LP Roundng for k-centers wth Non-unform Hard Capactes (Extended Abstract) Marek Cygan, MohammadTagh Hajaghay, Samr Khuller IDSIA, Unversty of Lugano, Swtzerland. Emal: marek@dsa.ch Department of Computer Scence, Unversty of Maryland, College Park, USA. Emal: {hajagha,samr}@cs.umd.edu Abstract In ths paper we consder a generalzaton of the classcal k-center problem wth capactes. Our goal s to select k centers n a graph, and assgn each node to a nearby center, so that we respect the capacty constrants on centers. The objectve s to mnmze the maxmum dstance a node has to travel to get to ts assgned center. Ths problem s NP -hard, even when centers have no capacty restrctons and optmal factor 2 approxmaton algorthms are known. Wth capactes, when all centers have dentcal capactes, a 6 approxmaton s known wth no better lower bounds than for the nfnte capacty verson. Whle many generalzatons and varatons of ths problem have been studed extensvely, no progress was made on the capactated verson for a general capacty functon. We develop the frst constant factor approxmaton algorthm for ths problem. Our algorthm uses an LP roundng approach to solve ths problem, and works for the case of non-unform hard capactes, when multple copes of a node may not be chosen and can be extended to the case when there s a hard bound on the number of copes of a node that may be selected. Fnally, for non-unform soft capactes we present a much smpler 11- approxmaton algorthm, whch we fnd as one more evdence that hard capactes are much harder to deal wth. Keywords-approxmaton algorthms; k-center; non-unform capactes; hard capactes; LP roundng; I. INTRODUCTION The k-center problem s a classcal faclty locaton problem and s defned as follows: gven an edge-weghted graph G = (V, E) fnd a subset S V of sze at most k such that each vertex n V s close to some vertex n S. More formally, once we choose S the objectve functon s max u V mn v S d(u, v), where d s the dstance functon (a metrc). The problem s known to be NP-hard [2]. Approxmaton algorthms for the k-center problem have been well studed and are known to be optmal [3] [6]. In ths paper we consder the k-center problem wth non-unform capactes. We have a capacty functon L defned for each vertex, hence L(u) denotes the capacty The full verson of ths work can be found at [1]. The frst author was partally supported by ERC grant NEWNET reference 279352, ERC Startng Grant PAAl 259515, Foundaton for Polsh Scence. The frst and second author were supported n part by NSF CAREER award 1053605, ONR YIP award N000141110662, DARPA/AFRL award FA8650-11-1-7162, and a Unversty of Maryland Research and Scholarshp Award (RASA). Research of the thrd author was supported by NSF CCF-0728839, NSF CCF-0937865 and a Google Research Award. of vertex u. The goal s to dentfy a set S of at most k centers, as well as an assgnment of vertces to nearby centers. No more than L(u) vertces may be assgned to a chosen center at vertex u. Under these constrants we wsh to mnmze the maxmum dstance between a vertex v and ts assgned center φ(v). Formally, the cost of a soluton S s mn S V, S =k max v V d(v, φ(v)) such that {v φ(v) = u} L(u) u S where φ : V S. For the specal case when all the capactes are dentcal, a 6 approxmaton was developed by Khuller and Sussmann [7] mprovng the prevous bound of 10 by Bar-Ilan, Kortsarz and Peleg [8]. In the specal case when multple copes of the same vertex may be chosen, the approxmaton factor was mproved to 5. No mprovements have been obtaned on these results n the last 15 years. The assumpton that the capactes are dentcal s crucal for both these approaches as t allows one to select centers and then shft to a neghborng vertex. In addton, one can use arguments such as N L s a lower bound on the optmal soluton; wth non-unform capactes we cannot use such a bound. Ths problem has ressted any progress at all, and no constant approxmaton algorthm was developed for the non-unform capacty verson. In ths work we present the frst constant factor approxmatons for the k-center problem wth arbtrary capactes. Moreover, our algorthm satsfes hard capacty constrants and only one copy of any vertex s chosen. When multple copes of a vertex can be chosen then a constant factor approxmaton s mpled by our result for the hard capacty verson. For convenence, we dscuss the algorthm for the case when at most one copy of a vertex may be chosen. Our algorthms use a novel LP roundng method to obtan the result. In fact ths s the frst tme that LP technques have been appled for any varaton of the k-center problem. Whle our constants are large, we do show va ntegralty gap examples that the problem wth non-unform capactes s sgnfcantly harder than the basc k-center problem. In addton we establsh that f there s a (3 ɛ)-approxmaton for the k-center problem wth non-unform capacty constrants then P = NP. Such a result s known for the cost k- center problem [9] and from that one can nfer the result for the unt cost capactated k-center problem wth non-unform capactes, but our reducton s a drect reducton from Exact
Cover by 3-Sets and consderably smpler. We would lke to note that for the k-suppler problem, whch s k-center wth dsjont sets of clents and potental centers, a smple proof of (3 ɛ) approxmaton hardness under P NP was obtaned by Karloff and can be found n [5]. In all cases of studyng coverng problems, the hard capacty restrcton makes the problems very challengng. For example, for the smple capactated vertex cover problem wth soft capactes, a 2 approxmaton can be obtaned by a varety of methods [10], [11] however mposng a hard capacty restrcton makes the problem as hard as set cover [12]. In the specal case of unweghted graphs t was shown that a 3 approxmaton s possble [12], whch was subsequently mproved to 2 [13]. A. Related Faclty Locaton Work The faclty locaton problem s a central problem n operatons research and computer scence and has been a testbed for many new algorthmc deas resultng a number of dfferent approxmaton algorthms. In ths problem, gven a metrc (va a weghted graph G), a set of nodes called clents, and openng costs on some nodes called facltes, the goal s to open a subset of facltes such that the sum of ther openng costs and connecton costs of clents to ther nearest open facltes s mnmzed. When the facltes have capactes, the problem s called the capactated faclty locaton problem. The frst constant-factor approxmaton algorthm for the (uncapactated) verson of ths problem was gven by Shmoys, Tardos, and Aardal [14] and was based on LP roundng and a flterng technque due to Ln and Vtter [15]. A long seres of mprovements culmnated n a 1.5 approxmaton due to Byrka [16]. Up to now, the best known approxmaton rato s 1.488, due to L [17] who uses a randomzed selecton n Byrka s algorthm [16]. Guha and Khuller [18] showed that ths problem s hard to approxmate wthn a factor better than 1.463, assumng NP DT IME[n O(log log n) ]. Capactated faclty locaton has also receved a great deal of attenton n recent years. Two man varants of the problem are soft-capactated faclty locaton and hardcapactated faclty locaton: n the latter problem, each faclty s ether opened at some locaton or not, whereas n the former, one may specfy any nteger number of facltes to be opened at that locaton. Soft capactes make the problem easer and by modfyng approxmaton algorthms for the uncapactated problems, we can also handle ths case [14], [19]. Korupolu, Plaxton, and Rajaraman [20] gave the frst constant-factor approxmaton algorthm that handles hard capactes, based on a local search procedure, but ther approach works only f all capactes are equal. Chudak and Wllamson [21] mproved ths performance guarantee to 5.83 for the same unform capacty case. Pál, Tardos, and Wexler [22] gave the frst constant performance guarantee for the case of non-unform hard capactes. Ths was recently mproved by Mahdan and Pál [23] and Zhang, Chen, and Ye [24] to yeld a 5.83-approxmaton algorthm. All these approaches are based on local search. The only LP-relaxaton based approach for ths problem s due to Lev, Shmoys and Swamy [25] who gave a 5-approxmaton algorthm for the specal case n whch all faclty openng costs are equal (otherwse the LP does not have a constant ntegralty gap). The above approxmaton algorthms for hard capactes are focused on the unform demand case or the splttable case n whch each unt of demand can be served by a dfferent faclty. Recently, Baten and Hajaghay [26] consdered the unsplttable hard-capactated faclty locaton problem when we allow volatng faclty capactes by a 1+ɛ factor (otherwse, t s NP-hard to obtan any approxmaton factor) and obtan an O(log n) approxmaton algorthm for ths problem. A problem very close to both faclty locaton and k- center s the k-medan problem n whch we want to open at most k facltes (lke n the k-center problem) and the goal s to mnmze the sum of connecton costs of clents to ther nearest open facltes (lke faclty locaton). If facltes have capactes the problem s called capactated k-medan. The approaches for uncapactated faclty locaton often work for k-medan. In partcular, Charkar, Guha, Tardos, and Shmoys [27] gave the frst constant factor approxmaton for k-medan based on LP roundng. The best approxmaton factor for k-medan s 3 + ɛ, for an arbtrary postve constant ɛ, va the local search algorthm of Arya et al. [28]. Unfortunately obtanng a constant factor approxmaton algorthm for capactated k-medan stll remans open despte consstent effort. The methods used to solve uncapactated k-medan or even the local search technque for capactated faclty locaton all seem to suffer from serous drawbacks when tryng to apply them for capactated k-medan. For example standard LP relaxaton s known to have an unbounded ntegralty gap [27]. The only prevous attempts wth constant approxmaton factors for ths problem volate the capactes wthn a constant factor for the unform capacty case [27] and the non-unform capacty case [29] or exceed the number k of facltes by a constant factor [30]. CAPACITATED k-center PROBLEM Input: An undrected graph G = (V, E), a capacty functon L : V N and an nteger k. Output: A set S V of sze k, and a functon φ : V S, such that for each u S, φ 1 (u) L(u). Goal: Mnmze max v V dst G (v, φ(v)). Removng the metrc: We employ the standard thresholdng method used for bottleneck optmzaton problems. We can assume that we guess the optmal soluton, snce there are polynomally many dstnct dstances between pars of nodes. Once we guess the dstance correctly, we create an unweghted graph consstng of those edges uv such
that d(u, v) OP T. We henceforth assume that we are consderng the problem for an undrected graph G. By a c-approxmaton algorthm we denote a polynomal tme algorthm, that for an nstance for whch there exsts a soluton wth objectve functon equal to 1, returns a soluton usng dstances at most c. Note that the dstance functon dst(u, v), measures the dstance n the unweghted undrected graph. In the soft-capactated verson S can be a multset, that s one can open more than one center at a vertex. To avod confuson we call the standard verson of the problem hardcapactated. B. Our results Whle LP based algorthms have been wdely used for uncapactated faclty locaton problems as well as capactated versons of faclty locaton wth soft capactes, these methods are not of much use for problems n dealng wth hard capactes due to the fact that they usually have an unbounded ntegralty gap [22], [27]. For general undrected graphs ths s also the case for the capactated k-center problem. Consder the LP relaxaton for the natural IP, whch we denote as LP1. We use y u as an ndcator varable for open centers. u V y u = k; (1) x u,v y u u, v V (2) v V x u,v L(u)y u u V (3) u V x u,v = 1 v V (4) 0 y u 1 u V (5) x u,v = 0 u, v V dst G (u, v) > 1 (6) x u,v 0 u, v V (7) For the sake of presentaton we have ntroduced varables x u,v for all u, v, even f the dstance between u and v n G s greater than one. We wll use those varables n our roundng algorthm. Furthermore n constrants (1) and (4) we used equalty nstead of nequalty to make our roundng algorthm and lemma formulatons smpler. In the soft-capactated verson the y u 1 part of constrant (5) should be removed. Note that we are only nterested n feaslbty of LP1, and there s no objectve functon. For an undrected graph G = (V, E) and a postve nteger δ, by G δ we denote the graph (V, E ), where uv E ff dst G (u, v) δ. By an ntegralty gap of LP1 we mean the mnmum postve nteger δ such that f LP1 has a feasble soluton, then the graph G δ admts a capactated k-center soluton. As ths s usually the case for capactated problems, by a smple example we prove LP1 has unbounded ntegralty gap for general graphs. Due to space lmtatons, proofs of theorems marked wth a spade symbol ( ) are postponed to the full verson of ths paper. Theorem I.1 ( ). LP1 has unbounded ntegralty gap, even for unform capactes. However, nterestngly, f we assume that the gven graph s connected, the stuaton changes dramatcally. Our man result s, that both for hard and soft capactated verson of the k-center problem, even for non-unform capactes, LP1 has constant ntegralty gap for connected graphs. Moreover by usng novel technques we show a correspondng polynomal tme roundng algorthm, whch conssts of several steps, descrbed at hgh level n the followng subsecton. The actual algorthm s somewhat complex, although t can be mplemented qute effcently. Theorem I.2. There s a polynomal tme algorthm, whch gven an nstance of the hard-capactated k-center problem for a connected graph, and a fractonal feasble soluton for LP1, can round t to an ntegral soluton that uses non-zero x u,v varables for pars of nodes wth dstance at most c. Corollary I.3. The ntegralty gap of LP1 for connected graphs s bounded by a constant, and there s a constant factor approxmaton algorthm for connected graphs. To smplfy the presentaton we do not calculate the exact constant proved n the above corollary, but t s n the order of hundreds. As a counterposton, for soft capactes n the full verson we present a much smpler 11-approxmaton algorthm, whch we fnd as one more evdence that hard capactes are much harder to deal wth. Theorem I.4 ( ). For connected graphs there s a polynomal tme roundng algorthm, upper boundng the ntegralty gap of LP1 by 11 for soft-capactes. By usng standard technques one can restrct the capactated k-center problem to connected graphs. Theorem I.5 ( ). If there exsts a polynomal tme c- approxmaton algorthm for the capactated k-center problem n connected graphs, then there exsts a polynomal tme c-approxmaton algorthm for general graphs. Therefore we prove there s a constant factor approxmaton algorthm for the hard-capactated k-center problem 1. Our results easly extend to the case when there s an upper bound U(u) of the number of tmes vertex u may be chosen as a center. Constrant 5 should be modfed to be 0 y u U(u) to yeld a relaxaton LP2. We can employ the same roundng procedure as dscussed for the hard capacty case wth U(u) = 1. The proof of the followng theorem s omtted. Theorem I.6 ( ). There s a polynomal tme algorthm, whch gven an nstance of the hard-capactated k-center problem for a connected graph, and a fractonal feasble soluton for LP2, can round t to an ntegral soluton that 1 Wth some care, perhaps some of the constants can be mproved, however our focus was to show that a constant approxmaton s obtanable usng LP roundng.
uses non-zero x u,v varables for pars of nodes wth dstance at most c. Whle our constants are large, we do show va ntegralty gap examples that the problem wth non-unform capactes s sgnfcantly harder than the basc k-center problem. Theorem I.7 ( ). For connected graphs the ntegralty gap of LP1 s at least 5 for unform-hard-capactes and at least 4 for unform-soft-capactes. Moreover n the non-unform hard-capactated case, the ntegralty gap of LP1 for connected graphs s at least 7, even f all the non-zero capactes are equal. Despte the fact, that the algorthm of [7] for unform capactes was obtaned more than a decade ago, no lower bound for the capacty verson (nether soft nor hard), better than the trval 2 ɛ, derved from the uncapactated verson, s known. We beleve that the ntegralty gap examples, presented n ths paper, are of ndependent nterest snce they may help n provng a stronger lower bound for the capactated k-center problem wth unform capactes. To make a step n ths drecton we nvestgate lower bounds for the non-unform case. By a reducton from the cost k-center problem [9] one can show that there s no (3 ɛ)-approxmaton for the capactated k-center problem wth non-unform capactes. By a smple reducton from Exact Cover by 3-Sets, n the full verson, we prove the same result under the assumpton P NP. Fnally we gve evdence that our LP approach mght be the proper tool for solvng the capactated k-center problem. The proof of the followng theorem shows that when the Khuller-Sussmann algorthm fals to fnd a soluton then n fact there s no feasble LP soluton for that guess of dstance. The smallest radus guess for whch the algorthm succeeds, proves an ntegralty gap on the LP. Consderng the result of Theorem I.7, t follows that for unform capactes the gap n the analyss s small, snce our bounds are tght up to an addtve +1 error. Theorem I.8 ( ). For connected graphs the ntegralty gap of LP1 s at most 6 for unform-hard-capactes and at most 5 for unform-soft-capactes. C. Our technques We assume that G s connected and that LP1 has a feasble soluton for the graph G. We call two functons x : V V R + {0} and y : V R + {0} an assgnment even f (x, y) s potentally nfeasble for LP1. In other words ntally we have a feasble fractonal soluton, n the end we wll obtan a feasble ntegral soluton, although durng the executon of our roundng algorthm an assgnment (x, y) s not requred to be feasble. Furthermore wthout loss of generalty we assume that for a vertex v wth L(v) = 0 we have y v = 0. We need to show that there exsts a constant δ such that f for a connected component LP1 has a feasble soluton, then one can (n polynomal tme) fnd an ntegral feasble soluton for G δ. Defnton I.9 (δ-feasble soluton). An assgnment s called δ-feasble f t s feasble for the graph G δ. Note that the only dfference between LP1 s for the graphs G and G δ s constrant (6). Defnton I.10 (radus (x,y) ). For a δ-feasble soluton (x, y) to LP1 we defne a functon radus (x,y) : V {0,..., δ} whch for a vertex u assgns the greatest nteger such that there exsts a vertex v wth dst G (v, u) = and x u,v > 0 (f no such exsts then radus (x,y) (u) = 0). We gve a bref overvew of the followng sectons. Intally we start wth a 1-feasble (fractonal) soluton (x, y) to LP1 and our goal s to make t ntegral. We perform several steps where n each step we get more structure on the δ-feasble soluton but at the same tme the value of δ wll ncrease. In Sectons II-A-II-D n four non-trval steps we round the y-values of a feasble soluton. Frst, n Secton II-A we defne a caterpllar structure whch s a key structure n the roundng process. In Secton II-B we defne the y- flow and chan shftng operatons whch allow for transferrng y-values between dstant vertces usng ntermedate vertces on the caterpllar structure. Unfortunately, because the capactes are non-unform and hard, to fnd a roundng flow for a caterpllar structure we need more assumptons. To overcome ths dffculty n the most challengng part of the roundng process, that s n Secton II-C, we defne a safe caterpllar structure and show how to splt a gven caterpllar structure nto a set of safe caterpllar structures (at the cost of ncreasng radus of the δ-feasble soluton). In Secton II-D we desgn a roundng procedure for a safe caterpllar structure, obtanng a c-feasble soluton wth ntegral y-values, for some constant c. We would lke to note, that for unform capactes every caterpllar structure s safe, therefore for non-unform capactes we have to desgn much more nvolved tools comparng to the prevously known unform capactes case. Fnally n Secton II-E we show, that usng standard technques, when we have ntegral y-values then roundng x- values s smple, obtanng a constant factor approxmaton algorthm. II. LP ROUNDING FOR HARD-CAPACITIES A. Group shftng and caterpllar structure In the frst phase of our procedure we obtan a pathlke structure contanng all vertces wth non-ntegral y- values. We frst defne the noton of shftng values between varables of LP1 relaxaton.
Defnton II.1 (shftng). For an assgnment (x, y) for the LP, two dstnct vertces a, b V and a postve real α mn(y a, 1 y b ) such that L(a) L(b) by shftng α from a to b we consder the followng operaton: 1) Let ɛ = α y a ; for each v V let v = ɛx a,v, decrease x a,v by v and ncrease x b,v by v. 2) Increase y b by α, and decrease y a by α. Lemma II.2 ( ). Let (x, y) be a δ-feasble soluton to LP. Let (x, y ) be a result of shftng α from a to b, for some α, a, b such that L(a) L(b), 0 < α mn(y a, 1 y b ). Then (x, y ) s a (δ + dst G (a, b))-feasble soluton and for each vertex v b we have radus (x,y )(v) radus (x,y) (v) whereas radus (x,y )(b) max(radus (x,y) (a) + dst G (a, b), radus (x,y) (b)). Defnton II.3 (group shftng). For a δ-feasble soluton (x, y) and a set V 0 V by a group shftng we denote the followng operaton. Assume V 0 = {v 1,..., v l }, where L(v ) L(v +1 ) for 1 < l. As long as there are at least two vertces n V 0 wth fractonal y-values, let a be the smallest, and b the greatest nteger such that v a, v b V 0 are vertces wth fractonal y-values. Shft mn(y a, 1 y b ) from a to b. Lemma II.4. Let (x, y) be a δ-feasble soluton, V 0 be a subset of V and d = max a,b V0 dst G (a, b). After group shftng on V 0 we obtan a (δ + d)-feasble soluton (x, y ), where there s at most one vertex n V 0 wth fractonal y-value and moreover for v V \ V 0 we have radus (x,y )(v) radus (x,y) (v). To make a graph Hamltonan we use the followng lemma known from 1960 [31], [32]. Lemma II.5. For any undrected connected graph G there always exsts a Hamltonan path n G 3 and one can fnd t n polynomal tme. We defne a caterpllar structure whch s one of the key ngredents of our roundng process. Intutvely we want to defne an auxlary path-lke tree, where adjacent vertces are close n the orgnal graph G, vertces wth fractonal y- values are leaves of the tree, and all non-leaf vertces have y-values equal to 1. Defnton II.6 (caterpllar structure). By a δ-caterpllar structure for an assgnment (x, y) we denote a sequence of dstnct vertces P = (v 1,..., v p ) together wth a sequence P = (v 0,..., v p+1) where: 1) for each = 1,..., p we have y v = 1, 2) for each = 1,..., p 1 we have dst G (v, v +1 ) δ, 3) for each = 0,..., p + 1 ether v = nl or v V \ {v j : j = 1,..., p}, 4) for each = 1,..., p f v nl then L(v ) L(v ), 0 < < 1, dst G (v, v ) δ, 5) f v 0 nl then dst G (v 0, v 1 ) δ, 0 < 0 < 1, 6) f v p+1 nl then dst G (v p+1, v p ) δ, 0 < p+1 < 1, 7) for each 0 < j p + 1 f v nl and v j nl then v v j, 8) v V (P ) y v s ntegral. We sometmes omt δ and smply wrte caterpllar structure when the value of δ s rrelevant. 0 = 0.2 L(v 0) = 15 L(v 1) = 5 y v1 = 1 1 = 0.4 L(v 1) = 5 L(v 2) = 1 y v2 = 1 L(v 3) = 10 y v3 = 1 3 = 0.9 L(v 3) = 1 L(v 4) = 2 y v4 = 1 4 = 0.5 L(v 4) = 1 Fgure 1. Example of a δ-caterpllar structure ((v 1, v 2, v 3, v 4 ), (v 0, v 1, nl, v 3, v 4, nl)). Vertces connected by edges are wthn dstance δ n the graph G. Note that the sum of y-values over all vertces s ntegral. Lemma II.7. For a gven feasble LP soluton (x, y) we can fnd a 5-feasble soluton (x, y ) together wth a 21- caterpllar structure (P, P ) such that each vertex v V \ (V (P ) V (P )) has an ntegral y-value n (x, y ), and the frst and last element of the sequence P equals nl. Proof: Consder the followng algorthm for constructng sets S, S and a functon Φ : V S. The set S wll be an nclusonwse maxmal ndependent set n G 2 and moreover we ensure that L(Φ(v)) L(v), for any v V. 1) Set V 0 := V and S := S :=. 2) As long as V 0 let v be a hghest capacty vertex n V 0. Let f(v) be a hghest capacty vertex n N G [v] (potentally f(v) V 0 ). Add f(v) to S and for each u N G [N G [v]] V 0 set Φ(u) = f(v). Add v to S and set V 0 := V 0 \ N G [N G [v]]. Observe that each tme we remove from the set V 0 all vertces that are wthn dstance two from v, hence the set S s an ncluson maxmal ndependent set n G 2. For ths reason vertces n the set S have dsjont neghborhoods and moreover by constrants (4) and (2) of the LP1 we nfer that for each v V we have: u N[v] y u u N[v] x u,v = 1 (8) We perform shftng operatons to make sure all vertces n the set S have y-value equal to one. Consder a vertex v S and the correspondng vertex f(v) chosen by the algorthm. As long as y f(v) < 1 take any u N[v], u f(v) such that y u > 0 and shft mn(y u, 1 y f(v) ) from u to f(v). Note that L(u) L(f(v)) by the defnton of f(v) and for ths reason shftng s possble. By Lemma II.2 after all the
shftng operatons we have a 3-feasble soluton (x, y), snce before a shft from u to f(v) we have radus (x,y) (u) 1, radus (x,y) (f(v)) 3 and dst G (u, f(v)) 2. Moreover by Inequalty (8) we nfer, that all the vertces n the set S have y-value equal to one, snce otherwse a shftng operaton from some u N[v] to f(v) would be possble. Observe that by the maxmalty of the ndependent set S n G 2 the graph G 5 [S] s connected, otherwse we could add a vertex to S stll obtanng an ndependent set n G 2. Moreover for any two adjacent vertces u, v S n G 5 [S], the vertces f(u), f(v) are adjacent n G 7 [S ]. By the connectvty of G 5 [S], the graph G 7 [S ] s also connected. By Lemma II.5 we can n polynomal tme order the vertces of S to obtan a Hamltonan path P n G 21 [S ]. Currently for each vertex v from the set V \ S we have radus (x,y) (v) 1. For each v S we use group shftng on the set Φ 1 (f(v)) \ S. Snce max dst G(a, b) a,b Φ 1 (f(v))\s max dst G(a, v) + dst G (v, b) 4, a,b Φ 1 (f(v))\s by Lemma II.4 we obtan a 5-feasble soluton (x, y) such that all vertces n the set S have y-value equal to one and moreover for each f(v) S the set Φ 1 (f(v)) \ S contans at most one vertex wth fractonal y-value. Let us assume that the already constructed path P s of the form P = (v 1,..., v p ). We construct a sequence P = (nl, v 1,..., v p, nl) where as v we take the only vertex from Φ 1 (v ) \ S that has fractonal y-value, or we set v := nl f Φ 1 (v ) \ S has no vertces wth fractonal y-value. Note that snce the way we select vertces to the sets S, S s capacty drven (recall as v we select the hghest capacty vertex n V 0 and as f(v) we select a hghest capacty vertex n N[v]), for each vertex u Φ 1 (v ) we have L(u) L(v ). In ths way we have constructed a 5- feasble soluton (x, y) together wth a desred 21-caterpllar structure (P, P ). As the reader mght notce n the above proof we always construct a caterpllar structure wth v 0 = v p+1 = nl. The reason why the defnton of a caterpllar structure allows for v 0 and v p+1 have non-nl values s that n Secton II-C we wll splt a caterpllar structure nto two smaller peces and n order to have those peces satsfy Defnton II.6 we need v 0 and v p+1. B. y-flow and chan shftng In the prevous secton we defned a group shftng operaton. Unfortunately we can only perform such an operaton f vertces are close. In ths secton we defne notons of y-flow and chan shftng whch allow us to transfer y- value between dstant vertces. We wll use those tools n Sectons II-C and II-D. Defnton II.8 (y-flow). For a gven assgnment (x, y) let S V and T V be two dsjont sets and let F be a set contanng sequences of the form (α, v 1,..., v t ) representng paths, where α s a postve real, each v V s a vertex (for = 1,..., t), v 1 S, v t T, L(v 1 ) L(v t ) and for = 2,..., t 1 we have v S T, y v = 1, L(v ) L(v 1 ). We call (α, v 1,..., v t ) a path transferrng α from v 1 to v t through v 2,..., v t 1. We denote v 2,..., v t 1 as nternal vertces of the path (α, v 1,..., v t ). The set F s a y-flow from S to T ff: for each v S the sum of values transferred from v n F s at most y v, for each v T the sum of values transferred to v n F s at most 1 y v, for each v V \ (S T ) the sum of values transferred through v n F s at most 1. For a gven y-flow F from S to T we defne G F = (V, A) as an auxlary drected graph wth the same vertex set as G, where an arc (u, v) belongs to A ff there s a path n F contanng u and v as consecutve vertces n exactly ths order. We call the y-flow F acyclc ff the drected flow graph G F s acyclc. Furthermore we defne a functon f F : A (0, 1], whch for an arc (u, v) assgns the sum of α values n all the paths n F that contan u and v as consecutve vertces. Moreover by fl F : A R + we denote a functon, whch for an arc (u, v) assgns the sum of terms L(s)α over all paths from F that start wth α and s S and contan u, v as consecutve elements. Intutvely by f F ((u, v)) we denote the fractonal number of centers that are transferred from u to v, whereas by fl F ((u, v)) we denote the fractonal number of vertces (clents) that were prevously covered by u and wll be covered by v after the shftng operaton (see Fg. 2). L(s 1 ) = 5 f F ((s 1, a)) = 0.2 fl F ((s 1, a)) = 1 f F ((s 2, a)) = 0.8 fl F ((s 2, a)) = 1.6 L(s 2 ) = 2 f F ((a, b)) = 1 fl F ((a, b)) = 2.6 L(a) = 12 L(b) = 6 L(t 1 ) = 4 f F ((b, t 1 )) = 0.6 fl F ((b, t 1 )) = 1.2 f F ((b, t 2 )) = 0.2 fl F ((b, t 2 )) = 0.4 f F ((b, t 3 )) = 0.2 fl F ((b, t 3 )) = 1 L(t 3 ) = 5 L(t 2 ) = 20 Fgure 2. The graph G F for an acyclc y-flow F = {(0.2, s 1, a, b, t 3 ), (0.6, s 2, a, b, t 1 ), (0.2, s 2, a, b, t 2 )} from S = {s 1, s 2 } to T = {t 1, t 2, t 3 }, where y s1 = 0.4, y s2 = y a = y b = 1, y t1 = 0, y t2 = 0.8, y t3 = 0.1. Note that even though each path n F has startng pont capacty not greater than ts endng pont capacty the vertex t 1 T s reachable from s 1 S n G F despte the fact that L(s 1 ) > L(t 1 ). Now we show that f we are gven an acyclc y-flow F then we can transfer y-values usng a chan shftng method wthout ncreasng the radus of vertces by too much. Formal defntons and lemmas follow. Defnton II.9 (chan shftng). Let F be an acyclc y-
flow from S to T and let (x, y) be a δ-feasble soluton. Let G F = (V, A) be the auxlary acyclc flow graph. By chan shftng we denote the followng operaton: For each u, v V, set u,v = 0. For each arc (u, a) A n reverse topologcal orderng of G F : For each v V, let = x u,v fl F (u, a)/(l(u)y u ), set a,v = a,v + and u,v = u,v. For each u, v V, set x u,v = x u,v + u,v. For each s S decrease y s by (s,u) A f F((s, u)). For each t T ncrease y t by (u,t) A f F((u, t)). For a drected graph G = (V, A), for a vertex v, we denote N n (v) = {u : (u, v) A} and N out (v) = {u : (v, u) A}. Lemma II.10 ( ). Let (x, y ) be the result of the chan shftng operaton on a δ-feasble soluton (x, y) accordng to an acyclc y-flow F from S to T. If d s the greatest dstance n G between two adjacent vertces n G F, then (x, y ) s a (δ + d)-feasble soluton, and for each vertex v of ndegree zero n G F, we have radus (x,y )(v) radus (x,y) (v), whereas for other vertces v, we have radus (x,y )(v) max(radus (x,y) (v), max a N n GF (v)(radus (x,y) (a) + dst G (a, v))). Furthermore for each v V \ (S T ) ts y-value s the same n (x, y) and (x, y ). C. Separable caterpllar structure If we knew that n the caterpllar structure (P, P ) produced by Lemma II.7 the capacty of each vertex n P s not smaller than the capacty of each vertex n P then we could skp ths secton. Unfortunately some vertces of V (P ) may have smaller capacty than some vertces of V (P ) and for ths reason we defne the noton of dangerous, safe and separable caterpllar structures. Defnton II.11 (safe, dangerous). For a caterpllar structure P = (P = (v 1,..., v p ), (v 0,..., v p+1)), by Γ(P) V (P ) we denote the set contanng all vertces v, such that there exst 0 0 < < 1 p + 1, such that v 0 nl, L(v 0 ) > L(v ) and v 1 nl, L(v 1 ) > L(v ). A caterpllar structure P s safe f Γ(P) = and dangerous otherwse. Defnton II.12 (separable). Let (x, y) be a δ- feasble soluton and let P = (P = (v 1,..., v p ), P = (v 0,..., v p+1)) be a dangerous caterpllar structure. We call P separable ff there exsts 1 p such that v Γ(P), L(v ) = mn v Γ(P) L(v) and ether: S 1 S 2 S 2, where S 2 = j=+1,...,p+1 v j nl j and S 1 s the sum of values (1 y v ) where v V, v = v j, L(v) > L(v ) for some < j p + 1, or, 0 = 0.2 L(v 0) = 10 L(v 1) = 5 L(v 2) = 3 L(v 3) = 8 L(v 4) = 9 L(v 5) = 3 L(v 6) = 5 L(v 7) = 2 1 = 0.1 L(v 1) = 1 0.3 0.3 0.2 0.2 3 = 0.8 L(v 3) 4 = 0.9 = 2 L(v 4) = 8 0.1 0.2 6 = 0.7 L(v 6) 7 = 0.3 = 5 L(v 7) = 2 Fgure 3. A separable caterpllar structure (P, P ), where Γ((P, P )) = {v 1, v 2, v 5 } (note that v 7 Γ((P, P )), snce v 8 = nl). By dashed edges an acyclc flow F = {(0.1, v 2, v 3, v 4, v 4 ), (0.2, v 2, v 3, v 4, v 5, v 6, v 6 )} from {v 2 } to {v 4, v 6 } s marked wth values f F prnted n the mddle of each arc. S 1 S 2 S 2, where S 2 = j=0,..., 1 v j nl j and S 1 s the sum of values (1 y v ) where v V, v = v j, L(v) > L(v ) for some 0 j <. We call such as above a wtness of separablty of P. A caterpllar structure that s not separable s called nonseparable. The ntuton behnd the sums S 1, S 2 s as follows. The sum S 2 contans all the y-values of vertces of P to the rght (or left) of. Snce we want to round all the y-values of vertces of P, f we want to splt the caterpllar structure (P, P ) by removng the edge v v +1 (or v 1 v ), we need to send S 2 S 2 unts of flow to the part that does not contan v, n order to make the sum of y-values over all the leaves n both new caterpllar structures ntegral. That s to satsfy (8) of Defnton II.6. In S 1 we sum over all vertces, that can potentally receve flow f we start a path at v, and the value (1 y v ) s the y-value a vertex v may receve. An example of a separable caterpllar structure s depcted n Fg. 3. Observe that a non-separable path structure may be dangerous as n Fg. 4. Lemma II.13 ( ). Let P = ((v 1,..., v p ), (v 0,..., v p+1)) be a dangerous caterpllar structure and let be an ndex such that v Γ(P) and L(v ) = mn v Γ(P) L(v). Moreover let j be an ndex such that v j nl, L(v j ) > L(v ). Then for any a [mn(, j), max(, j)] we have L(v a ) L(v ). Lemma II.14. Let P = ((v 1,..., v p ), (v 0,..., v p+1 )) be a dangerous non-separable caterpllar structure and let l = mn v Γ(P) L(v). For I = { : 0 p + 1 v nl L(v ) > l} we have I (1 ) < 2. Proof: Consder any v Γ(P) such that L(v ) = l. Let I 1 = I [0, 1] and I 2 = I [ + 1, p + 1] (note that I = I 1 I 2 ). We know that v s not a wtness of separablty hence each of the two sums S 1 n Defnton II.12 s strctly smaller than 1, snce otherwse we would have S 1 1 S 2 S 2. Consequently I 1 (1 ) < 1 and smlarly I 2 (1 ) < 1. In the followng lemma we use a procedure whch gven a δ-caterpllar structure (P, P ) produces a set of nonseparable δ-caterpllar structures. At very hgh level t checks
L(v 1) = 7 L(v 2) = 8 L(v 3) = 3 L(v 4) = 9 L(v 5) = 3 L(v 6) = 9 L(v 7) = 4 0.6 0.6 0.6 0.4 0.4 0.4 0.4 0 = 0.4 L(v 0) 0.4 = 10 1 = 0.7 L(v 1) = 1 3 = 0.8 L(v 3) 4 = 0.2 = 2 L(v 4) 5 = 0.8 = 2 L(v 5) 6 = 0.6 = 1 L(v 6) 7 = 0.5 = 3 L(v 7) = 4 Fgure 4. A dangerous caterpllar structure (P, P ), where Γ((P, P )) = {v 3, v 5 }. The caterpllar structure s non-separable because both for = 3 and = 5 n Defnton II.12 the sum S 1 s at most 0.6, whle S 2 S 2 s equal to 0.9. By dashed edges an acyclc flow F = {(0.6, v 3, v 2, v 1, v 0 ), (0.4, v 3, v 4, v 5, v 6, v 7, v 7 )} from {v 3} to {v 0, v 7 } s marked wth values f F prnted n the mddle of each arc. whether (P, P ) s separable, and f yes t sets as a wtness from Defnton II.12 wth the smallest value of L(v ). Next an acyclc flow from v to leaves of (P, P ) s constructed (see Fg. 3), and afterwards the procedure s run on two caterpllar structures nduced by the parts to the left, and to the rght of v. Lemma II.15 ( ). For a gven feasble LP soluton (x, y) we can fnd a 68-feasble soluton (x, y ) together wth a set of vertex dsjont non-separable 21-caterpllar structures S such that each vertex v outsde of the set has an ntegral y-value n (x, y ). Furthermore for each vertex v that belongs to some caterpllar structure from S we have radus (x,y )(v) 47. In the followng lemma we transform non-separable caterpllar structures nto safe caterpllar structures. Lemma II.16. There exst constants c, δ such that for a gven feasble LP soluton (x, y) we can fnd a c-feasble soluton (x, y ) together wth a set of vertex dsjont safe δ-caterpllar structures S such that each vertex v outsde of the set has an ntegral y-value n (x, y ). Proof: We use Lemma II.15 to obtan a set S of vertex dsjont non-separable 21-caterpllar structures. Our goal s to transform each dangerous caterpllar structure n S nto a safe caterpllar structure. Consder a dangerous non-separable δ 0 -caterpllar structure P = ((v 1,..., v p ), (v 0,..., v p+1)) S and let v a be a mnmum capacty vertex n Γ(P). Moreover let I = { : 0 p + 1 v nl L(v ) > L(v a)}. Construct any acyclc y-flow whch sends mn(1, I (1 y v )) from {v a } to {v : I} (see Fg. 4). Such flow always exsts due to Lemma II.13. Let Y = {v a, v a, v a 1} \ {nl} and perform group shftng on Y (note that a 1, snce v a Γ(P)). Replace P n S wth the (2δ 0 )- caterpllar structure ((v 1,..., v a 1, v a+1,..., v p ), (v 0,..., v a 2, u, v a+1,..., v p+1)), where as u we set the only vertex from Y wth fractonal y-value after group shftng or we set u = nl f all vertces n Y have ntegral y-values. We need to argue, that when u nl, we have L(u) L(v a 1 ), n order to satsfy (4) of Defnton II.6. Observe, that f v a nl, then L(v a) L(v a ), and smlarly f v a 1 nl, then L(v a 1) L(v a 1 ). Hence to show L(u) L(v a 1 ) t s enough to show L(v a ) L(v a 1 ), but ths follows from Lemma II.13, snce v a Γ(P). Note, that each caterpllar structure wll be modfed accordng to the above procedure at most twce, snce after one teraton the sum I (1 y v ) ether equals zero or decreases by one, and by Lemma II.14 we have I (1 y v ) < 2. Consequently by Lemmas II.10, II.4 we obtan the desred set of vertex dsjont δ-caterpllar structure together wth a c-feasble soluton. D. Roundng safe caterpllar structures In ths secton we descrbe how to round the c-feasble soluton (x, y ) usng the set of vertex dsjont safe caterpllar structures S from Lemma II.16. In order to do that we ntroduce a noton of roundng flow whch s a specal knd of y-flow defned for a caterpllar structure. Defnton II.17 (roundng flow). For a caterpllar structure (P, P ) and an assgnment (x, y) we call F a roundng flow ff F s a y-flow from S to T where S T = V (P ), for each v S we have f F((v, v )) = and for each v T we have f F ((v, v )) = 1 y v. Furthermore each flow path from F can not go through a vertex from V \ (V (P ) V (P )). In order to obtan a roundng flow for each vertex of V (P ) (whch by defnton have fractonal y-values), we have to decde whether t wll be a source (member of S) or a snk (member of T ). After chan shftng accordng to F all sources should have y-value equal to zero whereas all snks should have y-value equal to one and consequently all vertces from the caterpllar structure wll have ntegral y-value. In the followng lemma we show that for each nonseparable caterpllar structure we can always fnd a roundng flow n polynomal tme. Lemma II.18. For any safe δ-caterpllar structure (P, P ) and an assgnment (x, y) there exsts a roundng flow F such that for any two adjacent vertces n G F ther dstance n G s at most δ. Furthermore we can fnd such a roundng flow n polynomal tme. Proof: We present a recursve procedure whch constructs a desred roundng flow. Note that some recursve calls of the procedure mght potentally nvolve nfeasble assgnments (x, y ), however we prove that f the ntal call gves the procedure a safe δ-caterpllar structure, then as a result we obtan a vald roundng flow. Let us descrbe a procedure whch s gven a caterpllar structure (P, P ) together wth an assgnment y (the procedure does not need the x part of an assgnment). Denote P = (v 1,..., v p ) and P = (v 0,..., v p+1). If V (P ) = then we smply return the empty roundng flow. Otherwse let be the smallest nteger such that the sum of y-values of X = {v 0,..., v } \ nl s at least one (such always
exsts snce the sum of all y-values n V (P ) s ntegral by (8) of Def. II.6). Note that snce all vertces n V (P ) have fractonal y-values we have > 0. Let 0 0 be an ndex such that v 0 nl and v 0 has the bggest capacty n X. Let α = v X y v. If α = 1 then we recursvely construct a roundng flow F from S to T for a smaller caterpllar structure ((v +1,..., v p ), (nl, v +1,..., v p+1)) and () add to S the set of vertces X \ {v 0 } () add to T the vertex v 0 () for each v j X \ {v 0 } add to F a flow path ( j, v j, v j,..., v 0, v 0 ). In ths case we return F as a desred roundng flow for (P, P ). Hence from now on we assume α > 1 and α 1 <. Consder two cases: 0 < and 0 =. Frst let us assume that 0 <. We store z := and temporarly set = α 1. Next recursvely construct a roundng flow F from S V (P ) to T V (P ) for a smaller caterpllar structure ((v,..., v p ), P ), where P = (nl, v,..., v p+1) (note that the sum of y-values n P s ntegral). Now consder two cases: f v S then: () add to S vertces from X \ {v, v 0 } () add to T the vertex v 0 () for each v j X \ {v 0, v } add to F a flow path (y v j, v j, v j,..., v 0, v 0 ) (v) add to F a flow path (z, v, v,..., v 0, v 0 ) (v) set := z (v) return F. f v T then: () add to S vertces from X \ {v, v 0 } () add to T the vertex v 0 () out of the flow paths n F that end n v leave only that many, that send exactly 1 z unts of flow and reroute the rest paths to v 0 through vertces v 1, v 2,..., v 0 (v) for each v j X \ {v 0, v } add to F a flow path ( j, v j, v j,..., v 0, v 0 ) (v) return F. Now assume that 0 =. We create a smaller caterpllar structure ((v a, v +1, v +2,..., v p ), (nl, v a, v +1,..., v p+1)), where v a, v a are two newly created vertces wth a := α 1 and L(v a) := L(v a ) := L(v 1 ), where v 1 s the second bggest capacty vertex n the set X. Next run recursvely our procedure on the newly created caterpllar structure to obtan a roundng flow F from S to T. Agan, consder two cases: f v a S then: () set S := (S \ {v a}) (X \ {v }) () set T := T {v } () change n F all the paths that start n v a to start n X \ {v } (v) add to F paths that start n X and transfer 1 unts of flow from X to v (v) return F. f v a T then: () set S := S (X \ {v, v 1 }) () set T := (T \ {v a}) {v, v 1 } () reroute some of the flow paths from F that end n v a to that transfer exactly 1 unts of flow to v (that s remove v a as the last vertex on those paths and extend the paths by v, v ) (v) reroute all the remanng flow paths n F that end n v a to v 1 (that s remove v a and extend those paths by v, v 1,..., v 1, v 1 ) (v) for each v j X\{v, v 1 } add to F a flow path ( j, v j, v j,..., v 1, v 1 ) (v) return F. Fnally we prove that f the procedure receves a safe caterpllar structure then t returns a desred roundng flow. The only property of the roundng flow that needs detaled analyss s the assumpton that each nternal vertex of a flow path has capacty not smaller than ts the capacty of ts startng pont. Let us assume that there exsts a path n F that starts n v a, goes though v b and ends n v c, where L(v c) L(v a) > L(v b ). Ths contradcts the assumpton that P s safe because v b Γ(P). The followng theorem summarzes Sectons II-A, II-B, II-C, II-D. Theorem II.19. For a connected graph G, f LP1 has a feasble soluton then we can fnd a c-feasble soluton wth ntegral y-values. Proof: Usng a feasble soluton to LP1, by Lemma II.16, we obtan a c-feasble soluton (x, y ), together wth a set of vertex dsjont safe δ-caterpllar structures S, such that vertces that do not belong to any caterpllar structure n S have ntegral y-value n (x, y ). Next by Lemma II.18 for each δ-caterpllar structure (P, P ) S we fnd a roundng flow F (P,P ). Fnally for each δ-caterpllar structure (P, P ) we perform chan shftng wth respect to F (P,P ), and by Lemma II.10 we obtan a c -feasble soluton (x, y ) to LP1. By Lemma II.16, vertces outsde of S have ntegral y- value n (x, y ). Moreover by Defnton II.17, after chan shftng all the vertces n each caterpllar structure of S have ntegral y-values n (x, y ). E. Roundng x-values In ths secton we show how to extend Theorem II.19 to obtan not only ntegral y-values, but also ntegral x-values. The followng lemma s standard (usng network flows). Lemma II.20. Let (x, y) be a δ-feasble soluton such that all y-values are ntegral. There s a polynomal tme algorthm that creates a δ-feasble soluton whch has both x- and y-values ntegral. As a consequence of Theorem II.19 and the above lemma the proof Theorem I.2 follows. III. CONCLUSIONS AND OPEN PROBLEMS We have obtaned the frst constant approxmaton rato for the k-center problem wth non-unform hard capactes. The approxmaton rato we obtan s n the order of hundreds (however we do not calculate t explctly), so the natural open problem s to gve an algorthm wth a reasonable approxmaton rato. Moreover, we have shown that the ntegralty gap of the standard LP formulaton for connected graphs n the unform capactes case s ether 5 or 6, whch we thnk mght be an evdence, that t should be possble to narrow the gap between the known lower bound of (2 eps) and upper bound 6 n the unform capactes case.
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