD3 a and PD2 µ=1 satisfy exatly the same onditions and therefore it an be guaranteed that PD3 a always leads to an f a -aroximate solution as well. PD3 b algorithm: We hoose to set a = + and no further differenes between PD3 b and PD2 µ=1 exist. This has the following imortant result: the solution x k+1 rodued at the urrent iteration an never assign the air of labels, b (i.e. the air exluded by RESOLVE) to the verties, resetively. To rove this, it is enough to reall the reassign rule and also observe that there will always be a ossibility of inreasing the flow through direted without that edge ever beoming saturated (sine a = + ). Indeed, if label is assigned to by x k+1 (whih, aording to the reassign rule, means that there is an unsaturated ath s ) then label b an never be assigned to sine in that ase the ath s would also be unsaturated and so, by the reassign rule again, would have to be assigned label as well. Put otherwise, if the labels exluded by RESOLVE are assigned to, an infinite overestimation of the searation ost takes lae and so we imliitly revent those labels from being assigned to the onfliting air. 7 instead of using the aaity definition (44) whih would now be invalid. 8 and also their onjugate variables 21

25 Unfortunately the rie we ay for this infinite overestimation is that no guarantees about the algorithm s otimality an be rovided. The reason is that the balane variables may now inrease without bound (sine a = + ) and so we annot make sure that the generated rimal-dual solutions satisfy a not too far away from feasibility ondition like (43). This in turn imlies that no dual-fitting tehniue an be alied in this ase. However, the PD3 b algorithm has a nie interretation in the rimal domain. This an be seen in the following theorem whih is analogous to theorem 4.6 and an be roved using similar reasoning with the roof of that theorem. Theorem 5.1. The label assignment x k+1 seleted during a -iteration of the PD3 b algorithm, has the minimum rimal ost among all label assignments that may result after a -exansion of x k, disregarding the ones that assign labels exluded by RESOLVE to onfliting airs. This theorem designates the rie we ay for d ab being a semimetri: in the metri ase we an hoose the best assignment among all -exansion moves while in the semimetri ase we are only able to hoose the best one among a ertain subset of these -exansion moves. Desite this fat, the onsidered subset ontains a very large number of -exansion moves whih makes the algorithm a good andidate as a loal minimizer. Another interesting thing to note is that the the hoie of the -exansion moves to be inluded in this subset an be ontrolled by the RESOLVE sheme that will be seleted. This sheme an be aliation seifi and eah time ould reflet a riori knowledge about the onsidered roblem (exluding for examle onfigurations whih are a riori highly unlikely to aear). 5.2 Algorithm PD3 Contrary to the revious two algorithms PD3 a and PD3 b, the algorithm of this setion may differ with PD2 µ=1 even at non-onfliting airs. In addition, PD3 does not make use of any RESOLVE sheme at all. Instead, it alies the following modifiations to the PD2 µ=1 algorithm. It first adjusts (if needed) the dual solution y k so that the following ineuality holds for any neighboring verties, : load xk,y k w (d a + d b ) (55) If this is not the ase (i.e. if load xk,y k = y,a k + y,b k is greater than w (d a + d b )) then in order to restore (55) one an simly derease y,a, k y,b k so that,y k loadxk = w (d a + d b ). After this initial adjustment of the dual solution, the algorithm then ontinues in exatly the same way as the PD2 µ=1 algorithm with the only differene being that instead of using euation (44) to define aaity a, that aaity is set by the algorithm as follows: a = w (d a + d b d ab ) In other words, the algorithm has simly relaed the distane d ab in euation (44) with a new distane d ab whih is defined as:,y k d ab = loadxk (56) w Due to (55) it is obvious that d ab d a + d b and so the above definition of aaity a is valid i.e. a 0. No other differenes between PD3 and PD2 µ=1 exist. Based on this definition of PD3, it an then be shown that if d ab is a metri then the distanes d ab, dab oinide (i.e. d ab = d ab ) while, in addition, ondition (55) always holds true and so no initial adjustment of y k is needed. These two fats imly that PD3 is omletely euivalent to PD2 µ=1 in this ase. It is now interesting to examine what haens if, is a onfliting air (with x k = a, x k = b ). In that ase it holds that d a + d b < d ab and so by ombining this ineuality with (56) and (55) one an onlude that d ab < d ab as follows:,y k d ab = loadxk w (d a + d b ) < w d ab = d ab w w w Furthermore, it an be roved that if either the air (a, ) or (, b) is assigned to, by x k+1 during the urrent iteration, then the resulting load (i.e. load xk+1,y k+1 ) will reresent exatly the atual searation ost of, (i.e. either w d a or w d b ). However, if none of, is assigned a new label by x k+1 (i.e. they both retain their urrent labels a, b) then it an also be shown that load xk+1,y k+1 = w dab and so the load onstitutes an underestimation of the atual searation ost w d ab sine d ab < d ab as was shown above. 22

26 Based on these observations, one an see that the PD3 algorithm works in a omlementary way to the PD3 a algorithm: in order to restore the triangle ineuality for the onfliting label-trilet (a, b, ), instead of overestimating the distane between either the labels (a, ) or (, b) (like PD3 a did), it hooses to underestimate the distane between (a, b). Again one may view this as a dynami aroximation of the d semimetri by a onstantly adating metri d, however this time we set d ab = load xk,y k /w < d ab, d a = d a and d b = d b. It an be shown that the intermediate rimal-dual solutions generated by both algorithms PD3 and PD2 µ=1 satisfy exatly the same onditions exet for ondition (40). In lae of that ondition, the intermediate solutions of PD3 an be shown to satisfy: load xk,y k w ˆdx k x (57) k where ˆd ab = min L ( da + d b ). By alying then the same (as in PD2µ=1 ) dual fitting fator to the last dual solution of PD3, one an easily rove that PD3 leads to an f a-aroximate solution where: f a = f a 0 with 0 = max a b d ab ˆd ab (58) Sine it is always true that ˆd ab d ab (due to d ab = d ab + d bb ), this has as a result that 0 1 with euality holding only if d ab is a metri. Therefore it will always hold that f a f a and so PD3 annot guarantee a better aroximation fator. It should be noted at this oint that in the ase of d ab being a semimetri the hoie (between PD3 a, PD3 b, PD3 ) of the algorithm that will be alied an be deided on an iteration by iteration basis. 6 Algorithmi roerties of the resented rimal-dual algorithms As imlied by the Primal-Dual Prinile in setion 2, eah ratio r = T x/b T y (where x, y is any air of integral-rimal, dual feasible solutions) rovides a subotimality bound for the ost of the urrent rimal solution, in the sense that x is guaranteed to be an r-aroximation to the otimal integral solution. This roerty (whih is a very signifiant advantage of any rimal-dual algorithm) leads to an imortant onseuene that roves to be very useful in ratie: By onsidering the seuene of rimal-dual solutions {x k, y k } t k=1 generated throughout the rimal-dual shema, a series of subotimality bounds {r k = T x k /b T y k } t k=1 an be obtained. The minimum of these bounds is muh tighter (i.e. muh loser to 1) than the orresonding worst-ase bound and so this allows one to have a muh learer view about the goodness of a solution. This has been verified exerimentally by alying the resented algorithms to the stereo mathing roblem. In this ase, the labels reresent image ixel disarities and they an be hosen from a set L = {0, 1,..., K} of disretized disarities where K denotes the maximum allowed disarity. The verties of the grah G are the image ixels and the edges of G onnet eah ixel to its 4 immediate neighbors in the image. During our tests, the label ost for assigning disarity a to the image ixel has been set eual to:,a = I right ( + a) I left () (59) where I left, I right reresent the intensities of the left and right images resetively. We have alied our algorithms to the well-known Tsukuba stereo data set [16] setting the maximum disarity value eual to K = 14 based on the rovided ground truth data. A samle from the results rodued when using the algorithms PD2 µ=1 (whih is also euivalent to the -exansion algorithm) and PD1 are shown in Fig. 7. It should be noted that no attemt has been made to model olusions during the stereo mathing roedure while, in addition, all edge weights w have been set eual to eah other instead of roerly adjusting their values based on the intensity differenes I left () I left () (something whih would imrove the uality of the estimated disarity sine, in ratie, disarity disontinuities usually oinide with intensity edges). The reason for that as well as for using the very simle label ost resented in (59) is beause our main goal was not to rodue the best ossible disarity estimation but only to test the tightness of the subotimality bounds that are rovided by the onsidered algorithms i.e. to test the ability of these algorithms to effiiently minimize the objetive funtion of a Metri Labeling roblem. To this end, 3 different distanes d ab have been used during our exeriments. These are the Potts distane d (a metri), 23

27 (a) (b) () (d) Fig. 7: (a) The left and (b) right images for one stereo air from the Tsukuba data set. () The disarity estimated by the PD1 algorithm. (d) The orresonding disarity of the PD2 µ=1 algorithm. This algorithm was shown to be euivalent to the exansion Grah Cuts algorithm. In both of the above ases the Potts metri has been used as the distane between labels. Sine this distane is a metri the algorithms PD3 a, PD3 b, PD3 oinide with PD2 µ=1 in this ase and therefore rodue the same result as that in figure (d). the trunated linear distane d l (also a metri) and the trunated uadrati distane d (a semimetri), defined as follows: d ab = 1 a b (60) d l ab = min(m, a b ) a, b (61) d ab = min(m, a b 2 ) a, b (62) In the above euations the onstant M denotes the maximum allowed distane. Eah exeriment onsisted of seleting an aroximation algorithm and a distane funtion and then using them for omuting disarities for eah one of the Tsukuba stereo airs. The average values of the obtained subotimality bounds are dislayed in table 1. The olumns fa P D1, f P D2 µ=1 a, f P D3 a a, f P D3 b a, f P D3 a of that table list these averages for the algorithms P D1, P D2 µ=1, P D3 a, P D3 b and P D3 resetively. In addition, the last olumn lists the value of the orresonding aroximation fator f a whih, as already roved, onstitutes a worst-ase subotimality bound for most of the above algorithms. By observing table 1 one an onlude that the er-instane subotimality bounds are muh tighter (i.e. muh loser to 1) than the worst-ase bounds redited in theory. This holds for all ases i.e. for all ombinations of algorithms and distanes and so this fat indiates that the resented algorithms are always able to extrat a nearly otimal solution (with this being true even for the more diffiult ase where d ab is merely a semimetri). Sine the -exansion algorithm has roved to be euivalent to one of the resented algorithms this gives yet another exlanation for the great suess that grah-ut tehniues exhibit in ratie. Besides the tightness of the er instane subotimality bounds, another imortant issue is their auray i.e. how well these bounds desribe the true subotimality of the generated solutions. To investigate this issue we modified our exeriments in the following way: we alied our stereo mathing algorithms to one image sanline at a time (instead of the whole image). In this ase the grah G redues to a hain and the true otimum an be easily omuted using dynami rogramming. This in turn imlies that we are able to omute the true subotimality of a solution. If this fat is alied to our ase, it leads to the onstrution of table 2. Its olumns ftrue P D1, f P D2µ=1 true, f P D3 a true, f P D3 b true, f P D3 true ontain the true average subotimality of the solutions of P D1, P D2 µ=1, P D3 a, P D3 b and P D3 resetively where the average is taken over all image sanlines. Furthermore, by examining table 2 one an see that the true subotimality of a solution is always lose to the orresonding subotimality bound. This imlies that these bounds are relatively aurate and therefore reliable for judging the goodness of an algorithms s solution. More generally seaking, any rimal-dual aroximation algorithm laes an uer bound on the so-alled integrality ga IGAP rimal [14] of the LP relaxation of the rimal roblem. In artiular, an f-aroximation rimal-dual algorithm laes the following bound on the integrality ga: IGAP rimal f where the integrality ga IGAP rimal is defined as the suremum (over all instanes of the roblem) of the ratio of the otimal integral and frational solutions and is always 1. Obviously, the integrality ga is the best aroximation fator we an hoe to rove. In fat, for seial ases of the Metri Labeling Problem it an be shown that the resented rimal-dual algorithms yield aroximation fators that are essentially eual to the resulting integrality ga i.e. these are the best ossible aroximation fators. 24