Faster Exact and Approximate Algorithms for k-cut

Transcription

1 2018 IEEE 59th Annual Symposum on Foundatons of Computer Scence Faster Exact and Approxmate Algorthms for k-cut Anupam Gupta Computer Scence Department CMU Pttsburgh, USA Euwoong Lee Courant Insttute of Mathematcal Scences NYU New York Cty, USA Jason L Computer Scence Department CMU Pttsburgh, USA ml@cs.cmu.edu Abstract In the k-cut problem, we are gven an edgeweghted graph G and an nteger k, and have to remove a set of edges wth mnmum total weght so that G has at least k connected components. The current best algorthms are an O(n (2o(1))k ) randomzed algorthm due to Karger and Sten, and an Õ(n2k ) determnstc algorthm due to Thorup. Moreover, several 2-approxmaton algorthms are known for the problem (due to Saran and Vazran, Naor and Raban, and Rav and Snha). It has remaned an open problem to (a) mprove the runtme of exact algorthms, and (b) to get better approxmaton algorthms. In ths paper we show an O(k O(k) n (2ω/3+o(1))k )-tme algorthm for k-cut. Moreover, we show an (1 + ε)-approxmaton algorthm that runs n tme O((k/ε) O(k) n k+o(1) ), and a approxmaton n fxed-parameter tme O(2 O(k2) poly(n)). Keywords-mnmum k-cut, graph algorthms, parameterzed algorthm, approxmaton algorthms I. INTRODUCTION In ths paper we consder the k-cut problem: gven an edge-weghted graph G =(V,E,w) and an nteger k, delete a mnmum-weght set of edges so that G has at least k connected components. Ths problem s a natural generalzaton of the global mn-cut problem, where the goal s to break the graph nto k =2peces. Ths problem has been actvely studed n theory of both exact and approxmaton algorthms, where each result brought new nsghts and tools on graph cuts. It s not a pror clear how to obtan poly-tme algorthms for any constant k, snce guessng one vertex from each part only reduces the problem to the NPhard MULTIWAY CUT problem. Indeed, the frst result along these lnes was the work of Goldschmdt and Hochbaum [1] who gave an O(n (1/2o(1))k2 )-tme exact algorthm for k-cut. Snce then, the exact exponent n terms of k has been actvely studed. The current best runtme s acheved by an Õ(n2(k1) ) randomzed algorthm due to Karger and Sten [2], whch performs random edge contractons untl the remanng graph has k nodes, and shows that the resultng cut s optmal wth probablty at least Ω(n 2(k1) ). The asymptotc runtme of Õ(n2(k1) ) was later matched by a determnstc algorthm of Thorup [3]. Hs algorthm was based on tree-packng theorems; t showed how to effcently fnd a tree for whch the optmal k-cut crosses t 2k 2 tmes. Enumeratng over all possble 2k 2 edges of ths tree gves the algorthm. These elegant O(n 2k )-tme algorthms are the state-ofthe-art, and t has remaned an open queston to mprove on them. An easy observaton s that the problem s closely related to k-clique, so we may not expect the exponent of n to go below (ω/3)k. Gven the nterest n fne-graned analyss of algorthms, where n the range [(ω/3)k, 2k 2] does the correct answer le? Our man results gve faster determnstc and randomzed algorthms for the problem. Theorem I.1 (Faster Randomzed Algorthm). Let W be a postve nteger. There s a randomzed algorthm for k-cut on graphs wth edge weghts n [W ] wth runtme Õ(k O(k) n k+ (k2)/3 ω+1+((k2) mod 3) W ) O(k O(k) n (1+ω/3)k ), that succeeds wth probablty 1 1/poly(n). Theorem I.2 (Even Faster Determnstc Algorthm). Let W be a postve nteger. For any ε>0, there s a determnstc algorthm for exact k-cut on graphs wth edge weghts n [W ] wth runtme k O(k) n (2ω/3+ε)k+O(1) W O(k O(k) n (2ω/3)k ). In the above theorems, ω s the matrx multplcaton constant, and Õ hdes polylogarthmc terms. Whle the determnstc algorthm from Theorem I.2 s asymptotcally faster, the randomzed algorthm s better for small values of k. Indeed, usng the current best value of ω < [4], Theorem I.1 gves a randomzed algorthm for exact k-cut on unweghted graphs whch mproves upon the prevous best Õ(n 2k2 )-tme algorthm of Karger and Sten for all k [8,n o(1) ].For k 6, faster algorthms were gven by Levne [5] /18/$ IEEE DOI /FOCS

2 Approxmaton algorthms: The k-cut problem has also receved sgnfcant attenton from the approxmaton algorthms perspectve. There are several 2(1 1/k)-approxmaton algorthms that run n tme poly(n, k) [6], [7], [8], whch cannot be mproved assumng the Small Set Expanson Hypothess [9]. Recently, we gave an approxmaton algorthm that runs n 2 O(k6) n O(1) [10]. In ths current paper, we gve a (1+ε)-approxmaton algorthm for ths problem much faster than the current best exact algorthms; pror to our work, nothng better was known for (1 + ε)- approxmaton than for exact solutons. Theorem I.3 (Approxmaton). For any ε>0, there s a randomzed (combnatoral) algorthm for k-cut wth runtme (k/ε) O(k) n k+o(1) tme on general graphs, that outputs a (1 + ε)-approxmate soluton wth probablty 1 1/poly(n). The technques from the above theorem, combned wth the prevous deas n [10], mmedately gve an mproved FPT approxmaton guarantees for the k-cut problem: Theorem I.4 (FPT Approxmaton). There s a determnstc 1.81-approxmaton algorthm for the k-cut problem that runs n tme 2 O(k2) n O(1). Due to page restrctons n ths extended abstract, we defer the proofs of Theorems I.3 and I.4 to the full verson of ths paper. Lmtatons: Our exact algorthms rase the natural queston: how fast can exact algorthms for k-cut be? We gve a smple reducton showng that whle there s stll room for mprovement n the runnng tme of exact algorthms, such mprovements can only mprove the constant n front of the k n the exponent, assumng a popular conecture on algorthms for the CLIQUE problem. Clam I.5 (Relatonshp to Clque). Any exact algorthm for the k-cut problem for graphs wth edge weghts n [n 2 ] can solve the k-clique problem n the same runtme. Hence, assumng k-clique cannot be solved n faster than n ωk/3 tme, the same lower bound holds for the k-cut problem. A. Our Technques Our algorthms buld on the approach poneered by Thorup: usng tree-packngs, he showed how to fnd a tree T such that t crosses the optmal k-cut at most 2k 2 tmes. (We call such a tree a Thorup tree, or T-tree.) Now brute-force search over whch edges to delete from the T-tree (and how to combne the resultng parts together) gave an Õ(n2k2 )-tme determnstc algorthm. Ths last step, however, rases the natural queston havng found such a T-tree, can we use the structure of the k-cut problem to beat brute force? Our algorthms answer the queston n the affrmatve, n several dfferent ways. The man deas behnd our algorthm are dynamc programmng and fast matrx-multplcaton, carefully combned wth the fxed-parameter tractable algorthm technque of colorcodng, and random samplng n general. Fast matrx multplcaton: Our dea to apply fast matrx multplcaton starts wth the crucal observaton that f () the T-tree T s tght and crosses the optmal k- cut only k 1 tmes, and () these edges are ncomparable and do not le on a root-leaf path, then the problem of fndng these k 1 edges can be modeled as a max-weght clque-lke problem! (And hence we can use matrx-multplcaton deas to speed up ther computaton.) An mportant property of ths specal case s that choosng an edge e to cut fxes one component n the k-cut soluton by ncomparablty, the subtree below e cannot be cut anymore. The cost of a k-cut can be determned by the weght of edges between each par of components (ust lke beng a clque s determned by parwse connectvty), so ths case can be solved va an algorthm smlar to k-clique. Randomzed algorthm: Our randomzed algorthm removes these two assumptons step by step. Frst, whle the above ntuton crucally reles on assumpton (), we gve a more sophstcated dynamc program usng color-codng schemes for the case where the edges are not ncomparable. Moreover, to remove assumpton (), we show a randomzed reducton that gven a tree that crosses the optmal cut as many as 2k 2 tmes, fnds a tght tree wth only k 1 crossngs (whch s the least possble), at the expense of a runtme of O(k 2 n) k1. Note that guessng whch edges to delete s easly done n n k1 tme, but addng edges to regan connectvty whle not ncreasng the number of crossngs can navely take a factor of m k1 more tme. We lose only a k 2(k1) factor usng our random-samplng based algorthm, usng that n an optmal k-cut a splt cluster should have more edges gong to ts own parts than to other clusters. Determnstc algorthm: The determnstc algorthm proceeds along a dfferent drecton and removes both assumptons () and () at once. We show that by deletng some O(log k) carefully chosen edges from the T-tree T, we can break t nto three forests such that we only need to delete about 2k/3 edges from each of these forests. Such a deleton s not possble when T s a star, but approprately extendng T by ntroducng Stener nodes admts ths deleton. (And Θ(log k) s tght n ths extenson.) For each forest, there are n 2k/3 ways 114

3 to cut these edges, and once a choce of 2k/3 edges s made, the forest wll not be cut anymore. Ths property allows us to bypass () and establsh desred parwse relatonshps between choces to delete 2k/3 edges n two forests. Indeed, we set up a trpartte graph where one part corresponds to the choces of whch 2k/3 edges to cut n one forest and the cost of the mn k- cut s the weght of the mn-weght trangle, whch we fnd effcently usng fast matrx multplcaton. Some techncal challenges arse because we need to some components for some forests may only have Stener vertces, but we overcome these problems usng colorcodng. Approxmaton schemes: The (1 + ε)-approxmaton algorthm agan uses the O(k 2 n) k1 -tme randomzed reducton, so that we have to cut exactly k 1 edges from a tght T-tree T. An exact dynamc program for ths problem takes tme n k as t should, snce even ths tght case captures clque, when T s a star and hence these k 1 edges are ncomparable. And agan, we need to handle the case where these k 1 edges are not ncomparable. For the former problem, we replace the problem of fndng clques by approxmately fndng partal vertex covers nstead. (In ths new problem we fnd a set of k 1 vertces that mnmze the total number of edges ncdent to them.) Secondly, n the DP we cannot afford to mantan the boundary of up to k edges explctly any more. We show how to mantan an ε-net of nodes so that carefully roundng the DP table to only track a small f(k)-szed set of these rounded subproblems ncurs only a (1 + ε)-factor loss n qualty. Our approxmate DP technque turns out to be useful to get a 1.81-approxmaton for k-cut n FPT tme, mprovng on our prevous approxmaton of [10]. In partcular, the lamnar cut problem from [10] also has a tght T-tree structure, and hence we can use (a specal case of) our approxmate DP algorthm to get a (1+ε)-approxmaton for lamnar cut, nstead of the 2ε-factor prevously known. Combnng wth other deas n the prevous paper, ths gves us the 1.81-approxmaton. Agan, the full detals of the (1 + ε)-approxmaton algorthm that the 1.81-approxmaton FPT algorthm are deferred to the full verson. B. Related Work The frst non-trval exact algorthm for the k-cut problem was by Goldschmdt and Hochbaum, who gave an O(n (1/2o(1))k2 )-tme algorthm [1]; ths s somewhat surprsng because the related MULTIWAY CUT problem s NP-hard even for k = 3. They also proved the problem to be NP-hard when k s part of the nput. Karger and Sten mproved ths to an O(n (2o(1))k )- tme randomzed Monte-Carlo algorthm usng the dea of random edge-contractons [2]. Thorup mproved the O(n 4k+o(1) )-tme determnstc algorthm of Kamdo et al. [11] to an Õ(n2k )-tme determnstc algorthm based on tree packngs [3]. Better algorthms are known for small values of k [2, 6] [12], [13], [14], [15], [16], [17], [5]. Approxmaton algorthms: The frst such result for k- CUT was a 2(1 1/k)-approxmaton of Saran and Vazran [6]. Later, Naor and Raban [7], and also Rav and Snha [8] gave 2-approxmaton algorthms usng tree packng and network strength respectvely. Xao et al. [18] extended Kapoor [19] and Zhao et al. [20] and generalzed Saran and Vazran to gve an (2 h/k)- approxmaton n tme n O(h). On the hardness front, Manurangs [9] showed that for any ε > 0, t s NP-hard to acheve a (2 ε)-approxmaton algorthm n tme poly(n, k) assumng the Small Set Expanson Hypothess. In recent work [10], we gave a approxmaton for k-cut n FPT tme f(k)poly(n); ths does not contradct Manurangs s work, snce k s polynomal n n for hs hard nstances. We mprove that guarantee to 1.81 by gettng a better approxmaton rato for the lamnar k-cut subroutne, mprovng from 2 ε to 1+ε. Ths follows as a specal case of the technques we develop n the proof of Theorem I.3, and s also deferred to the full verson. FPT algorthms: Kawarabayash and Thorup gve the frst f(opt) n 2 -tme algorthm [21] for unweghted graphs. Chtns et al. [22] used a randomzed colorcodng dea to gve a better runtme, and to extend the algorthm to weghted graphs. Here, the FPT algorthm s parameterzed by the cardnalty of edges n the optmal k-cut, not by the number of parts k. For more detals on FPT algorthms and approxmatons, see the book [23], and the survey [24]. C. Prelmnares For a graph G =(V,E,w), consder some collecton of dsont sets S = {S 1,...,S r }. Let E G (S) = E G (S 1,...,S r ) denote the set of edges n E G [ r S r] (.e., among the edges both of whose endponts le n these sets) whose endponts belong to dfferent sets S. For any vertex set S, let S denote the edges wth exactly one endpont n S; hence E G (S) = S P S. For a collecton of edges F E, let w(f ) := e F w(e) be the sum of weghts of edges n F.In partcular, for a k-cut soluton {S 1,...,S k }, the value of the soluton s w(e G (S 1,...,S k )). 115

4 For a rooted tree T =(V T,E T ), let T v V T denote the subtree of T rooted at v V T. For an edge e E T wth chld vertex v, let T e := T v. Fnally, for any set S V T, T S = v S T v. For some sectons, we make no assumptons on the edge weghts of G, whle n other sectons, we wll assume that all edge weghts n G are ntegers n [W ], for a fxed postve nteger W. We default to the former unrestrcted case, and explctly menton transtonng to the latter case when needed. II. A FAST RANDOMIZED ALGORITHM In ths secton, we present a randomzed algorthm to solve k-cut exactly n tme Õ(kO(k) n (1+ω/3)k ), provng Theorem I.1. Secton II-A ntroduces our hghlevel deas based on Thorup s tree packng results. Secton II-B shows how to refne Thorup s tree to a good tree that crosses the optmal k-cut exactly k 1 tmes, and Secton II-C presents an algorthm gven a good tree. A. Thorup s Tree Packng and Thorup s Algorthm Our startng pont s a transformaton from the general k-cut problem to a problem on trees, nspred by Thorup s algorthm [3] based on greedy tree packngs. We wll be nterested n trees that cross the optmal partton only a few tmes. We fx an optmal k-cut soluton, S := {S1,...,S k }. Let OPT := E G(S1,...,S k ) be edges n the soluton, so that w(opt) s the soluton value. Defnton II.1 (T-trees). AtreeT of G s a l-t-tree f t crosses the optmal cut at most l tmes;.e., E T (S1,...,S k ) l. Ifl = 2k 2, we often drop the quantfcaton and call t a T-tree. If l = k 1, the mnmum value possble, then we call t a tght T-tree. Our frst step s the same as n [3]: we compute a collecton T of n O(1) trees such that there exsts a T-tree,.e., a tree T T that crosses OPT at most 2k 2 tmes. Theorem II.2 ([3], Theorem 1). For α (0, 9 10 ), let T be a greedy tree packng wth at least 3m(k/α) 3 ln(nmk/α) trees. Then, on the average, the trees T T cross each mnmum k-cut less than 2(k 1+2α) tmes. Furthermore, the greedy tree packng algorthm takes Õ(k3 m 2 ) tme. The runnng tme comes from the executon of Õ(k3 m) mnmum spannng tree computatons. Note that, snce our results are only nterestng when k 7, resultng n algorthms of runnng tme Ω(n 7+2ω ), we can completely gnore the runnng tme of the greedy tree packng algorthm, whch s only run once. Lettng α := 1/8, we get the followng corollary. Corollary II.3. We can fnd a collecton of Õ(k 3 m) trees such that for a random tree T T, E T (S1,...,S k ) 2k 3/2 n expectaton. In partcular, there exsts a T-tree T T. In other words, f we choose such a T-tree T T, we get the followng problem: fnd the best way to cut 2k 2 edges of T, and then merge the connected components nto exactly k components S 1,...,S k so that E G (S 1,...,S k ) s mnmzed. Thorup s algorthm accomplshes ths task usng brute force: try all possble O(n 2k2 ) ways to cut and merge, and output the best one. Ths gves a runtme of Õ(k 3 n 2k2 m), oreven Õ(n 2k2 m) wth a more careful analyss [3]. The natural queston s: can we do better than brute-force? For the mn-cut problem (when k = 2), Karger was able to speed up ths step from O(n 2k2 )=O(n 2 ) to Õ(n) usng dynamc tree data structures [15]. However, ths case s specal: snce there are 3 components produced from cuttng the 2k 2=2tree edges, only one par of components need to be merged. For larger values of k, t s not clear how to generalze the use of clever data structures to handle multple merges. Our randomzed algorthm gets the mprovement n three steps: Frst, nstead of tryng all possble trees T T, we only look at a random subset of Ω(k log n) trees. By Corollary II.3 and Markov s nequalty, the probablty that a random tree satsfes E T (S1,...,S k ) 2k 1 s (2k 3/2)/(2k 1) = 1 Ω(1/k). Therefore, by tryng Ω(k log n) random trees, we fnd a T-tree T w.h.p. Next, gven a T-tree T from above, we show how to fnd a collecton of n k1 trees such that, wth hgh probablty, one of these trees T s a tght T-tree,.e., t ntersects OPT n exactly k1 edges. We show ths n II-B. Fnally, gven a tght T-tree T from the prevous step, we show how to solve the optmal k-cut n tme O(n (ω/3)k ), much lke the k-clique problem [25]. The runtme s not concdental; the W [1] hardness of k-cut derves from k-clique, and hence technques for the former must work also for the latter. We show ths n II-C. B. A Small Collecton of Tght Trees In ths secton we show how to fnd a collecton of n k1 trees such that, wth hgh probablty, one of 116

5 analyss may only gve Ω(1/m) for the second part. v r Fgure II.1: The red edges are deleton-worthy edges n ths T-tree; the dashed lnes mark the optmal components. these trees T s a tght T-tree. Formally, Lemma II.4. There s an algorthm that takes as nput atreet such that E T (S1,...,S k ) 2k 2, and produces a collecton of k O(k) n k1 log n trees, such that one of the new trees T satsfes E T (S1,...,S k ) = k 1 w.p. 1 1/poly(n). The algorthm runs n tme k O(k) n k1 m log n. The algorthm proceeds by teratons. In each teraton, our goal s to remove one edge of T and then add another edge back n, so that the result s stll a tree. In dong so, the value of E T (S1,...,S k ) can ether decrease by 1, stay the same, or ncrease by 1. We call an teraton successful f E T (S1,...,S k ) decreases by 1. Throughout the teratons, we wll always refer to T as the current tree, whch may be dfferent from the orgnal tree. Fnally, f E T (S1,...,S k ) = l ntally, then after l (k 1) consecutve successful teratons, we have the desred tght T-tree T. Assume we know l beforehand; we can easly dscharge ths assumpton later. For an ntermedate tree T n the algorthm, we say that component S s unsplt f S nduces exactly one connected component n T, and splt otherwse. Intally, there are at most (k1)l splt components, possbly fewer f some components nduce many components n T. Moreover, f all l (k 1) teratons are successful, all components are unsplt at the end. Lemma II.5. The probablty of any teraton beng successful,.e., reducng the number of tree-edges belongng to the optmal cut, s at least Ω(1/nk 2 ). Proof: Each successful teraton has two parts: frst we must delete a deleton-worthy edge (whch happens wth probablty 1/(n 1)), and then we add a good connectng edge (whch happens wth probablty Ω(1/k 2 )). The former ust uses that a tree has n 1 edges, but the latter must use that there are many good edges crossng the resultng cut a nave We frst descrbe the edges n T that we would lke to delete. These are the edges such that f we delete one of them, then we are lkely to make a successful teraton (after selectvely addng an edge back n). We call these edges deleton-worthy. Let us frst root the tree T =(V,E T ) at an arbtrary, fxed root v r V.For any edge e, let T e denote the subtree below t obtaned by deletng the edge e. Defnton II.6. A deleton-worthy edge e E T satsfes the followng two propertes: (1) The edge crosses between two parts of the optmal partton,.e., e E T (S 1,...,S k ). (2) There s exactly one part S S satsfyng S T e and S T e. In other words, exactly one component of S ntersects T e but s not completely contaned n T e. Note that, by condton (1), S s necessarly splt. Clam II.7. If there s a splt component S, there exsts a deleton-worthy edge e E T. Proof: For each S, contract every connected component of S nduced n T, so that splt components contract to multple vertces. Root the resultng tree at v r, and take a vertex v V of maxmum depth whose correspondng component S s splt. It s easy to see that v v r and the parent edge of v n the rooted tree s deleton-worthy. Fnally, we descrbe the deleton part of our algorthm. The procedure s smple: choose a random edge n T to delete. Wth probablty 1/(n 1), we remove a deleton-worthy edge n T. Ths gves rse to the n 1 factor n the probablty of a successful teraton. Now we show that, condtoned on deletng a deletonworthy edge, we can selectvely add an edge to produce a successful teraton wth probablty k O(1). In partcular, we add a random edge n E G (T e,v T e ).e., an edge from subtree under e to the rest of the vertces where the probablty s weghted by the edge weghts n E G (T e,v T e ). We show that ths makes the teraton successful wth probablty Ω(1/k 2 ). (Recall that the teraton s successful f the number of tree edges lyng n the optmal cut decreases by 1.) Frst of all, t s clear that addng any edge n E G (T e,v T e ) wll get back a tree. Next, to lower bound the probablty of success, we begn wth an auxlary lemma. Clam II.8. Gven a set of k + 1 components 117

6 S 1,...,S k+1 ( that( partton ) V, we have 1 ) k +1 w(opt) 1 w(e G (S 1,...,S k+1 )). 2 Proof: Consder mergng two components S,S unformly at random. Every edge n E(S 1,...,S k+1 ) has probablty ( ) k of dsappearng from the cut, so the expected ( ( new cut ) s 1 ) k +1 1 E(S 1,...,S k+1 ), 2 and w(opt) can only be smaller. For convenence, defne C := S T e, where S s the splt component correspondng to the deleton-worthy edge e we ust deleted. Observe that the only edges n E(T e,v T e ) that are not n OPT must be n E(C, S C); ths s because, of the components S ntersectng T e, only S s splt. Therefore, w(e(t e,v T e )) w(opt)+w(e(c, S C)), and the probablty of selectng an edge n E(C, S C) s w(e(c, S C)) w(e(t e,v T e )) w(e(c, S C)) w(opt)+w(e(c, S C)). (II.1) Clam ( II.9. w(e(c, S C)) ( 1 ( ) k+1 1 ) ) w(opt) = Ω(1/k 2 ) w(opt). Proof: The set of edges OPT E(C, S C) cuts the graph G nto k +1 components. ( Clam II.8 mples ths set has total weght 1 ( ) k+1 1 ) 1 2 w(opt). Observng that the edges of OPT and E(C, S C) are dsont from each other completes the proof. Usng the above clam n (II.1) means the probablty of selectng an edge n E(C, S C) s Ω(1/k 2 ). Hence the probablty of an teraton beng successful s Ω(1/(nk 2 )), completng the proof of Lemma II.5. Snce we have l teratons, the probablty that each of them s successful s l O(l) n l. If we repeat ths algorthm l O(l) n l log n tmes, then wth probablty 1 1/poly(n), one of the fnal trees T wll satsfy E T (S1,...,S k ) = k1. We can remove the assumpton of knowng l by tryng all possble values of l [k 1, 2k 2], gvng a collecton of k O(k) n k1 log n trees n runnng tme k O(k) n k1 m log n. Ths completes the proof of Lemma II.4. C. Solvng k-cut on Tght Trees In the prevous secton, we found a collecton of n k trees such that, wth hgh probablty, the ntersecton of one of these trees wth the optmal k-cut OPT conssts of only k 1 edges. In ths secton, we show that gven ths tree we can fnd the optmal k-cut n tme n ωk/3. Ths wll follow from Lemma II.10 below. In ths secton, we restrct the edge weghts of our graph G to be postve ntegers n [W ]. Lemma II.10. There s an algorthm that takes a tree T and outputs, from among all parttons {S 1,...,S k } that satsfy E T (S 1,...,S k ) = k 1, a partton S := {S 1,...,S k } mnmzng the number of nter-cluster edges E G (S 1,...,S k ), n tme Õ(k O(k) n (k2)/3 ω+2+(k2) mod 3 W ). Gven a tree T =(V,E T ) and a set F E T of tree edges, deletng these edges gves us a vertex partton S F = {S 1,...,S F +1 }. Let Cut(F ) be the set of edges n G that go between the clusters n S F ;.e., Cut(F ):=E(S 1,...,S F +1 ). (II.2) Put another way, these are the edges (u, v) E such that the unque u-v path n T contans an edge n F. Note that Lemma II.10 seeks a set E E T of sze k 1 that mnmzes w(cut(f )). 1) A Smple Case: Incomparable Edges: Our algorthm bulds upon the algorthm of Nešetřl and Polak [25] for k-clique, usng Boolean matrx multplcaton to obtan the speedup from the nave O(n k ) brute force algorthm. It s nstructve to frst consder a restrcted settng to hghlght the smlarty between the two algorthms. Ths settng s as follows: we are gven a vertex v r V and the promse that f the nput tree T =(V,E T ) s rooted at v r, then the optmal k1 edges E := E T (S 1,...,S k ) to delete are ncomparable. By ncomparable, we mean any root-leaf path n T contans at most one edge n E. Lke the algorthm of [25], our algorthm creates an auxlary graph H =(V H,E H ) on O(n k/3 ) nodes. Our graph constructon dffers slghtly n that t always produces a trpartte graph, and that ths graph has edge weghts. In ths auxlary graph, we wll call the vertces nodes n order to dfferentate them from the vertces of the tree. The nodes n graph H wll form a trpartton V 1 V 2 V 3 = V H. For each r, let F r 2 E be the famly of all sets of exactly r edges n E T that are parwse ncomparable n T. For each =1, 2, 3, defne r := (k1)+(1) 3 so that r 1 + r 2 + r 3 = k 1. For each =1, 2, 3 and each F F r, add a node v F to V representng set F. Consder a par (V a,v b ) of parts n the trpartton wth (a, b) (1, 2), (2, 3), (3, 1). Consder a par of sets F a := {e a 1,...,e a r a } F ra, F b := {e b 1,...,e b r b } F rb ; recall these are sets of r a 118

7 and r b ncomparable edges n T. If the edges n F a are also parwse ncomparable wth the edges n F b, then add an edge (v F a a,v F b b ) V a V b of weght w H (v F a a,v F b b ):= r a r a w(e(t e a,v T e a )) r a =+1 r a r b =1 w(e(t e a,t e a )) w(e(t e a,t e b )). Observe that every trple of nodes n graph H that form a trangle together represent r 1 + r 2 + r 3 = k 1 many ncomparable edges. Moreover, the weghts are set up so that for any trangle (v F 1 1,v F 2 2,v F 3 3 ) V 1 V 2 V 3 such that F := F 1 F 2 F 3 = {e 1,...,e k1 }, the total weght of the edges s equal to = w H (v F 1 1,v F 2 2 )+w H (v F 2 2,v F 3 3 )+w H (v F 3 3,v F 1 k1 k1 k1 w(e(t e,v T e )) =+1 1 ) w(e(t e,t e )). (II.3) A straghtforward countng argument shows that ths s exactly w(e(t e1,...,t ek1 )) = Cut(F ), the soluton value of cuttng the edges n F. Hence, the problem reduces to computng a mnmum weght trangle n graph H. Whle the mnmum weght trangle problem s unlkely to admt an O(N 3ε ) tme algorthm on a graph wth N vertces wth arbtrary edge weghts, the problem does admt an Õ(MNω ) tme algorthm when the graph has ntegral edge weghts n the range [M,M] [26]. Snce the orgnal graph G has ntegral edge weghts n [W ], the edge weghts n H must be n the range [O(Wm),O(Wm)]. Therefore, we can set N := O(n (k1)/3 ) and M := O(Wm) to obtan an Õ(Wn (k1)/3 ω m) tme algorthm n ths restrcted settng. 2) The General Algorthm: Now we prove Lemma II.10 n full generalty, and show how to fnd E. The deas we use here wll combne the matrx-multplcaton dea from the restrcted case of ncomparable edges, together wth dynamc programmng. Gven a tree edge e E T, and an nteger s [k 2], let State(e, s) denote a set of edges F n subtree T e such that F = s 1 and Cut({e} F ) s mnmzed. In other words, State(e, s) represents the optmal way to cut edge e along wth s 1 edges n T e. For ease of presentaton, we assume that ths value s unque. Observe that, once all of these states are computed, the remanng problem bols down to choosng an nteger l [k 1], ntegers s 1,...,s l whose sum s k 1, and ncomparable ( edges e 1,...,e l that mnmzes l ) k1 Cut State(e,s ) = State(e,s ) k1 k1 =+1 w(e(t e,t e )). Comparng ths expresson to (II.3) suggests that ths problem s smlar to the ncomparable case n II-C1, a connecton to be made precse later. We now compute states for all edges e E T, whch we do from bottom to top (leaf to root). When e s a leaf edge, the states are straghtforward: State(e, 1) = Cut({e}) and State(e, s) = for s > 1. Also, for each edge e E T, defne desc(e) to be all descendant edges of e, formally defned as all edges f E T e whose path to the root contans edge e. Fx an edge e E T and an s [k 2], for whch we want to compute State(e, s). Suppose we order the edges n T e n an arbtrary but fxed order. Let us now fgure out some propertes for ths (unknown) value of State(e, s). As a thought experment, let F be the lst of all the maxmal edges n State(e, s) n other words, f F ff f State(e, s) and f / desc(f ) for all f State(e, s). Let l := F and F =(e 1,...,e ) l be the sequence n the defned order, and for each e, let s := 1+ desc(e ) State(e, s). Observe that s = s 1, and that we must satsfy l ) State(e, s) = ({e } State(e,s ). (II.4) Also, w(state(e, s)) = E(T e,v T e ) l ) + w(e(g[t e ]) Cut ({e } State(e,s ) ) l l =+1 w(e G[Te][T e,t e ]), snce the only edges double-counted n the frst summaton of w(state(e, s)) are those connectng dfferent T e,t e. Gven these deal values l and {s }, our algorthm repeats the followng procedure multple tmes: Pck a number l unformly at random n [s 1]. Then, let functon σ : [l] [s 1] be chosen unformly at random among all (s1) l possble functons satsfyng l σ() =s1. Wth probablty (s 1) (l +1) = k O(k), we correctly 119

8 guess l = l and σ() =s for each [l].1 Construct an auxlary graph H as follows. As n II-C1, H has a trpartton V 1 V 2 V 3 = V H, and assume there s an arbtrary but fxed total orderng on the edges of the tree. For each r, let F r 2 E be the famly of all sets of exactly r edges n E T that are parwse ncomparable n T. For each =1, 2, 3, let r := l+(1) 3 so that r 1 + r 2 + r 3 = l, and for each F F r, add a node v F to V representng the edges F as a sequence n the total order. Also, defne R := 1 =1 r for = 1, 2, 3, 4. Note that R 1 = 0 and R 4 = r 1 + r 2 + r 3 = l. Our ntenton s map the nteger values {σ(r +1),σ(R +2),...,σ(R +1 )} to the sequences represented by nodes n V, as we wll see later. Consder each trpartton par (V a,v b ) wth (a, b) (1, 2), (2, 3), (3, 1). For each par F a F ra, F b F rb represented as ordered sequences F a =(e a 1,...,e a r a ) and F b =(e b 1,...,e b r b ),fthe edges n F a are parwse ncomparable wth the edges n F b, then add an edge (v F a a,v F b b ) V a V b n the auxlary graph of weght r a w H (v F a a,v F b b ):= w (State ( e a,σ(r a + ) )) r a r a =+1 r a r b =1 w(e G[Te](T e a,t e a )) w(e G[Te](T e a,t e b )). For any trangle (v F 1 1,v F 2 2,v F 3 3 ) V 1 V 2 V 3 such that F := F 1 F 2 F 3 has ordered sequence (e 1,...,e l ), the total weght of the edges s equal to w H (v F 1 1,v F 2 2 )+w H (v F 2 2,v F 3 3 )+w H (v F 3 3,v F 1 1 ) l = w(state(e a,σ())) l l =+1 w(e G[Te](T e,t e )). A straghtforward countng argument shows that ths s exactly ( w ( l Cut {e} State(e,σ()) )) w(e(t e,v T e )). Thus, the weght of each trangle, wth w(e(t e,v T e )) added to t, corresponds to the cut value of one possble soluton to State(e, s). Moreover, f we guess l and σ : [l] [s 1] correctly, then ths 1 Of course, we could nstead brute force over all k O(k) possble choces of l and σ. trangle wll exst n auxlary graph H, and we wll compute the correct state f we compute the mnmum weght trangle n Õ(Wn l/3 ω m) tme. Snce the probablty of guessng l, σ( ) correctly s k O(k), we repeat the guessng k O(k) log n tmes to succeed w.h.p. n tme Õ(kO(k) n (k2)/3 ω mw ). Ths concludes the computaton of each State(e, s); snce there are O(kn) such states, the total runnng tme becomes Õ(k O(k) n (k2)/3 ω+1 mw ). Lastly, to compute the fnal k-cut value, we let s := k 1 and construct the same auxlary graph H, except that k 2 s replaced by k 1 and the relevant graph G[T e ] becomes the entre G. By the same countng arguments, the weght of trangle (v F 1 1,v F 2 2,v F 3 3 ) V 1 V 2 V 3 such that F := F 1 F 2 F 3 has ordered sequence (e 1,...,e l ) s exactly ( w ( l Cut {e} State(e,σ()) )). Agan, by repeatng the procedure k O(k) log n, we compute an optmal k-cut w.h.p., n tme Õ(k O(k) n (k1)/3 ω mw ). Note that ths tme s domnated by the runnng tme Õ(k O(k) n (k2)/3 ω+1 mw ) of computng the states. In order to get the runtme clamed n Theorem I.1, we need a couple more deas however, they can be skpped on the frst readng, so we defer them to the full verson of ths paper. III. A FASTER DETERMINISTIC ALGORITHM In ths secton, we show how to buld on the randomzed algorthm of the prevous secton and mprove t n two ways: we gve a determnstc algorthm, wth a better asymptotc runtme. (The algorthm of the prevous secton has a better runtme for smaller values of k.) Formally, the man theorem of ths secton s the followng: Theorem I.2 (Even Faster Determnstc Algorthm). Let W be a postve nteger. For any ε>0, there s a determnstc algorthm for exact k-cut on graphs wth edge weghts n [W ] wth runtme k O(k) n (2ω/3+ε)k+O(1) W O(k O(k) n (2ω/3)k ). Our man dea s a more drect applcaton of matrx multplcaton, wthout payng the O(n k ) overhead n the prevous secton. Instead of convertng a gven T-tree to a tght tree where matrx multplcaton can be combned wth dynamc programmng, wth only n O(log k) overhead, we partton the gven T-tree to subforests that are amenable to drect matrx multplcaton approach. As n II we buld on the framework of Thorup [3], where the k-cut problem reduces to n O(1) nstances 120

9 of the followng problem: gven the graph G and a spannng tree T, fnd a way to cut 2k 2 edges from T, and then mergng the connected components of T nto k connected components, that mnmzes the number of cut edges n G. Agan, the optmal k-cut s denoted by S = {S 1,...,S k }. For the rest of ths secton, let T be some spannng tree n the nstance that crosses the optmal k-cut n (r 1) 2k 2 edges. If we delete these r 1 edges from T, ths gves us r components, whch we denote by C1,...,Cr these are a refnement of S, and hence can be then be merged together to gve us S. Let ET := E T (C1,...,Cr )=E T (S1,...,S k ) be these r 1 cut edges n T. A. Balanced Separators We frst show the exstence of a small-sze balanced separator n the followng sense: there exst forests F 1,F 2,F 3 whose vertces partton V (T ), such that () we can delete O(log k) edges n T to get the forests,.e., E(T ) 3 E(F ) = O(log k), and () we want to cut few edges from each forest,.e., E(F ) ET 2k/3 for each. Of course, small-sze balanced edge separators typcally do not exst n general trees, such as f the tree s a star. So we frst apply a degree-reducng step. Ths operaton reduces the maxmum degree of the tree to 3, at a cost of ntroducng Stener vertces, whch are handled later. Lemma III.1 (Degree-Reducton). Gven a tree T = (V T,E T ), we can construct a tree T = (V T,E T ), where V T = V T X, where X are called the Stener vertces, such that 1. T has maxmum degree V (T ) 2 V (T ) 3. For every way to cut r edges n T and obtan components C 1,...,C r+1, there s a way to cut r edges n T and obtan components C 1,...,C r+1 such that each C s precsely C V T. Proof: Root the tree T at an arbtrary root, and select any non-stener vertex v V T wth more than two chldren. Replace the star composed of v and ts chldren wth an arbtrary bnary tree wth v as the root and ts chldren as the leaves. Ths process does not ntroduce any new vertex wth more than two chldren, so we can repeat t untl t termnates, gvng us a tree T of maxmum degree 3. Every star of z edges adds exactly z 1 Stener nodes, and there are V T 1 edges ntally, so V T 2 Stener vertces are added throughout the process, and V T 2 V T. Fnally, f we cut some r edges (u,v ) E T where v s the parent of u, then we can cut the parent edge of each u n T to obtan the requred components. Havng appled Lemma III.1 to T to get T, Property (3) shows that we can stll delete 2k 2 edges n T to obtan the components of the optmal soluton before mergng. To avod excess notaton, we assume that T tself s a tree of degree 3, possbly wth Stener nodes. From now on, our task s to delete 2k 2 edges of T and merge them nto k components, each of whch contanng at least one non-stener vertex, that mnmzes the number of cut edges n G. To show that the aforementoned forests F 1,F 2,F 3 exst n the new tree T, we ntroduce the followng easy lemma: Lemma III.2. Let T beatreeofdegree 3 and F E(T ) be a subset of the edges. For any nteger r [1, F 1], there exsts a vertex partton A, B of V (T ) such that E T (A, B) = O(log(r +1)), and the nduced subgraphs T [A] and T [B] have at most r and at most F r edges from F, respectvely. Proof: We provde an algorthm that outputs a collecton of O(log r) dsont subtrees whose unon comprses A. Root T at a degree-1 vertex, and fnd a vertex of maxmal depth whose rooted subtree contans >redges from F. The degree condton ensures that v has 2 chldren, and by maxmalty, all of v s chldren have r edges n F n ther subtrees. Moreover, the edges n T v s precsely the unon of the edge sets E(T u ) {(u, v)} for all chldren u of v. For convenence, defne E + (T u ):=E(T u ) {(u, v)} for a chld u of v. So there must be a chld u of v satsfyng E + (T u ) F (r/2,r]. If E + (T u ) F = r, then (A, B) =(V (T u ),V(T ) V (T u )) s a satsfyng partton wth E T (A, B) =1, and we are done. Otherwse, recurse on the tree T := T [V (T )V (T u )] where we remove (u, v) and the subtree below t, wth the parameters r := r E + (T u ) F and F := F \ E + (T u ) to get partton (A,B ), and set A := A V (T u ) and B := B. By recurson, we guarantee that E(T [A]) F E(T [A ]) F + E + (T u ) F (r E + (T u ) F )+ E + (T u ) F = r and E(T [B]) F = E(T [B ]) F F (r E + (T u ) F ) =( F E + (T u ) F ) (r E + (T u ) F ) = F r. Snce the value of r drops by at least half each tme, there are O(log r) steps of the recurson. Each step can 121

10 only add the addtonal edge (u, v) to E T (A, B), so E T (A, B) = O(log r). Corollary III.3. There exst forests F 1,F 2,F 3 whose vertces partton V (T ) such that () the number of crossng edges s E(T ) 3 E(F ) = O(log ET ), and () E(F ) ET E T /3 for each. Proof: We apply Lemma III.2 wth F := ET and r := ET /3 to obtan the separaton (A, B), and then set F 1 := T [A]. Before applyng the lemma agan on B, we frst connect the connected components of B arbtrarly nto a tree; let F + denote the added edges. Then, we apply wth F := ET E[F 1] and r := ET /3 to obtan separaton (A,B ), and then set F 2 := T [A ] F + and F 3 := T [B ] F +. Gven ths result, our algorthm starts by tryng all possble n O(log k) ways to delete O(log k) edges of T and partton the connected components nto three forests. By Corollary III.3, one of these attempts produces the desred F 1,F 2,F 3 satsfyng the two propertes. B. Matrx Multplcaton The balanced parttonng procedure from the prevous secton gves us three forests F 1,F 2,F 3, such that the optmal soluton cuts at most 2k/3 edges n each and then combnes the resultng peces together. The algorthm now computes these solutons separately for each forest, and then uses matrx multplcaton to combne these solutons together. Indeed, for each F {F 1,F 2,F 3 }, the algorthm computes all O(n 2k/3 ) ways to cut 2k/3 edges n F, followed by all 3 O(k) ways to label each of the 2k/3 + O(log k) connected components wth a label n [k]. For each one forest, note that some of these components mght only contan Stener vertces of the tree; we call these the Stener components, and the other the normal components. For each subset S [k], let F S denote all possble ways to cut and label F n the aforementoned manner such that the set of labels that are attrbuted to at least one normal component s precsely S. The algorthm now enumerates over every possble trple of subsets S 1,S 2,S 3 [k] (not necessarly dsont) whose unon s exactly [k]. Note that there are at most 7 k of these trples. For each trple S 1,S 2,S 3, we construct the followng trpartte auxlary graph H =(V H,E H ) on O(k O(k) n 2k/3 ) vertces, wth trpartton V H = V 1 V 2 V 3. For each =1, 2, 3, each element n F S s a tuple (X,σ ) where X F s a set of edges that we cut from F, and σ s a labelng of the normal components n the resultng forest so that the label set s exactly S. Now for each (X, σ) F S, add a node v (X,σ) to V. Moreover, for each trpartton par (V a,v b ) wth (a, b) (1, 2), (2, 3), (3, 1), and for each way (X a,σ a ) Fa Sa to cut F a nto components C1 a,...,cr a a wth labels σ a (1),...,σ a (r a ), and for each way (X b,σ b ) F S b b to cut F b nto components C1,...,C b r b b wth labels σ b (1),...,σ b (r b ), we add an edge (v a (Xa,σa),v (X b,σ b ) b ) V a V b of weght w H (v a (Xa,σa),v (X b,σ b ) := r a r a =+1 r a r b + =1 b ) 1[σ a () σ a ()] w(e G [C a,c a ]) 1[σ a () σ b ()] w(e G [C a,c b ]), (III.5) where 1 s the ndcator functon, takng value 1 f the correspondng statement s true and 0 otherwse. Fnally, the algorthm computes the mnmum weght trangle n H. A straghtforward countng argument shows that the weght of each trangle (v (X1,σ) 1,v (X2,σ2) 2,v (X3,σ3) 3 ) n H s exactly the value of the cut n G obtaned by mergng all components n F 1,F 2,F 3 wth the same label together. In partcular, for the correct trple S 1,S 2,S 3 for ET, there s a trangle n H whose weght s the cost of the optmal soluton, and the algorthm wll fnd t, provng the correctness of the algorthm. As for runnng tme, the algorthm has an n O(log k) 7 k overhead for the guesswork of fndng the forests (F 1,F 2,F 3 ) and the correct trple (S 1,S 2,S 3 ) of subsets of labels. Ths s followed by computng matrx multplcaton on a graph wth k O(k) n 2k/3 nodes, wth edge weghts n [Wm,Wm]. Altogether, ths takes k O(k) n (2ω/3+ε)k+O(1) W for any ε>0, provng Theorem I.2. REFERENCES [1] O. Goldschmdt and D. S. Hochbaum, A polynomal algorthm for the k-cut problem for fxed k, Math. Oper. Res., vol. 19, no. 1, pp , [Onlne]. Avalable: [2] D. R. Karger and C. Sten, A new approach to the mnmum cut problem, Journal of the ACM (JACM), vol. 43, no. 4, pp , [3] M. Thorup, Mnmum k-way cuts va determnstc greedy tree packng, n Proceedngs of the forteth annual ACM symposum on Theory of computng. ACM, 2008, pp

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