Approximation Algorithms and Hardness of the k-route Cut Problem

Transcription

1 Appoximation Algoithms and Hadness of the k-route Cut Poblem Julia Chuzhoy Yuy Makaychev Aavindan Vijayaaghavan Yuan Zhou Novembe 26, 2011 Abstact We study the k-oute cut poblem: given an undiected edge-weighted gaph G = V, E), a collection {s 1, t 1 ), s 2, t 2 ),..., s, t )} of souce-sink pais, and an intege connectivity equiement k, the goal is to find a minimum-weight subset E of edges to emove, such that the connectivity of evey pai s i, t i ) falls below k. Specifically, in the edge-connectivity vesion, EC-kRC, the equiement is that thee ae at most k 1) edgedisjoint paths connecting s i to t i in G \ E, while in the vetex-connectivity vesion, VC-kRC, the same equiement is fo vetex-disjoint paths. Pio to ou wok, poly-logaithmic appoximation algoithms have been known fo the special case whee k 3, but no nontivial appoximation algoithms wee known fo any value k > 3, except in the single-souce setting. We show an Ok log 3/2 )-appoximation algoithm fo ECkRC with unifom edge weights, and seveal polylogaithmic bi-citeia appoximation algoithms fo EC-kRC and VC-kRC, whee the connectivity equiement k is violated by a constant facto. We complement these uppe bounds by poving that VC-kRC is had to appoximate to within a facto of k ɛ fo some fixed ɛ > 0. We then tun to study a simple vesion of VC-kRC, whee only one souce-sink pai is pesent. We give a simple bi-citeia appoximation algoithm fo this case, and show evidence that even this esticted vesion of the poblem may be had to appoximate. Fo example, we pove that the single souce-sink pai vesion of VCkRC has no constant-facto appoximation, assuming Feige s Random κ-and assumption. Toyota Technological Institute, Chicago, IL cjulia@ttic.edu. Suppoted in pat by NSF CAREER awad CCF and Sloan Reseach Fellowship. Toyota Technological Institute, Chicago, IL yuy@ttic.edu. Depatment of Compute Science, Pinceton Univesity. aavindv@cs.pinceton.edu. Wok done while visiting Toyota Technological Institute, Chicago Compute Science Depatment, Canegie Mellon Univesity, Pittsbugh, PA. yuanzhou@cs.cmu.edu. Wok done while visiting Toyota Technological Institute, Chicago 1 Intoduction Multi-commodity flows and cuts ae among the most extensively studied gaph optimization poblems. Due to thei ich connections to many combinatoial optimization poblems, algoithms fo vaious vesions of flow and cut poblems ae a poweful and a widely used toolkit. One of the cental poblems in this aea is minimum multicut: given an n-vetex gaph G = V, E) with non-negative weights w e on edges e E and a collection {s 1, t 1 ), s 2, t 2 ),..., s, t )} of souce-sink pais, find a minimum-weight subset E of edges to delete, so that each pai s i, t i ) is disconnected in the esulting gaph G \ E. The dual to minimum multicut is the maximum multi-commodity flow poblem, whee the goal is to find a maximum flow between the pais s i, t i ), with the estiction that each edge e caies at most w e flow units. It is easy to see that minimum multicut can be viewed as evealing a bottleneck in the outing capacity of G, as the value of any multi-commodity flow cannot exceed the value of the minimum multicut in G. A fundamental esult, due to Leighton and Rao [LR99] and Gag, Vaziani and Yannakakis [GVY95] shows that the value of minimum multicut is within an Olog ) facto of that of maximum multicommodity flow in any gaph, whee is the numbe of the souce-sink pais. This esult can be seen as an extension of the famous min-cut maxflow theoem to the multicommodity setting, and it also gives an efficient Olog )-appoximation algoithm fo minimum multicut the best cuently known appoximation guaantee fo it. In this pape we study a natual genealization of minimum multicut - the minimum k-oute cut poblem. In this poblem, the input again consists of an n-vetex gaph G = V, E) with nonnegative weights w e on edges e E, and a collection {s 1, t 1 ), s 2, t 2 ),..., s, t )} of souce-sink pais. Additionally, we ae given an integal connectivity theshold k > 0. The goal is to find a minimum-weight subset E E of edges to delete, such that the connectivity of each pai s i, t i ) falls below k in the esulting gaph G \ E. We study two vesions of this poblem: in the edge-connectivity vesion EC-kRC), the equiement is

2 that fo each 1 i, the numbe of edge-disjoint paths connecting s i to t i in gaph G \ E is less than k. In the vetex-connectivity vesion VC-kRC), the equiement is that the numbe of vetex-disjoint paths connecting s i to t i is less than k. It is not had to see that VC-kRC captues EC-kRC as a special case see the full vesion of the pape fo details ), and hence is moe geneal. It is also easy to see that minimum multicut is a special case of both EC-kRC and VC-kRC, with the connectivity equiement k = 1. We also conside a special case of EC-kRC, whee all edges have unit weight, and we efe to it as the unifom EC-kRC. We note that fo VC-kRC, the unifom and the non-unifom edge-weight vesions ae equivalent up to a small loss in the appoximation facto, and so we do not distinguish between them. The pimay motivation fo studying k-oute cuts comes fom multi-commodity flows in fault toleant settings, whee the esilience to edge and node failues is impotant. An elementay k-oute flow between a pai s and t of vetices is a set of k disjoint paths connecting s to t. A k-oute st)-flow is just a combination of such elementay k-oute flows, whee each elementay flow is assigned some factional value. This is a natual genealization of the standad st) flows, which ensues that the flow is esilient to the failue of up to k 1) edges o vetices. Multi-oute flows wee fist intoduced by Kishimoto [Kis96], and have since been studied in the context of communication netwoks [BCSK07, BCK03, ACKN07]. In a seies of papes, Kishimoto [Kis96], Kishimoto and Takeuchi [KT93] and Aggawal and Olin [AO02] have developed a numbe of efficient algoithms fo computing maximum multi-oute flows. As in the case of standad flows, we can extend k- oute st)-flows to the multi-commodity setting, whee the goal is to maximize the total k-oute flow among all souce-destination pais. It is easy to see that the minimum k-oute cut is a natual uppe bound on the maximum k-oute flow just like minimum multicut uppe-bounds the value of the maximum multicommodity flow. Hence, as in the case with the standad multicut, multi-oute cuts can be seen as evealing the netwok bottleneck, and so the minimum k-oute cut in a gaph captues the obustness of eal-life compute and tanspotation netwoks. The fist appoximation algoithm fo the EC-kRC poblem, due to Chekui and Khanna [CK08], achieved a facto Olog 2 n log )-appoximation fo the special case whee k = 2, by ounding a Linea Pogamming elaxation. This was impoved by Baman and Chawla [BC10] to give an Olog 2 )-appoximation algoithm fo the same vesion, by genealizing the egiongowing LP-ounding scheme of [LR99, GVY95]. They note that it seems unlikely that thei algoithm can be extended to handle highe values of k using simila techniques. Vey ecently, Kolman and Scheidele [KS11] obtained a Olog 3 ) appoximation to EC-3RC k = 3 case) fom the linea pogam of [BC10] by using a multi-level ball gowing ounding. To the best of ou knowledge, no appoximation algoithms with subpolynomial in n) guaantees ae known fo any vaiant of the poblem, fo any value k > 3, except in the single-souce setting that we discuss late. Ou fist esult is an Ok log 1.5 )-appoximation algoithm fo the unifom vesion of EC-kRC. Since the poblem appeas to be computationally difficult, it is natual to tun to bi-citeia appoximation, by slightly elaxing the connectivity equiement. Given paametes α, β > 1, we say that an algoithm is an α, β)-bi-citeia appoximation fo EC-kRC o VC-kRC), if it is guaanteed to poduce a valid k - oute cut of weight at most β OPT, whee k αk, and OPT is the value of the optimal k-oute cut. Indeed, we can do much bette in the bi-citeia setting: we obtain a 1 + δ, O 1 δ log1.5 ))-bi-citeia appoximation fo any constant 0 < δ < 1, fo the unifom EC-kRC poblem notice that the factos do not depend on k). When edge weights ae abitay, we obtain a 2, Õlog2.5 ) ) -bi-citeia appoximation in n Ok) time, and an Olog ), Olog 3 ) ) -bi-citeia appoximation in time polynomial in n and k. We also show an Olog 1.5 )-appoximation fo the special case whee k = 2, thus slightly impoving the esult of [BC10]. The peviously known uppe bounds and ou esults fo ECkRC ae summaized in Table 1. Pevious esults Cuent pape k = 2 Olog 2 ) [BC10] Olog 1.5 ) k = 3 abitay k, unifom abitay k, geneal Olog 3 ) [KS11] - Ok log 1.5 ), 1 + δ, O 1 δ log1.5 )) - fo any constant 0 < δ < 1 2, Olog 2.5 log log ) ) in time n Ok) ; Olog ), Olog 3 ) ) in polyn)-time Table 1: Uppe bounds fo EC-kRC. Running time is polynomial in n and k unless stated othewise. We note that on the inappoximability side, it is easy to show that fo any value of k, EC-kRC is at least as had as minimum multicut, up to small

3 constant factos 1. While multicut is known to be had to appoximate up to any constant facto assuming the Unique Games Conjectue [KV05, CKK + 06], it is only known to be NP-had to appoximate to within a small constant facto [DJP + 94]. In fact one of the motivations fo studying k-oute cuts is that inappoximability esults may yield insights into appoximation hadness of multicut. We now tun to the moe geneal VC-kRC poblem. The Olog 2 n log )-appoximation of [CK08], and the Olog 2 )-appoximation of [BC10] fo 2-oute cuts extend to the vetex-connectivity vesion as well, as does ou Olog 1.5 )-appoximation algoithm. Pio to ou wok, no non-tivial appoximation algoithms wee known fo any highe values of k. In this pape, we show a 2, Õkd log2.5 ) ) -bi-citeia appoximation algoithm fo VC-kRC, with unning time n Ok), whee d is the maximum numbe of demand pais in which any teminal paticipates. We note that, as in the case of EC-kRC, fo any value of k, VC-kRC is at least as had to appoximate as minimum multicut up to small constant factos), and to the best of ou knowledge, no othe inappoximability esults have been known fo this poblem. We show that VC-kRC is had to appoximate up to any facto bette than Ωk ɛ ), fo some constant ɛ > 0. Ou esults fo VC-kRC ae summaized in Table 2. Pevious esults Cuent pape k = 2 Olog 2 ) [BC10] Olog 1.5 ) abitay k APXhad [DJP + 94] no constant facto appoximation unde UGC [KV05, CKK + 06] Table 2: Results fo VC-kRC. 2, Odk log 2.5 log log ) ) - appoximation algoithm, unning time n Ok), whee d is the maximum numbe of demand pais in which any teminal paticipates Ωk ɛ )-hadness fo some constant ɛ > 0 In ode to bette undestand the multi-oute cut poblem computationally, it is instuctive to conside a simple special case, whee we ae only given a single souce-sink pai s, t). We efe to this special case of VC-kRC and EC-kRC as st)-vc-krc and st)-ec-krc, espectively. As in the geneal case, it is easy to see that st)-ec-krc can be cast as a special case of st)-vc-krc. When the connectivity equiement k is constant, both 1 A simple eduction eplaces evey vetex v of the multicut instance by a set S v of M vetices, whee M k, and evey edge u, v) by a set of M 2 edges connecting evey vetex of S v to evey vetex of S u. poblems can be solved efficiently as follows: guess a set E of k 1) edges, and compute the minimum edge st) cut in gaph G \ E. The algoithm fo st)-vckrc is simila except that we guess a set V of k 1) vetices, and compute the minimum edge st) cut in gaph G \ V. Howeve, fo lage values of k, only a 2k 1)-appoximation is known fo st)-ec-krc, fo the special case whee the edge weights ae unifom, due to Buhn et al [BČH+ 08] 2. Baman and Chawla [BC10] show that a genealization of st)-ec-krc whee edges ae allowed to have capacities is NP-had. As no good appoximation guaantees ae known fo the poblem, it is natual to tun to bi-citeia appoximation. Fo geneal values of k, Baman and Chawla [BC10] have shown a 4, 4)-bi-citeia appoximation algoithm fo st)-ec-krc, and a 2, 2)-bi-citeia appoximation fo unifom st)-ec-krc. In fact all these algoithms extend to a single-souce multiple-sink scenaio, except that the facto 4, 4)-appoximation equies that the numbe of teminals is constant. In this pape we focus on the moe geneal node-connectivity vesion of the poblem. We stat by showing a simple facto k + 1)- appoximation algoithm fo st)-vc-krc, and a facto c, 1 + c) -bi-citeia appoximation fo any constant c. We then complement these uppe bounds by poviding evidence that the poblem is had to appoximate. Specifically, we show that fo any constant C, thee is no 1 + γ, C)-bi-citeia appoximation fo st)-vc-krc, assuming Feige s Random κ-and Hypothesis, whee γ is some small constant depending on C. We also show that a facto ρ appoximation algoithm fo st)-vckrc would lead to a facto 2ρ 2 -appoximation fo the Densest κ-subgaph poblem. These inappoximability esults ae inspied by the ecent wok of Aoa et al. [AAM + 11], who have uled out a constant facto appoximation fo Densest κ-subgaph assuming Feige s Random κ-and hypothesis. Recall that the Densest κ-subgaph poblem takes as input a gaph GV, E) on n vetices and a paamete κ, and asks fo a subgaph of G on at most κ vetices containing the maximum numbe of edges. While it is a fundamental gaph optimization poblem, thee is a huge gap between the best known appoximation algoithm and the known inappoximability esults. The cuent best appoximation algoithm due to [BCC + 10] gives an On 1/4+ɛ )-facto appoximation algoithm which uns in time n O1/ɛ) fo any constant ɛ > 0. On the inappoximability side, Feige [Fei02] initially showed a small constant facto inappoximability using the andom 3-SAT assumption, and late 2 This esult also extends to the single-souce multiple-sinks setting.

4 Khot [Kho04] used quasi-andom PCPs to ule out a PTAS, assuming NP ɛ>0 BPTIME2nɛ ). Raghavenda and Steue [RS10] and Alon et al. [AAM + 11] uled out constant facto appoximation algoithms fo Densest κ-subgaph unde othe less standad complexity assumptions. The Densest κ-subgaph poblem can also be genealized to λ-unifom hypegaphs, whee the goal is again to find a subset of κ vetices containing maximum possible numbe of hypeedges. We show that fo any constant λ 2, a facto ρ appoximation algoithm fo st)-vc-krc would lead to a facto 2ρ λ )- appoximation fo the λ-unifom Densest κ-subgaph. We note that Applebaum [App11] has ecently shown that fo λ 3, the λ-unifom Densest κ-subgaph poblem is had to appoximate to within n ɛ -facto fo some constant ɛ > 0 assuming the existence of a cetain family of one-way functions. All ou inappoximability esults fo st)-vc-krc ae poved using a poxy poblem, Small Set Vetex Expansion SSVE). In this poblem, we ae given a bipatite gaph G = U, V, E) and a paamete 0 α 1. The goal is to find a subset S U of α U vetices, while minimizing the numbe of its neighbos ΓS). We show an appoximation-peseving eduction fom SSVE to st)-vc-krc, and then pove inappoximability esults fo the SSVE poblem. In paticula, we show that appoximating SSVE is almost as had as appoximating Densest κ-subgaph poblem that is, if thee is a ρ appoximation algoithm fo SSVE then thee is a 2ρ 2 )-appoximation algoithm fo the Densest κ- subgaph poblem). This esult suggests that although the SSVE poblem looks simila to the Small Set Expansion SSE) poblem [RS10], it might be much hade than SSE. On the othe hand, the SSVE poblem is of independent inteest besides its application to the st)-vc-krc poblem, Applebaum et al. [ABW10] used a planted vesion of SSVE as a hadness assumption to constuct a public key encyption scheme. Othe Related Wok Anothe vesion of the ECkRC poblem that has eceived a significant amount of attention ecently is the single-souce setting. In this setting we ae given a single souce s and a set T of teminals. The souce-sink pais ae then set to be {s, t)} t T. Buhn et al. [BČH+ 08] have shown a facto 2k 1)-appoximation fo the unifom vesion of this poblem, and Baman and Chawla [BC10] have shown a facto 6, O ln ))-bi-citeia appoximation fo the geneal vesion, a facto 4, 4)-bi-citeia appoximation fo the geneal vesion whee is a constant, and a facto 2, 4)-appoximation fo the unifom vesion and abitay. The st)-ec-krc and st)-vc-krc poblems captue two natual budgetted cut minimization poblems. The fist is the Minimum Unbalanced cut poblem [HKPS05], in which we ae given a gaph G with a souce vetex s and a budget B. The goal is to find a cut S, S) in G with s S and ES, S) B, while minimizing S. Hayapetyan et al. [HKPS05] obtain a 1 + 1/λ, λ)-bi-citeia appoximation algoithm fo any λ > 1, by ounding a Lagangean elaxation fo the poblem. Given an instance G = V, E) of the Minimum Unbalanced cut poblem, we can tansfom it into an instance of st)-ec-krc, by setting the weights of all edges in E to, adding a sink t, that connects to evey vetex in V with a unit-weight edge, and setting k = B. The othe poblem is the Minimum k-size st)-cut poblem, whee we ae given a gaph G = V, E) with a special souce vetex s and a paamete k, and the goal is to find a cut S, S) in G with s S and S k, minimizing the size of the cut ES, S). Li and Zhang [LZ10] give an Olog n)-appoximation to this poblem using Räcke s gaph decomposition [Räc08]. This poblem can be educed to st)-ec-krc by assigning unit weights to the edges of E, and adding a sink t with infinity-weight edges v, t) fo each v V ; the paamete k emains unchanged. Ou esults and techniques The following two theoems summaize ou esults fo the EC-kRC poblem. Theoem 1.1. Thee is an efficient Ok log 1.5 )- appoximation algoithm, and a 1 + δ, O 1 δ log1.5 )) - bi-citeia appoximation algoithm fo any constant δ 0, 1), fo the unifom EC-kRC poblem. Theoem 1.2. Thee is a 2, Olog 2.5 log log ) ) -biciteia appoximation algoithm with unning time n Ok) and an Olog ), Olog 3 ) ) -bi-citeia appoximation algoithm with unning time polyn) fo the ECkRC poblem. We now poceed to discuss ou techniques. Ou algoithms ae based on a simple iteative appoach: find a spase cut that sepaates some demand pais, emove all cut edges except fo the k 1) most expensive ones fom the gaph, also emove all demand pais that ae no longe k-connected, and then ecusively solve the obtained instance. The main challenge in this appoach is to ensue that the cost of the emoved edges is bounded by the cost of the optimal solution. In fact, in the fist step of the algoithm, we use a modified notion of spasity we use the k-oute spasity of a cut, which is the cost of all but k 1) most expensive edges of the cut divided by the numbe of sepaated teminals see below fo fomal definitions). This is necessay since

5 the standad spasest cut can be pohibitively expensive; its cost cannot be bounded in tems of the cost of the optimal solution. We pove howeve that the cost of the k-oute spasest cut can be bounded in tems of the cost of the optimal solution and thus obtain guaantees on the pefomance of ou algoithms. This is the most technically challenging pat of the analysis of ou algoithms. We extend ou bi-citeia appoximation fo EC-kRC to the moe geneal VC-kRC poblem in the following theoem. Theoem 1.3. Thee is a 2, Odk log 2.5 log log ) ) - bi-citeia appoximation algoithm fo VC-kRC with unning time n Ok), whee d is the maximum numbe of demand pais in which any teminal paticipates. We also pove the following hadness of appoximation esult fo VC-kRC, whose poof uses ideas simila to those used by Kotsaz et al. [KKL04] and Chakaboty et al. [CCK08] to pove hadness of vetex-connectivity netwok design: Theoem 1.4. Thee ae constants 0 < ɛ < 1, k 0 > 1, such that fo any constant η, fo any k = O 2 log n)1 η), whee k > k 0, thee is no k ɛ - appoximation algoithm fo VC-kRC, unde the assumption that P NP fo constant k, and unde the assumption that NP DTIMEn poly log n ) fo supeconstant k. Finally, fo the special case of k = 2, we obtain a slightly impoved appoximation algoithm: Theoem 1.5. Thee is an efficient facto Olog 1.5 )- appoximation algoithm fo both VC-kRC and EC-kRC, when k = 2. We now tun to the single st) pai vesion of the poblems. We stat with a simple appoximation algoithm, summaized in the next theoem. Theoem 1.6. Thee is an efficient facto k + 1)- appoximation algoithm, and fo evey constant c > 0, thee is an efficient c, 1 + c) -bi-citeia appoximation algoithm fo both st)-vc-krc and st)-ec-krc. We then poceed to show inappoximability esults fo the single st) pai vesion of the poblem. Ou fist inappoximability esult uses Feige s andom κ- AND assumption [Fei02]. Given paametes n and, a andom κ-and instance is defined to be a κ-and fomula on n vaiables and m = n clauses, whee each clause chooses κ liteals unifomly at andom fom the set of 2n available liteals. We say that a fomula Φ is α-satisfiable iff thee is an assignment to the vaiables that satisfies an α-faction of the clauses. Notice that a andom assignment satisfies a 1/2 κ -faction of the clauses in expectation, and we expect that this is a typical numbe of simultaneously satisfiable clauses fo a andom κ-and fomula. We next state the Random κ-and conjectue of Feige [Fei02] and ou inappoximability esult fo st)-vc-krc. Hypothesis 1.1. Random κ-and assumption: Hypothesis 3 in [Fei02]). Fo some constant c 0 > 0, fo evey κ, thee is a value of 0, such that fo evey > 0, thee is no polynomial time algoithm that fo andom κ-and fomulas with n vaiables and m = n clauses, outputs typical with pobability 1/2, but neve outputs typical on instances with m/2 c0 κ simultaneously satisfiable clauses. Theoem 1.7. Fo evey constant C > 0, thee exists a small constant 0 < γ < 1 which depends only on C, such that assuming Hypothesis 1.1, thee is no polynomial time algoithm which obtains a 1 + γ, C)-bi-citeia appoximation fo the st) VC-kRC poblem. We also pove a slightly diffeent inappoximability esult based on the slightly weake Random 3-SAT assumption of Feige. Given paametes n and, a andom 3-SAT fomula on n vaiables and m = n clauses is constucted as follows: each clause chooses 3 liteals unifomly at andom among all available liteals. Notice that a andom assignment satisfies a 7/8-faction of clauses in expectation. Below is Feige s 3-SAT assumption and ou inappoximability esult fo st)- VC-kRC. Hypothesis 1.2. Random 3-SAT assumption: Hypothesis 2 fom [Fei02]). Fo evey fixed ɛ > 0, fo a sufficiently lage constant independent of n, thee is no polynomial time algoithm that on a andom 3CNF fomula with n vaiables and m = n clauses, outputs typical with pobability at least 1/2, but neve outputs typical when the fomula is 1 ɛ)-satisfiable i.e. thee is an assignment satisfying simultaneously 1 ɛ)m clauses). Theoem 1.8. Assuming Hypothesis 1.2, fo any constant ɛ > 0, no polynomial-time algoithm achieves a ɛ, 1.1 ɛ) -bi-citeia appoximation fo st) VCkRC.

6 Finally, we show that an existence of a good appoximation algoithm fo st)-vc-krc would imply a good appoximation fo the λ-unifom Hypegaph Densest κ-subgaph poblem. Recall that in the λ-unifom Hypegaph Densest κ-subgaph poblem, we ae given a gaph GV, E) whee E is the set of λ-unifom hypeedges, and a paamete κ. The goal is to find a subset S V G) of κ vetices, maximizing the numbe of hype-edges e S. Notice that fo λ = 2, this is the standad Densest κ-subgaph poblem. Theoem 1.9. Fo any constant λ 2, and fo any appoximation facto ρ that may depend on n), if thee is an efficient facto ρ appoximation algoithm fo the st) VC-kRC poblem, then thee is an efficient facto 2ρ λ )-appoximation algoithm fo the λ-unifom Hypegaph Densest κ-subgaph poblem. We note that Theoem 1.9, combined with the ecent esult of [AAM + 11] immediately implies supeconstant inappoximability fo st)-vc-krc, unde Hypothesis 1.1. Howeve, ou poof of Theoem 1.7 is conceptually simple, and also leads to a bi-citeia inappoximability. Oganization We pesent notation and definitions and pove some esults that we use thoughout the pape in Section 2. We study the unifom case of ECkRC in Section 3, and the non-unifom case in Section 4. We descibe ou esults fo VC-kRC in Section 5. Then we pesent an algoithm fo 2-oute cuts in Section 6. We pove n ɛ hadness of VC-kRC in Section 7. Finally, we study the single souce-sink case in Section 8, whee we pesent an appoximation algoithm and pove seveal hadness esults fo the poblem. Due to lack of space, some details ae defeed to the full vesion of the pape, available fom the authos web pages. 2 Peliminaies In all ou poblems, the input is an undiected n-vetex gaph G = V, E) with non-negative weights we) on edges e E and a paamete k. Additionally, we ae given a set D = {s 1, t 1 ),..., s, t )} of souce-sink pais, that we also efe to as demand pais. We let T V be the subset of vetices that paticipate in any demand pais, and we efe to the vetices in T as teminals. Fo evey vetex v V, let D v be the numbe of demand pais in which v paticipates. Given a subset S V of vetices, let DS) = v S D v be the total numbe of teminals in S, counting multiplicities. We also denote by DS, S) the numbe of demand pais s i, t i ) with s i S, t i S, o s i S and t i S. Given any subset E E of edges, we denote by we ) = e E we) its weight. Thoughout the pape, we denote by E the optimal solution to the given ECkRC o VC-kRC poblem instance, and by OPT its value. One of the main ideas in ou algoithms is to elate the value of the appopiately defined spasest cut in gaph G to the value of the optimal solution to the k-oute cut poblem. We now define the diffeent vaiations of the spasest cut poblem that we use. The Spasest Cut Poblem. Given any cut S, S) in gaph G, its unifom spasity is defined to be ΦS) = wes, S)) min { }. DS), D S) The unifom spasity ΦG) of the gaph G is the minimum spasity of any cut in G, ΦG) = min {ΦS)}. S V : DS),D S)>0 We use the O log )-appoximation algoithm fo the unifom spasest cut poblem due to Aoa, Rao and Vaziani [ARV04]. Let A ARV denote this algoithm, and let α ARV ) = O log ) denote its appoximation facto. Given an edge-weighted gaph G and a set D of demand pais, algoithm A ARV finds a subset S V of vetices with ΦS) α ARV ) ΦG). Given any cut S, S) in gaph G, its non-unifom spasity is defined to be ΦS) = ΦG) = wes, S)) DS, S). The non-unifom spasity ΦG) of the gaph G is: } { ΦS). min S V : DS, S)>0 We also use the O log log log )-appoximation algoithm fo the non-unifom spasest cut poblem of Aoa, Lee and Nao [ALN05]. Let A ALN denote this algoithm, and let α ALN ) = O log log log ) denote its appoximation facto. Given an edge-weighted gaph G with a set D of demand pais, algoithm A ALN finds a subset S V of vetices with ΦS) α ALN ) ΦG). We next genealize the notion of the spasest cut to the multi-oute setting. Given a subset S V of vetices, let F denote the set of k 1) most expensive edges of ES, S), beaking ties abitaily, and we efe to F as the set of fee edges fo cut S, S). We then define w k) S, S) = e ES, S)\F w e. be: The unifom k-oute spasity of set S is defined to Φ k) S) = w k) S, S) min { }, DS), D S)

7 and the unifom k-oute spasity of the gaph G is: { } Φ k) G) = Φ k) S). min S V : DS),D S)>0 Similaly, the non-unifom k-oute spasity of S is: Φ k) S) = wk) S, S) DS, S), and the non-unifom k-oute spasity of the gaph G is: Φ k) { Φk) G) = S)}. min S V : DS, S)>0 Note that Φ 1) G) = ΦG) and Φ 1) G) = ΦG) ae the standad unifom and non-unifom spasest cut values, espectively. We now show that thee is an efficient algoithm to find an appoximate k-oute spasest cut when k is a constant. Theoem 2.1. Thee is an algoithm that, given a gaph G = V, E) with souce-sink pais and an intege k, computes in time n Ok) a cut S V, with Φ k) S) α ARV ) Φ k) G). Similaly, thee is an algoithm that computes in time n Ok) a cut S, with Φ k) S) α ALN ) Φ k) G). Poof. We stat with the unifom k-oute spasest cut. We go ove all subsets F E of k 1 edges. Fo each such subset F, we compute the α ARV )-appoximate spasest cut in the gaph G \ F using the algoithm A ARV, and output the best cut ove all such subsets F. The algoithm fo the non-unifom spasest k-oute cut is simila, except that we use the algoithm A ALN fo the non-unifom spasest cut. The above theoem woks well fo constant values of k. Howeve, when k is supe-constant, the unning time of the algoithm is no longe polynomial. Fo such cases, we use a bi-citeia appoximation algoithm fo the k-oute spasest cut poblem, summaized in the next theoem. Theoem 2.2. Thee is an efficient algoithm that, given an edge-weighted gaph G = V, E), an intege k > 1, and a set D = {s i, i )} i=1 of demand pais, finds a cut S V with Φ k ) S) = Olog ) Φ k) G), whee k = Ck log fo some absolute constant C. Poof. We use as a sub-outine the appoximation algoithm of Englet et al. [EGK + 10] fo the l-multicut poblem. In the l-multicut poblem, we ae given a gaph G = V, E) with weights on edges, a set D of demand pais, and an intege l. The goal is to find a minimum-weight subset E E of edges, such that at least l of the demand pais ae disconnected in the gaph G \ E. Enget et al. [EGK + 10] give an efficient Olog )-appoximation algoithm fo this poblem. We denote thei algoithm by A EGK+, and the appoximation facto it achieves by α EGK+ = Olog ). Let S, S ) be the optimal non-unifom k-oute cut in G, and let F E G S, S ) be the subset of the k 1) most expensive edges in this cut. Then wes, S ) \ F ) = Φ k) G) DS, S ). Let W = wes, S ) \ F ) and let = DS, S ). Assume fist that ou algoithm is given the values of W and. We define new edge weights as follows: fo each edge e E, w e = min {w e, W /k 1)}. We use the algoithm A EGK+ on the esulting instance of the l-multicut poblem, with l =. Let S, S) be the output of this algoithm, and let F be the set of 2α EGK+ )k 1) most expensive edges of ES, S), with espect to the oiginal weights w e, beaking ties abitaily. The output of ou algoithm is the cut S, S). In ode to complete the poof, it is enough to show that wes, S) \ F ) Olog ) Φ k) G) DS, S). Note that the value of the optimal solution to the l-multicut poblem instance is at most wes, S )) wes, S ) \ F ) + F W k 1 wes, S ) \ F ) + W = 2W. Theefoe, wes, S)) 2α EGK+ )W. In paticula, ES, S) may contain at most 2α EGK+ )k 1) edges e with w e = W /k 1), and so all such edges lie in F. Fo edges e / F, w e < W /k 1) must hold, and theefoe, w e = w e. We conclude that wes, S) \ F ) = wes, S) \ F ) wes, S)) as equied. 2α EGK+ )W = 2α EGK+ ) Φ k) G) Olog ) Φ k) G)DS, S) Of couse, ou algoithm does not know the values of W and. Instead, we pefom the pocedue descibed above fo all possible values of {1,..., } and say) all values of W in {τw e : e E, 1 τ E }, and then output the best of the cuts found. One of the values of will be equal to, and one the values of W will be within a facto of 2 of W : if e is the most expensive edge in ES, S ) \ F, and τ = W /w e, then W τw e 2 W /w e )w e 2W. Fo these

8 values of and W, the algoithm will find a cut that satisfies the conditions of the lemma. Lamina Families of Minimum Cuts Ou main tool in establishing the connection between the values of the k-oute spasest cut and the cost of the optimal solution to the k-oute cut poblem is the following theoem, which shows that thee is a lamina family of minimum cuts disconnecting the souce-sink pais in the gaph G. Lemma 2.1. Thee is an efficient algoithm, that, given any edge-weighted gaph G = V, E) with a set D = {s i, t i )} i=1 of souce-sink pais, finds a lamina family S = {S 1,, S } of vetex subsets, such that fo all 1 i : S i, V \ S i ) is a minimum cut sepaating s i fom t i in G, and DS i ) so S i contains at most half the teminals, counting multiplicities). Poof. We use a Gomoy Hu tee T GH fo the gaph G. Recall that it is a weighted tee, whose vetex set is V. Let c e denote the costs of the edges e ET GH ). Tee T GH has the following key popety: fo evey pai u, v) V of vetices, the value of the minimum cut sepaating u fom v in gaph G equals the value of the minimum cut sepaating u fom v in T GH. Note that the latte cut contains only one edge the minimumcost edge on the unique path connecting u to v in the tee. We stat with a Gomoy Hu tee T GH fo the gaph G. Fo each 1 i, let L i, R i ) be a minimum cut sepaating s i fom t i in T GH. If DL i ) < DR i ), then we set S i = L i. If DR i ) < DL i ), we set S i = R i. Othewise, if DR i ) = DL i ), we let S i to be the side containing the vetex s 1. We use this tie-beaking ule that enfoces consistency acoss diffeent souce-sink pais late. This finishes the definition of the family S = {S 1,... S } of vetex subsets. It is immediate to see that fo each 1 i, S i, V \ S i ) is a minimum cut sepaating s i fom t i in G, and that DS i ). It now only emains to show that S 1,..., S fom a lamina family. Assume fo contadiction that fo some i j, S i S j, but S i \S j, and S j \S i. Let e i be the unique edge of T GH lying in the cut S i, V \ S i ) in tee T GH, and let e j be the unique edge of T GH lying in the cut S j, V \ S j ). Obseve that T GH \ {e i, e j } consists of thee non-empty connected components. Let C 1 denote the component that is incident on both e i and e j, C 2 the component incident on e j only, and C 3 the emaining component. We claim that S i = C 1 C 2. Othewise, since edge e i sepaates S i fom V \ S i in T GH, S i = C 3 must hold. But then eithe S j = C 2 and so S i S j =, o S j = C 1 C 3 and then S i S j, a contadiction. Theefoe, S i = C 1 C 2 and V \ S i = C 3. Similaly, S j = C 1 C 3 and V \ S j = C 2. Fom the definition of S i, eithe DS i ) < DV \S i ), o DS i ) = DV \ S i ) and s 1 S i. Assume fist that DS i ) < DV \ S i ). Then V \ S j = C 2 S i, and so DV \ S j ) < DS j ), contadicting the definition of S j. We each a simila contadiction if DS j ) < DV \ S j ). Theefoe, DS i ) = DV \ S i ) and DS j ) = DV \ S j ) must hold. In othe wods, DV \S i ) = DC 3 ) =, and DV \ S j ) = DC 2 ) =. Since C 2 and C 3 ae disjoint, this means that DC 1 ) = 0. But fom the definitions of S i and S j, s 1 S i S j must hold, a contadiction. Fist Algoithmic Famewok Most ou algoithms belong to one of two simple algoithmic famewoks. The fist famewok uses a divide-and-conque paadigm: We stat with the gaph G = V, E) and a set D of 1 demand pais, and then find a cut S, S) in G, with DS), D S) 1. We then select a subset E 0 ES, S) of edges to delete, and apply the algoithm ecusively to the sub-instances induced by G[S] and G[ S]. Hee, the sub-instance induced by G[S] consists of the gaph G[S] and the collection of the oiginal demand pais s i, t i ), with both s i, t i S. The sub-instance induced by G[ S] is defined similaly. Let E 1 and E 2 be the solutions etuned by the two ecusive calls, espectively. The final solution is E = E 0 E 1 E 2. The specific cut S, S), and the subsets E 0 ES, S) of edges computed will diffe fom algoithm to algoithm, and we will need to select them in a way that ensues the feasibility of the final solution. Howeve, the analysis of the solution cost is simila in all these algoithms, and is summaized in the following theoem. Theoem 2.3. Let A be any algoithm in the above famewok, and assume that the algoithm guaantees that we 0 ) α OPT min { DS), D S) }, fo some facto α. Then we ) 4α ln1 + ) OPT. Poof. The poof is by induction on. If = 1 then E = E 0, and the statement tivially holds. Assume now that the statement holds fo instances with fewe than demand pais, fo some > 1. Conside the cut S, S) computed by the algoithm A on the cuent instance. Let a be the numbe of demand pais s i, t i ) with s i, t i S, let b be the numbe of

9 demand pais s i, t i ) with s i, t i S, and assume w.l.o.g. that a b. Then DS) 2a, D S) 2b, and so DS) 2 D S) 2 b) and D S) 2 a). Theefoe, we 0 ) α OPT min { DS), D S) } 2α OPT a). The optimal solutions to the EC-kRC instances on gaphs G[S] and G[ S] have costs at most we ES)) and we E S)), espectively. By the induction hypothesis, the total cost of solutions E 1 and E 2 on gaphs G[S] and G[ S] is at most 4αwE ES)) ln1 + a) + 4αwE E S)) ln1 + b) 4α we ES)) + we E S)) ) ln1 + a) 4α OPT ln1 + a). The total solution cost is then bounded by: we 2α OPT ) 4α OPT ln1 + a) + a) 4α OPT ln1 + a) + a ). 2 The theoem follows fom the following inequality: ln1 + a) + a ) 1 + a = ln1 + ) + ln + a ln1 + ) a a 2 ln1 + ), whee we have used the fact that ln ) ln 1 a 1+ 1+a 1+ a 1+, since ln1+x) x fo all x > 1. Second Algoithmic Famewok The algoithmic famewok pesented above has some limitations. Specifically, we can only use it in scenaios whee thee is a cheap collection E of edges with cost oughly compaable to OPT), whose emoval decomposes ou instance G into two disjoint sub-instances, G[S], G[ S], which can then be solved sepaately. This is the case fo the unifom EC-kRC, and the non-unifom EC-kRC and VC-kRC when k = 2. Fo highe values of k in the non-unifom setting, such a decomposition may not exist. Instead, we use the following famewok. Given a gaph G and a set D of 1 demand pais, we find a collection E 0 of edges to delete, togethe with a subset D 0 of demand pais to emove, whee D 0 1. We then solve the poblem ecusively on the gaph G = G \ E 0, and the ) = set D \ D 0 of the emaining demand pais. Let E 1 be the subset of edges etuned by the ecusive call. Then the solution computed by the algoithm is E = E 0 E 1. The specific subset E 0 of edges to emove and the subset D 0 of demands will again be computed diffeently by each algoithm, in a way ensues that the final solution is feasible. The analysis of the solution cost of such algoithms is summaized in the next theoem. Theoem 2.4. Let A be any algoithm in the above famewok, and assume that we ae guaanteed that we 0 ) α OPT D0, fo some facto α. Then we ) 2α ln1 + ) OPT. Poof. The poof is by induction on. If = 1 then E = E 0, and the statement tivially holds. Assume now that the statement holds fo instances with fewe than demand pais, fo some > 1. We pove the theoem fo instances with demand pais. Let a = D 0. Then by the induction hypothesis, we 1 ) 2α OPT ln1+ a). Theefoe, we ) 2α OPT ln1 + a) + α OPT a ) 1 + a = 2α OPT ln + 1) + ln + a ) α OPT ln + 1) a a ) 2 2α ln + 1)OPT. 3 Unifom EC-kRC This section is dedicated to poving Theoem 1.1. We fist show an Ok log 1.5 )-appoximation algoithm, and povide a bi-citeia algoithm late. Recall that we ae given an unweighted gaph G = V, E), a set {s i, t i )} i=1 of demand pais, and an intege k. Ou goal is to find a collection E of Ok log 3/2 ) OPT edges, such that fo each demand pai s i, t i ), thee ae at most k 1) edge-disjoint paths in gaph G \ E connecting them. We assume w.l.o.g. that each souce-sink pai s i, t i ) is k-edge connected in the cuent gaph G. Ou algoithm views the gaph G as an instance of the unifom spasest cut poblem. We use the algoithm A ARV to find a patition S, S) of V with ΦS) α ARV ) ΦG), add the edges in ES, S) to the solution E, and delete the demand pais s i, t i ) that ae no longe k-edge connected fom the list of souce-sink pais. Notice that each emaining souce-sink pai must be contained eithe in S o in S. We then ecusively

10 solve the EC-kRC poblem on the sub-instances induced by G[S] and G[ S]. The algoithm is summaized in Figue 1. Input: An unweighted gaph G = V, E) with demand pais {s i, t i )} 1 i Output: A set E of edges, such that each pai s i, t i ) has at most k 1) edge-disjoint paths connecting s i to t i in G \ E. 1. If = 0 etun E =. 2. Find a patition S, S) of V using the algoithm A ARV, with ΦS) α ARV ) ΦG). 3. Let E 0 = ES, S), G = G \ E Remove all demand pais s i, t i ) that ae no longe k-edge connected in G. 5. Solve the instances induced by G[S] and G[V \ S] ecusively, to obtain solutions E 1 and E 2, espectively. 6. Retun E = E 1 E 2 E 0. Appoximation algoithm fo unifom EC- Figue 1: krc. The heat of the analysis of the algoithm is the following theoem, that elates the value of the unifom spasest cut in gaph G to the value OPT of the optimal solution fo EC-kRC. Theoem 3.1. Suppose that we ae given an unweighted gaph G = V, E) with souce-sink pais {s i, t i )} i=1, such that fo each pai s i, t i ), thee ae at least k edge-disjoint paths connecting s i to t i in G, and let OPT be the cost of the optimal solution of ECkRC on G. Then ΦG) 2k OPT. Poof. Conside the gaph H = G \ E. We use Lemma 2.1 with edge weights w e = 1 on gaph H to obtain the lamina family S = {S i } i=1 of vetex subsets. Conside all maximal sets in the lamina family, that is, sets S i that ae not contained in othe sets. Assume w.l.o.g. that these sets ae S 1,..., S q, fo some q. Then q i=1 DS i) must hold. Note that fo each i, E H S i, V \ S i ) k 1 since s i and t i ae not k- edge connected in H, and S i, V \ S i ) is a minimum cut sepaating s i fom t i in H. On the othe hand, E G S i, V \S i ) k since s i and t i ae k-edge connected in G. Theefoe, 3.1) E G S i, V \ S i ) = E H S i, V \ S i ) + E G S i, V \ S i ) E k 1) + E G S i, V \ S i ) E k E G S i, V \ S i ) E. Note that evey edge e E belongs to at most two cuts E G S i, V \ S i ) and E G S j, V \ S j ). Theefoe, q E G S i, V \ S i ) i=1 On the othe hand, q E G S i, V \ S i ) = i=1 q k E G S i, V \ S i ) E i=1 2k OPT q ΦS i ) DS i ) i=1 q ΦG) DS i ) i=1 We conclude that ΦG) 2k OPT/. ΦG). We now analyze the algoithm. Since the algoithm emoves a demand pai s i, t i ) only when s i and t i ae no longe k-edge connected, and teminates when all demand pais ae emoved, the algoithm is guaanteed to find a feasible solution to the poblem. In ode to bound the solution cost, note that E 0 = ΦS) min { DS), D S) } α ARV ) ΦG) min { DS), D S) } 2kα ARV) OPT min { DS), D S) }. We can now use Theoem 2.3 with α = 2kα ARV ) to conclude that E = Okα ARV ) log )OPT = Ok log 3/2 )OPT. Bi-citeia appoximation algoithm We now slightly modify the algoithm fom Figue 1, to obtain a 1 + δ, O 1 δ log1.5 ) ) -bi-citeia appoximation algoithm fo any constant 0 < δ < 1. The algoithm woks exactly as befoe, except that it emoves a demand pai s i, t i ) in step 4 iff s i and t i ae no longe 1 + δ)k edge-connected. We also assume w.l.o.g. that in the oiginal instance G, evey demand pai s i, t i ) has at least 1 + δ)k edge-disjoint paths connecting s i to t i. As befoe, it is staightfowad to veify that if E is the final solution poduced by the algoithm, then each demand pai s i, t i ) pai has fewe than 1 + δ)k edge-disjoint paths connecting them in G \ E. In ode to bound the solution cost, we pove the following analogue of Theoem 3.1.

11 Theoem 3.2. Suppose that we ae given an unweighted gaph G with demand pais {s i, t i )} i=1, whee fo each pai s i, t i ), thee ae at least 1 + δ)k edge-disjoint paths connecting s i to t i in G. Then ΦG) 2OPT 1 + 1/δ). Poof. As befoe, we compute the lamina family of minimum cuts in gaph H = G \ E, using Lemma 2.1, and we conside the collection of all maximal cuts in this family. Assume w.l.o.g. that it is {S 1,..., S q }, fo q, and ecall that q i=1 DS i). As befoe, fo each 1 i q, E G S i, V \ S i ) k 1) + E G S i, V \ S i ) E. Since E G S i, V \ S i ) 1 + δ)k, we get that E G S i, V \ S i ) E δk, and so k 1) E G S i, V \ S i ) E /δ. We get that: E G S i, V \ S i ) k 1) + E G S i, V \ S i ) E 1 + 1/δ) E G S i, V \ S i ) E. On the othe hand, E G S i, V \ S i ) ΦG) DS i ). Summing up ove all 1 i q, we get that: 2OPT q E G S i, V \ S i ) E i=1 δ δ + 1 q E G S i, V \ S i ) i=1 δ q δ + 1 ΦG) DS i ) δ ΦG). δ + 1 i=1 We conclude that ΦG) 2OPT 1 + 1/δ). In ode to bound the final solution cost, obseve that E 0 = ΦS) min { DS), D S) } α ARV ) ΦG) min { DS), D S) } 2OPTα ARV) 1 + 1/δ) min { DS), D S) }. We now use Theoem 2.3 with α = 2α ARV )1 + 1/δ) to conclude that E = Oα ARV ) log /δ)opt = Olog 1.5 /δ)opt, when 0 < δ < 1. This concludes the poof of Theoem Non-unifom EC-kRC In this section we pove Theoem 1.2. We stat with a 2, Õlog2.5 ))-bi-citeia algoithm with unning time n Ok), and we show an algoithm whose unning time is polynomial in n and k late. Abusing the notation, fo each cut S, S) in gaph G, we denote by DS, S) both the set of demand pais s i, t i ) with {s i, t i } S = 1, and the numbe of such pais. 4.1 A 2, Õlog5/2 )) bi-citeia appoximation in time n Ok) We cannot employ the fist algoithmic famewok fo EC-kRC on weighted gaphs. A natual appoach in using it would be to find an appopiately defined spase cut S, S), emove all but k 1 most expensive edges of this cut, and then ecusively solve the poblem on instances G[S] and G[ S]. Let E 0 be the subset of edges emoved, and let G = G \ E 0 be the emaining gaph. This appoach does not wok because it is possible that a demand pai s i, t i ) with both s i, t i S is connected by a path that visits G[ S] in gaph G. So if we solve the poblem ecusively on G[S] and G[ S], then the combined solution is not necessaily a feasible solution to the poblem. On the othe hand, if, instead, we emove all o almost all edges of ES, S), then the esulting solution cost may be too high. Theefoe, we employ the second algoithmic famewok instead. We assume w.l.o.g. that in the input gaph G, each demand pai s i, t i ) has at least 2k 1) edge-disjoint paths connecting them. Ou algoithm, summaized in Figue 2, stats by finding an appoximate non-unifom 2k 1)-oute spase cut S, S) in G, using Theoem 2.1. That is, Φ 2k 1) S) α ALN ) Φ 2k 1) G). Let F be the set of the 2k 2) most expensive edges of ES, S), let E 0 = ES, S) \ F, and let G = G \ E 0. We emove all demand pais that ae no longe 2k 1) connected in G we denote the set of these demand pais by D 0 ), and then ecusively solve the esulting instance. It is immediate to veify that the algoithm etuns a feasible solution. The unning time of the algoithm is dominated by computing an appoximate 2k 1)- oute spasest cut, and is theefoe bounded by n Ok). In ode to bound the solution cost, we use the following lemma that elates the value of Φ 2k 1) G) to OPT. Theoem 4.1. Suppose that we ae given a gaph GV, E) with edge weights w e, and a set D = {s i, t i )} i [] of demand pais, whee evey pai s i, t i ) has at least 2k 1) edge-disjoint paths connecting s i to t i in G. Then OPT = Ω log ) Φ2k 1) G). Poof. Conside the gaph H = G \ E. Let S = {S 1,..., S } be the lamina family of minimum cuts in H, guaanteed by Lemma 2.1. Recall that fo all 1 i, E H S i, V \ S i ) k 1. We need the following lemma.

12 Input: A weighted gaph GV, E) with a set D = {s i, t i )} 1 i of demand pais, and edge weights {w e } e E. Output: A set E of edges, such that each demand pai s i, t i ) is no longe 2k 1)-edge connected in G \ E. 1. If = 0 etun E =. 2. Find an appoximate non-unifom 2k 1)-oute spasest cut S, S) with Φ 2k 1) S) α ALN ) Φ 2k 1) G), using Theoem 2.1. Let F be the set of the 2k 2) most expensive edges in ES, S), beaking ties abitaily. 3. Let E 0 = ES, S) \ F, G = G \ E 0, and let D 0 be the set of all demand pais that ae no longe 2k 1)-connected in G. 4. Recusively solve the poblem on G with the demand set D \ D 0, to obtain a solution E Retun E = E 0 E 1. Figue 2: A bi-citeia appoximation algoithm fo nonunifom EC-kRC in time n Ok). Lemma 4.1. We can efficiently find a collection P of mutually disjoint vetex subsets, such that: Fo each U P, DU) ; Fo each U P, E H U, U) 2k 1), and U P DU, U) 8 log. Poof. We will define each set U P to be eithe some set S S, o a diffeence of two sets, U = S \ S, fo S, S S. Since fo each set S S, DS), this will ensue the fist condition. Since E H U, U) E H S, S) + E H S, S ) 2k 1), this will also ensue the second condition. We now tun to define the sets U P so that the thid condition is also satisfied. Fo simplicity, if collection S contains identical sets, we discad them, keeping at most one copy of each set in S. Recall that fo each set S S, DS, S) is the set of all demand pais s j, t j ) with {s j, t j } S = 1. Let D S, S) be the union of DS, S ) fo all sets S S whee S S, and let qs) = DS, S) \ D S, S) We patition the family S into subsets S x, fo 1 x log 2 + 1, as follows: Collection S x contains all sets S S with 2 x 1 qs) < 2 x. Since S S qs) =, thee is at least one index x, fo which S S qs) x 2 log. Fix any such index x. Conside the decomposition foest F fo the sets in S x. The nodes of the foest ae the sets in S x, and fo a pai S, S S x, set S is the paent of S iff S S, and thee is no othe set S S x with S S S. Let S S x be the collection of sets that have at most one child in this foest. We ae now eady to define the collection P of vetex subsets. If S S is a leaf in F, then we add S to P. Othewise, if S is a non-leaf set in S, and S is the unique child of S in F, then we add S \ S to P. It now only emains to pove that U P DU, U) 8 log. In ode to do so, obseve that S S x /2, and ecall that fo each S S x, 2 x 1 qs) < 2 x. Theefoe, U P DU, U) S S qs) S x 2 x 1 = S 2 x 2 x 2. On the othe hand, S S qs) x 2 log, and so S x 2 x +1 log. We conclude that U P DU) 8 log. Let P be the set family computed by Lemma 4.1. Clealy, fo each U P, 4.2) w 2k 1) U, U) = Φ 2k 1) U) DU, U) Φ 2k 1) G) DU, U). On the othe hand, since E H U, U) 2k 2, we E G U, U)) w 2k 1) U, U) must hold. Theefoe, 4.3) U P w 2k 1) U, U) U P we E G U, U)) 2OPT. Combining Equations 4.2) and 4.3), we get that: 2OPT w 2k 1) U, U) Φ 2k 1) G) DU, U) U P U P Φ 2k 1) G) 8 log ) Theefoe, OPT = Ω Φ2k 1) log G). In ode to bound the cost we ) of the solution, we note that DS, S) D 0, and so we 0 ) = w 2k 1) S, S) = Φ 2k 1) S) DS, S) α ALN ) Φ 2k 1) G) D 0 = Oα ALN ) log ) D 0 OPT.

13 We can now use Theoem 2.4 with α = Oα ALN ) log ) to conclude that we ) = Oα ALN ) log 2 ) = Olog 2.5 log log ). 4.2 A polynomial-time bi-citeia appoximation algoithm In this section, we extend the algoithm fom Section 4.1 to handle highe values of k in polynomial time. Notice that the bottleneck in the algoithm fom Section 4.1 is computing the appoximate multi-oute spasest cut via Theoem 2.1, which is done in time n Ok). We use Theoem 2.2 instead, that gives an efficient algoithm fo computing the k-oute spasest cut, albeit with somewhat weake guaantees. Ou algoithm is identical to the algoithm in Figue 2, except fo the following changes. Fist, in step 2, we use Theoem 2.2 to find an appoximate 2k 1)-oute non-unifom spasest cut S. That is, Φk ) S) = Olog ) Φ 2k 1) G), whee k = C2k 1) log, and C is the constant fom Theoem 2.2. Note ) that Φ C log 2k 1)) G) Φ 2k 1) G) O log OPT by Theoem 4.1. Theefoe, we 0 ) Φ C log 2k 1)) S) DS, S) Olog ) Φ C log 2k 1)) G) D 0 O log 2 ) OPT D 0. Using Theoem 2.4 with α = Olog 2 ), we get that the algoithm etuns a bi-citeia Olog ), Olog 3 ))- appoximate solution to the poblem. 5 Vetex Connectivity In this section, we extend ou appoximation algoithms fo EC-kRC to handle vetex-connectivity and pove Theoem 1.3. We stat by extending some of ou technical definitions and esults to the vetex-connectivity setting. Let s, t) be any pai of vetices in gaph G, and let V be any subset of vetices. We say that is a sepaato fo s and t, o that sepaates s and t, iff s, t, and s and t belong to two distinct connected components of V \. We say that is a minimum cost sepaato fo s, t) iff fo each subset sepaating s fom t,. Given any pai S, T V of vetex subsets, let ES, T ) be the set of edges with one endpoint in S and the othe endpoint in T. Similaly, we say that sepaates S fom T iff S =, T =, and ES, T ) =. Notice that in geneal G \ may contain moe than two connected components. A vetex cut in gaph G = V, E) is a ti-patition S,, T ) of V, whee ES, T ) =. Fo any subset V, we will sometimes efe to as the cost of. We stat with the following lemma, which is an analogue of Lemma 2.1 fo vetex cuts. Fo technical easons, it is moe convenient to state it fo gaphs with costs on vetices. Given a gaph G = V, E) with costs c v on vetices v V, a cost of a subset V of vetices is v c v. Lemma 5.1. Lamina Family of Min. Vetex Cuts). Suppose we ae given a gaph G = V, E) with costs c v on vetices v V, and a set {s 1, t 1 ), s 2, t 2 ),, s, t )} of demand pais. Let T be the set of all vetices paticipating in the demand pais. Suppose additionally that fo evey demand pai s i, t i ), fo evey minimum-cost sepaato fo s i, t i ), T =. Then thee exists a family of vetex cuts S i, i, T i ) such that: 1. Fo evey i {1,, }, i is a minimum cost sepaato fo s i, t i ) note that s i may belong eithe to S i o T i ); and 2. Sets {S i } i=1 fom a lamina family. Fo the edge-connectivity case, we used the existence of Gomoy Hu tees to pove the coesponding lamina decomposition Lemma 2.1). Fo the vetexconnectivity case, we need a moe intensive case analysis, which we povide in the full vesion of the pape. Poof of Theoem 1.3 In this section, we pove Theoem 1.3, by showing a 2, Odk log 5/2 log log )) bi-citeia appoximation algoithm VC-kRC, whee d is the maximum numbe of demand pais in which any teminal paticipates. The unning time of the algoithm is n Ok). We stat by extending the definition of the k-oute spasest cut to the vetex connectivity vesion. Given two disjoint subsets S, T of vetices, let DS, T ) be the set of all demand pais s i, t i ) with exactly one of the vetices s i, t i lying in S, and the othe one lying in T. Given any pai S, ) of disjoint subsets, let Υ ) S) = e ES,V \S )) w e, whee ES, V \S )) is the subset of all edges with one endpoint in S and the othe endpoint in V \ S ). The k-oute vetex spasity of the set S is then defined to be: Ψ k) S) = { min V \S: k 1 Υ ) S) DS, V \ S )) }, and the k-oute vetex spasity of the gaph G is: { Ψ k) G) = min S)} Ψk) S V

14 It is easy to see that, similaly to the edge vesion of k-oute spasest cut, the k-oute vetex spasest cut can be appoximated in time n Ok) to within a facto of α ALN ), as we show in the next theoem. Theoem 5.1. Thee is an algoithm that finds, in time n Ok), disjoint subsets S, V of vetices, with k 1 such that Υ ) S) α ALN ) Ψ k) G) DS, V \ S )). Poof. Fo evey subset V of at most k 1 vetices, we use the algoithm A ALN to find an appoximate spasest cut in the gaph G\, and output the spasest cut among all such cuts. Ou algoithm fo VC-kRC is vey simila to the algoithm fo EC-kRC fom Section 4. The only diffeence is that we use Theoem 5.1 to find an appoximate k- oute vetex spasest cut. The algoithm is summaized in Figue 3. Input: A weighted gaph GV, E) with a set D = {s i, t i )} 1 i of demand pais, and edge weights {w e } e E. Output: A subset E of edges, such that no demand pai s i and t i is 2k 1)-vetex connected in G \ E. 1. If = 0 etun E =. 2. Find sets U and with 2k 1 and Υ ) U) α ALN ) Ψ 2k 1) G) DU, V \U )) using Theoem Let E 0 = EU, V \ U )), and let G = G \ E Let D 0 be the set of all demand pais s i, t i ) that ae no longe 2k 1)-vetex connected in G. 5. Solve the poblem ecusively on G with the set D \ D 0 of demand pais to obtain a solution E Retun E = E 0 E 1. Figue 3: Bi-citeia appoximation algoithm fo VCkRC in time n Ok). It is easy to veify that if E is the solution computed by the algoithm, then fo each demand pai s i, t i ) thee ae at most 2k 1) vetex-disjoint paths connecting them in G \ E. This is since the algoithm only emoves a demand pai s i, t i ) when the teminals s i and t i ae no longe 2k 1)-vetex connected, and it teminates, since it emoves at least one demand pai in each iteation. In ode to analyze the pefomance of the algoithm, we use the following theoem, that elates the value Ψ k) G) of the k-oute vetex spasest cut in gaph G to the value OPT of the optimal solution to VC krc. Theoem 5.2. ) dk log Ψ 2k 1) G) O OPT. Poof. Let H = G \ E. The poof oughly follows the poof of Theoem 3.1, except that we need one additional step, that is summaized in the following lemma. Lemma 5.2. Thee exists a subset D D of = Ω/dk)) demand pais, and a collection of vetex cuts {S i, i, T i )} si,t i) D, such that: Fo all s i, t i ) D, i is a sepaato fo s i, t i ) in H, i < k, and i T =, whee T is the set of all teminals paticipating in demand pais in D. {S i } si,t i) D is a lamina family of vetex subsets. Poof. Fo each 1 i, let i be a minimum vetex sepaato fo s i and t i in H. Since s i and t i ae not k-vetex connected in H, i < k. We constuct an auxiliay gaph Z, whose vetex set is [], that is, each vetex i of Z epesents the demand pai s i, t i ). We say that a demand i blocks anothe demand j iff i contains eithe s j o t j o both). We connect i and j with an edge in Z iff one of them blocks the othe. Since i k 1 and each vetex in i paticipates in at most d demand pais, demand i blocks at most d 1)k demands. Theefoe, thee ae at most d 1)k edges in Z. By Tuan s theoem, thee is an independent set I of size Ω/dk)) in Z. Let = I, and let D = {s i, t i ) i I}. Next, we apply Lemma 5.1 to gaph G with the set D of demand pais, whee we define the cost c u of evey vetex u as follows: c u = V if u = s i o u = t i fo some s i, t i ) D, and cv) = 1 othewise. Since demand pais in D do not block each othe, the minimum cost vetex cut fo each of them has cost at most k 1 < V. Let {S i, i, T i )} si,t i) D be the collection of cuts etuned by Lemma 5.1. It is easy to see that these cuts satisfy the conditions of the lemma. We apply Lemma 5.2 and find the subset D of demand pais and vetex cuts S i, i, T i ). We assume w.l.o.g. that D = {s 1, t 1 ),..., s, t )}. Now we need a countepat of Lemma 4.1.

15 Lemma 5.3. Thee is a family P = {U 1,..., U p } of disjoint vetex subsets, and a collection {U j, Λ j, R j )} p j=1 of vetex cuts in gaph H, such that: fo each 1 j p, Λ j < 2k 1, p j=1 DU j, R j ) 8 log. The poof closely follows the poof of Lemma 4.1 and can be found in the full vesion. Conside the family P = {U 1,..., U p } and the coesponding cuts U i, Λ i, R i ) as in Lemma 5.3. Since all sets in P ae mutually disjoint, and fo each such set U i P, E H U i, R i ) 2k 1, and so p Υ Λj) U j ) 2OPT, j=1 p j=1 ΥΛj) U j ) p j=1 DU j, R j ) O log O ) dk log OPT ) OPT. Theefoe, thee is an index 1 j p, such that Υ Λj) U j ) DU j, R j ) O dk log ) OPT. The left hand side of this inequality is at least Ψ 2k 1) G) since Λ j ) 2k 2. We conclude that Ψ 2k 1) G) O dk log OPT. In ode to complete the poof of Theoem 1.3, obseve that we 0 ) = Υ ) U), and by Theoem 5.2, we 0 ) α ALN )Ψ 2k 1) G) DU, V \ U )) ) dk log O OPT DU, V \ U )). Note that we emove all demand pais in DU, V \ U )) in step 4 of the algoithm. We can now use Theoem 2.4 with α = Odk log α ALN )) to conclude that the cost of the solution etuned by the algoithm is bounded by Odk log 5/2 log log ) OPT. 6 Algoithms fo 2-oute cuts In this section we pove Theoem 1.5. Since we pove in the full vesion that EC-kRC can be cast as a special case of VC-kRC, and the connectivity value k emains unchanged in this eduction, it is enough to pove the theoem fo VC-kRC, whee k = 2. In the est of this section we show an efficient Olog 3/2 )-appoximation algoithm fo VC-kRC with k = 2. Given a subset S of vetices in gaph G, the unifom vetex 2-oute spasity of S is: Ψ 2) S) = { min V \S: 1 Υ ) S) min {DS), DV \ S ))} }, and the unifom vetex 2-oute spasity of the gaph G is: { } Ψ 2) G) = min Ψ 2) S) S V As befoe, we can efficiently appoximate the unifom vetex 2-oute spasest cut in any gaph, as shown in the next theoem. Theoem 6.1. Thee is a polynomial time algoithm that finds disjoint subsets S V and V of vetices, with 1 and 0 < DS), such that Υ ) S) α ARV ) Ψ 2) G) DS). Poof. Fo evey subset V of size at most 1, we use the algoithm A ARV to find the α ARV )-appoximate unifom spasest cut in gaph G\, and output the cut with the smallest spasity. The appoximation algoithm fo VC-kRC with k = 2 is shown in Figue 4. In ode to analyze the algoithm, we stat by showing that it is guaanteed to poduce a feasible solution. Claim 6.1. The algoithm outputs a feasible solution to the poblem. Poof. The poof is by induction on the numbe of vetices in G. Assume that the algoithm outputs a feasible solution fo all gaphs containing fewe than n vetices, and conside a gaph G containing n vetices. Let s i, t i ) be any demand pai, and assume fo contadiction that thee ae at least two vetex-disjoint simple paths P 1, P 2 connecting s i to t i in G \ E. Obseve fist that eithe s i, t i S o s i, t i T must hold. Othewise, one

16 Input: A weighted gaph G = V, E) with demand pais {s i, t i )} 1 i, and edge weights {w e } e E, such that each demand pai has at least 2 vetex-disjoint paths connecting them in G Output: A set E of edges such that each pai s i, t i ) is no longe 2-vetex connected in G \ E. 1. If = 0 etun E =. 2. Find disjoint subsets S, of vetices with = 1, 0 < DS), such that Υ ) S) α ARV ) Ψ 2) G) DS), using Theoem 6.1. Let T = V \ S ). 3. Let E 0 = ES, T ); G = G \ E Remove all demand pais s i, t i ) that ae no longe 2-vetex connected in G. 5. Recusively solve the sub-instances induced by G[S ] and G[T ] to obtain solutions E 1 and E 2. The set of demand pais fo the instance induced by G[S ] is defined to be the subset of all emaining demand pais contained in S. The set of demand pais fo the instance induced by G[T ] is defined similaly. 6. Retun E = E 0 E 1 E 2. Figue 4: Appoximation algoithm fo VC-kRC, k = 2 weighted case). of the two vetices must belong to S and the othe to T. But is a sepaato fo S and T in gaph G, and since = 1, the paths P 1 and P 2 cannot be vetex-disjoint. Assume w.l.o.g. that s i, t i S. By the induction hypothesis, E 1 is a feasible solution to the instance induced by G[S ], and in paticula G[S ] \ E 1 cannot contain two vetex-disjoint paths connecting s i to t i. Theefoe, at least one of the two paths, say P 1, must contain a vetex of T. But since is a sepaato fo S and T, = 1, and both s i, t i T, path P 1 cannot be a simple path, a contadiction. It now emains to bound the cost of the solution poduced by the algoithm. As befoe, we do so by elating the value of the 2-oute vetex spasest cut to the value OPT of the optimal solution to the VC-kRC poblem. Theoem 6.2. Suppose that we ae given an undiected gaph G = V, E) with edge weights w e, and demand pais s 1, t 1 ),..., s, t ). Let OPT be cost of the optimal solution to the coesponding VC-kRC poblem instance, and assume that k = 2. Then Ψ 2) G) 4OPT. The poof poceeds by consideing a block decomposition of the gaph G \ OP T, and is defeed to the full vesion. Let a be the numbe of demand pais contained in S and b be the numbe of demand pais contained in V \ S) in gaph G. Fom Theoem 6.2, we 0 ) α ARV ) Ψ 2) G) DS) 4α ARV )OPT min {a, b} / 4α ARV )OPT min { a, b} / Theefoe, by setting α = 4α ARV ), we get the same ecuence as in the poof of Theoem 2.3: we ) we 1) + we 2) + 2α min { a, b} OPT Solving this ecuence as in Theoem 2.3, we get that we ) Olog 3/2 )OPT. 7 A facto k ɛ -hadness fo k -VC-kRC We pove Theoem 1.4 though a gadget eduction fom the 3SAT5) poblem each vaiable occus in at most five clauses). Given a 3SAT5) fomula φ, we say that it is a Yes-Instance if it is satisfiable, and it is a No- Instance iff the maximum faction of simultaneously satisfiable clauses is δ, fo some constant paamete δ. Fom the PCP theoem, thee is some constant δ, fo which it is NP-had to distinguish whethe an input 3SAT5) fomula φ is a Yes-Instance o a No-Instance. We use the Raz veifie fo 3SAT5) with l paallel epetitions. Given the 3SAT5) instance ϕ, the veifie chooses, independently at andom, l clauses C 1,..., C l, and fo each i : 1 i l, a vaiable x i paticipating in clause C i is chosen at andom. The veifie then sends one quey to each one of the two poves, while the quey to the fist pove consists of the indices of the vaiables x 1..., x l, and the quey to the second pove contains the indices of the clauses C 1,..., C l. The fist pove etuns an assignment to vaiables x 1,..., x l. The second pove is expected to etun an assignment to all the vaiables in clauses C 1,..., C l, which must satisfy the clauses. Finally, the veifie checks that the answes of the two poves ae consistent. The following theoem is obtained by combining the PCP theoem[as98, ALM + 98] with the paallel epetition theoem [Raz98].

17 ... ex- Theoem 7.1. [AS98, ALM + 98, Raz98]) Thee ists a constant γ > 0, such that: If ϕ is a Yes-Instance, then thee is a stategy of the poves, fo which the acceptance pobability is 1. If ϕ is a No-Instance, then fo any stategy of the poves, the acceptance pobability is at most 2 γl. We denote the set of all the andom stings of the veifie by R, R = 5n) l, and the sets of all the possible queies of the fist and the second pove by Q 1 and Q 2 espectively, Q 1 = n l, Q 2 = 5n/3) l. Fo each quey q Q, let Aq) be the collection of all the possible answes to q if q is a quey of the second pove, then Aq) only contains answes that satisfy all the clauses of the quey). Let A = 2 l, A = 7 l. Then fo each q Q 1, Aq) = A, and fo each q Q 2, Aq ) = A. Given a andom sting R, let q 1 ), q 2 ) be the queies sent to the fist and the second pove espectively, when the veifie chooses. Fo each q Q 1, let Nq) = {q Q 2 R : q 1 ) = q, q 2 ) = q }, and fo each q Q 2, let Nq ) = {q Q 1 R : q 1 ) = q, q 2 ) = q }. Notice that fo all q Q 1, Nq) = 5 l, and fo all q Q 2, Nq ) = 3 l. Constuction: We now tun to descibe ou eduction. Fo each quey q Q 1 of the fist pove, fo each answe a Aq), we have an edge eq, a), whose endpoints ae denoted by vq, a), uq, a), and whose cost is 5/3) l. We will think of vq, a) as the fist endpoint of eq, a) and of uq, a) as its second endpoint, even though the gaph is undiected. Similaly, fo each quey q Q 2 of the second pove, fo each answe a Aq), thee is an edge eq, a) = vq, a), uq, a)), of cost 1. As befoe, vq, a) is called the fist endpoint and uq, a) the second endpoint of eq, a). Let E 0 be the set of all esulting edges. Fo each q Q, let V q) = {vq, a), uq, a) a Aq)}. Fo each andom sting R of the veifie, we intoduce a souce-sink pai s), t), and two collections of edges E 1 ), E 2 ), whose costs ae. Let E 1 = R E 1) and E 2 = R E 2). The set of edges in the final gaph is E 0 E 1 E 2. We now fix some andom sting R, and define the set E 1 ) of edges. Let q = q 1 ), q = q 2 ). Let a 1, a 2,..., a A ) be any odeing of the set Aq) of answes to q 1. Fo each 1 i A, let b 1 a i ), b 2 a i ),..., b zi a i ) be the set of all answes to q that ae consistent with the answe a i to q. We stat by connecting the edges coesponding to b 1 a i ), b 2 a i ),..., b zi a i ) into a single path P i as follows: fo 1 j < z i, we connect the second endpoint of the edge eq, b j a i )) to the fist endpoint of edge eq, b j+1 a i )). We will efe to vq, b 1 a i )) as the fist vetex on path P i, and to uq, b zi a i )) as the last vetex. Next, we connect the souce s) to the fist vetex of eq, a 1 ) and the fist vetex of P 1. We also connect the second vetex of eq, a A ) and the last vetex of P A to the sink t). Finally, fo all 1 i < A, we connect the last vetex of P i to the fist vetices of eq, a i+1 ) and P i+1, and the second vetex of eq, a i ) to the fist vetices of eq, a i+1 ) and P i+1. This finishes the definition of the set E 1 ) of edges. Let G) be the gaph whose vetex set is V q) V q ) {s), t)}, and the edge set consists of E 1 ) and the edges of E 0 epesenting the answes to q and q, that is: {eq, a) a Aq)} {eq, a ) a Aq )}. Then G) is an almost layeed gaph, whee fo each 1 i A, laye i consists of the edge eq 1 ), a i ) and of the path P i see Figue 5). Notice that the only way to disconnect s) fom t) in gaph G), without deleting edges of E 1 ) whose cost is ), is to delete a pai eq, a), eq, a ) of edges, whee a and a ae matching answes to queies q and q, espectively. P i s) b 1 a i ) b 2 a i ) b zi a i ) a 1 a 2 P 1 P 2 a A P A Figue 5: Gaph G). Red edges belong to E 1 ) and have cost. Finally, we define the sets E 2 ) of edges fo all R. Given a andom sting R, let N) = Nq 1 )) Nq 2 )), and let U) = q N) V q). Notice that U) = Nq 1 )) 7 l + Nq 2 )) 2 l = 35 l + 6 l. We connect s) to evey vetex in U), and we connect evey vetex in U) to t). We denote the esulting set of edges by E 2 ), and we set the costs of these edges to be. Finally, we set the paamete k to be U) + 1 = 35 l + 6 l + 1 this value is identical fo all R). Let G be the final instance of the VC-kRC poblem. The completeness and soundness analysis ae included in the full vesion. 8 Single Souce-Sink Pai In this section we study the single souce-sink pai vesion of EC-kRC and VC-kRC, denoted by st) ECkRC and st) VC-kRC, espectively. We stat with algoithmic esults in Section 8.1, and complement them t)