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 July 10, 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) edge-disjoint paths connecting s i to t i in G \ E, while in the vetex-connectivity vesion, VC-kRC, the same equiement is fo vetexdisjoint paths. Pio to ou wok, poly-logaithmic appoximation algoithm has been known fo the special case whee k = 2, but no non-tivial appoximation algoithms wee known fo any value k > 2, except in the single-souce setting. We show an O(k log 3/2 )-appoximation algoithm fo EC-kRC 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 pesent 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 VC-kRC 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 CA- REER awad CCF and Sloan Reseach Fellowship. Toyota Technological Institute, Chicago, IL yuy@ttic.edu. Depatment of Compute Science, Pinceton Univesity. aavindv@cs.pinceton.edu. Compute Science Depatment, Canegie Mellon Univesity, Pittsbugh, PA. yuanzhou@cs.cmu.edu.

2 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 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, 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 multicommodity 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 O(log ) 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 max-flow theoem to the multicommodity setting, and it also gives an efficient O(log )-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 non-negative weights w e on edges e E, and a collection {(s 1, t 1 ), (s 2, t 2 ),..., (s, t )} of soucesink 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 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 Section A), 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, as shown in Section B, 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 1

3 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 multi-commodity 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 O(log 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 O(log 2 )-appoximation algoithm fo the same vesion, by genealizing the egion-gowing LPounding 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. In fact, no appoximation algoithms with sub-polynomial (in n) guaantees ae known fo any vaiant of the poblem, fo any value k > 2, except in the single-souce setting that we discuss late. Ou fist esult is an O(k 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, Õ(log2.5 ) ) -biciteia appoximation in n O(k) time, and an ( O(log ), O(log 3 ) ) -bi-citeia appoximation in time polynomial in n and k. We also show an O(log 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 EC-kRC ae summaized in Table 1. Pevious esults Cuent pape k = 2 O(log 2 ) [BC10] O(log 1.5 ) abitay k, unifom - O(k log 1.5 ), ( 1 + δ, O ( 1 δ log1.5 )) fo any constant 0 < δ < 1 abitay k, geneal - ( 2, O(log 2.5 log log ) ) in time n O(k) ; ( O(log ), O(log 3 ) ) in poly(n)-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 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 O(log 2 n log )-appoximation of [CK08], 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. 2

4 and the O(log 2 )-appoximation of [BC10] fo 2-oute cuts extend to the vetex-connectivity vesion as well, as does ou O(log 1.5 )-appoximation algoithm. As with EC-kRC, no non-tivial appoximation algoithms wee known fo any highe values of k. In this pape, we show a ( 2, Õ(log2.5 )dk ) - bi-citeia appoximation algoithm fo VC-kRC, with unning time n O(k), 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 Ω(k ɛ ), fo some constant ɛ > 0. Ou esults fo VC-kRC ae summaized in Table 2. Pevious esults Cuent pape k = 2 O(log 2 ) [BC10] O(log 1.5 ) abitay k APX-had [DJP + 94] no constant facto appoximation unde UGC [KV05, CKK + 06] ( 2, O(log 2.5 log log dk) ) -appoximation algoithm, unning time n O(k), whee d is the maximum numbe of demand pais in which any teminal paticipates Ω(k ɛ )-hadness fo specific constant ɛ > 0 Table 2: Results fo VC-kRC. 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 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)-vc-krc 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 2(k 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)-vc-krc 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 2 This esult also extends to the single-souce multiple-sinks setting. 3

5 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 G(V, 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 O(n 1/4+ɛ )-facto appoximation algoithm which uns in time n O(1/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 Khot [Kho04] used quasi-andom PCPs to ule out a PTAS. Raghavenda and Steue [RS10] have intoduced the Small Set Expansion conjectue and showed that Densest κ-subgaph has no constant facto appoximation algoithms unde this conjectue. Vey ecently, Aoa et al. [AAM + 11] poved that thee is no constant facto appoximation algoithm fo Densest κ-subgaph unde Feige s Random κ-and conjectue. 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), which may be of independent inteest. 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 Γ(S). We show an appoximationpeseving eduction fom SSVE to (st)-vc-krc, and then pove inappoximability esults fo the SSVE poblem. Othe Related Wok Anothe vesion of the EC-kRC 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 2(k 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 E(S, 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 E(S, S). Li and Zhang [LZ10] give an O(log n)-appoximation to this poblem using Räcke s gaph decomposition [Räc08]. This poblem can be educed to (st)-ec- 4

6 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 O(k 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, O(log 2.5 log log ) ) -bi-citeia appoximation algoithm with unning time n O(k) and an ( O(log ), O(log 3 ) ) -bi-citeia appoximation algoithm with unning time poly(n) fo the EC-kRC 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 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 the cut, which is the 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 the standad spasest cut can be pohibitively expensive; its cost cannot be bounded in tems of the cost of the optimal solution OPT. We pove howeve that the cost of the k-oute spasest cut can be bounded in tems of OPT 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, O(log 2.5 log log dk) ) -bi-citeia appoximation algoithm fo VCkRC with unning time n O(k), 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 DTIME(n poly log n ) fo supe-constant k. Finally, fo the special case of k = 2, we obtain a slightly impoved appoximation algoithm: Theoem 1.5. Thee is an efficient facto O(log 1.5 )-appoximation algoithm fo both VC-kRC and EC-kRC, when k = 2. 5

7 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 clauses 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 (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 c 0 k 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, 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 a 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 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 3CNFfomula 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 2, fo any constant ɛ > 0, no polynomial-time algoithm achieves a ( ɛ, 1.1 ɛ) -bi-citeia appoximation fo (st) VC-kRC. 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 G(V, 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. 6

8 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 supe-constant inappoximability fo (st)-vc-krc, unde Hypothesis 1. Howeve, ou poof of Theoem 1.7 is conceptually simple, and also leads to 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 EC-kRC 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. 2 Peliminaies In all ou poblems, the input is an undiected n-vetex gaph G = (V, E) with non-negative weights w(e) 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 D(S) = v S D v be the total numbe of teminals in S, counting multiplicities. We also denote by D(S, S) the set of all 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 w(e ) = e E w(e) its weight. Thoughout the pape, we denote by E the optimal solution to the given EC-kRC 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. to be Given any cut (S, S) in gaph G, its unifom spasity is defined Φ(S) = w(e(s, S)) min { }. D(S), D( S) The unifom spasity Φ(G) of the gaph G is the minimum spasity of any cut in G, Φ(G) = min {Φ(S)}. S V : D(S),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 ) 7

9 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 Φ(S) α ARV () Φ(G). Given any cut (S, S) in gaph G, its non-unifom spasity is defined to be Φ(S) = w(e(s, S)) D(S, S). The non-unifom spasity Φ(G) of the gaph G is: Φ(G) = min S V : D(S, S)>0 { Φ(S) }. 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 Φ(S) α ALN () Φ(G). 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 E(S, 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 E(S, S)\F w e. The unifom k-oute spasity of set S is defined to be: Φ (k) (S) = and the unifom k-oute spasity of the gaph G is: Φ (k) (G) = w (k) (S, S) min { }, D(S), D( S) min S V : D(S),D( S)>0 Similaly, the non-unifom k-oute spasity of S is: Φ (k) (S) = w(k) (S, S) D(S, S), and the non-unifom k-oute spasity of the gaph G is: Φ (k) (G) = min S V : D(S, S)>0 { } Φ (k) (S). { Φ(k) (S)}. Note that Φ (1) (G) = Φ(G) and Φ (1) (G) = Φ(G) 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 O(k) a cut S V, with Φ (k) (S) α ARV () Φ (k) (G). Similaly, thee is an algoithm that computes in time n O(k) a cut S, with Φ (k) (S) α ALN () Φ (k) (G). 8

10 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 a andomized 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) = O(log ) 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 andomized O(log )-appoximation algoithm fo this poblem. We denote thei algoithm by A EGK+, and the appoximation facto it achieves by α EGK+ = O(log ). 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 w(e(s, S ) \ F ) = Φ (k) (G) D(S, S ). Let W = w(e(s, S ) \ F ) and let = D(S, 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 E(S, 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 w(e(s, S) \ F ) O(log ) Φ (k) (G) D(S, S). Note that the value of the optimal solution to the l-multicut poblem instance is at most w(e(s, S )) w(e(s, S ) \ F ) + F W k 1 w(e(s, S ) \ F ) + W = 2W. Theefoe, w(e(s, S)) 2α EGK+ ()W. In paticula, E(S, 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 as equied. w(e(s, S) \ F ) = w(e(s, S) \ F ) w(e(s, S)) 2α EGK+ ()W = 2α EGK+ () Φ (k) (G) O(log ) Φ (k) (G)D(S, 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 9

11 {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 E(S, S ) \ F, and τ = W /w e, then W τw e (2 W /w e )w e 2W. Fo these 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 2.3. 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 D(S 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 E(T 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 minimum-cost 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 D(L i ) < D(R i ), then we set S i = L i. If D(R i ) < D(L i ), we set S i = R i. Othewise, if D(R i ) = D(L 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 D(S 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 1 and so S i S j =, o S j = C 2 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 D(S i ) < D(V \ S i ), o D(S i ) = D(V \ S i ) and s 1 S i. Assume fist that D(S i ) < D(V \ S i ). Then V \ S j = C 2 S i, and so D(V \ S j ) < D(S j ), contadicting the definition of S j. We each a simila contadiction if D(S j ) < D(V \S j ). Theefoe, D(S i ) = D(V \ S i ) and D(S j ) = D(V \ S j ) must hold. In othe wods, D(V \ S i ) = D(C 3 ) =, and D(V \ S j ) = D(C 2 ) =. Since C 2 and C 3 ae disjoint, this means that D(C 1 ) = 0. But fom the definitions of S i and S j, s 1 S i S j must hold, a contadiction. 10

12 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 D(S), D( S) 1. We then select a subset E 0 E(S, 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 E(S, 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 2.4. Let A be any algoithm in the above famewok, and assume that the algoithm guaantees that w(e 0 ) α OPT min { D(S), D( S) }, fo some facto α. Then w(e ) 4α ln(1 + ) 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 demand pais (s i, t i ) with s i, t i S, and assume w.l.o.g. that a b. Then D(S) 2a, D( S) 2b, and so D(S) 2 D( S) 2( b) and D( S) 2( a). Theefoe, w(e 0 ) α OPT min { D(S), D( S) } 2α OPT ( a). The optimal solutions to the EC-kRC instances on gaphs G[S] and G[ S] have costs at most w(e E(S)) and w(e 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 4α w(e E(S)) ln(1 + a) + 4α w(e E( S)) ln(1 + b) ( 4α w(e E(S)) + w(e E( S)) ) ln(1 + a) The total solution cost is then bounded by: w(e ) 4α OPT ln(1 + a) + ln(1 + a) + a 2 4α OPT ln(1 + a). The theoem follows fom the following inequality: ( 1 + a = ln(1 + ) + ln 1 + ( ) ( ) whee we have used the fact that ln 1+a 1+ = ln 1 a 1+ x > 1. ( 2α OPT ( a) 4α OPT ln(1 + a) + a ). 2 ) + a ln(1 + ) a a ln(1 + ), 2 11 a 1+, since ln(1 + x) x fo all

13 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 2.5. Let A be any algoithm in the above famewok, and assume that we ae guaanteed that w(e 0 ) α OPT D 0, fo some facto α. Then w(e ) 2α ln(1 + ) 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, w(e 1 ) 2α OPT ln(1 + a). Theefoe, w(e ) 2α OPT ln(1 + a) + α OPT a ( ( ) 1 + a = 2α OPT ln( + 1) + ln + a ) ( ( = 2α OPT ln( + 1) + ln 1 a ) + a ) ( 2α 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 O(k 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 O(k 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. 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 Φ(S) α ARV () Φ(G), add the edges in E(S, S) to the solution E, and delete the demand pais (s i, t i ) that ae no longe k-edge connected 12

14 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 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 Φ(S) α ARV () Φ(G). 3. Let E 0 = E(S, 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. Figue 1: Appoximation algoithm fo unifom EC-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 EC-kRC on G. Then Φ(G) 2k OPT. Poof. Conside the gaph H = G \ E. We use Lemma 2.3 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 D(S 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, 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. (1) 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 q E G (S i, V \ S i ) k E G (S i, V \ S i ) E 2k OPT. i=1 i=1 13

15 On the othe hand, q E G (S i, V \ S i ) = i=1 q Φ(S i ) D(S i ) i=1 q Φ(G) D(S i ) Φ(G). i=1 We conclude that Φ(G) 2k OPT/. 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 = Φ(S) min { D(S), D( S) } α ARV () Φ(G) min { D(S), D( S) } 2kα ARV() OPT min { D(S), D( S) }. We can now use Theoem 2.4 with α = 2kα ARV () to conclude that E = O(kα ARV () log )OPT = O(k 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. 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 Φ(G) 2OPT (1 + 1/δ). Poof. As befoe, we compute the lamina family of minimum cuts in gaph H = G \ E, using Lemma 2.3, 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 D(S 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 ) Φ(G) D(S i ). Summing up ove all 1 i q, we get that: 14

16 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 Φ(G) D(S i ) We conclude that Φ(G) 2OPT (1 + 1/δ). i=1 In ode to bound the final solution cost, obseve that δ Φ(G). δ + 1 E 0 = Φ(S) min { D(S), D( S) } α ARV () Φ(G) min { D(S), D( S) } 2OPTα ARV() (1 + 1/δ) min { D(S), D( S) }. We now use Theoem 2.4 with α = 2α ARV ()(1+1/δ) to conclude that E = O(α ARV () log /δ)opt = O(log 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, Õ(log2.5 ))-bi-citeia algoithm with unning time n O(k), 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 D(S, 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, Õ(log5/2 )) bi-citeia appoximation in time n O(k) 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 E(S, 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, 15

17 Φ (2k 1) (S) α ALN () Φ (2k 1) (G). Let F be the set of the (2k 2) most expensive edges of E(S, S), let E 0 = E(S, S) \ F, and let G = G \ E 0. We emove all demand pais that ae no longe (2k 1) connected in G, and then ecusively solve the esulting instance. Input: A weighted gaph G(V, 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 E(S, S), beaking ties abitaily. 3. Let E 0 = E(S, 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 non-unifom EC-kRC in time n O(k). 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 O(k). 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 G(V, 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 ) Φ(2k 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 2.3. Recall that fo all 1 i, E H (S i, V \ S i ) k 1. We need the following lemma. Lemma 4.2. We can efficiently find a collection P of mutually disjoint vetex subsets, such that: Fo each U P, D(U) ; Fo each U P, E H (U, U) 2(k 1), and U P D(U, 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, D(S), this will ensue the fist condition. Since E H (U, U) E H (S, S) + E H (S, S ) 2(k 1), this will also ensue the second condition. 16

18 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, D(S, 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 D(S, S ) fo all sets S S whee S S, and let q(s) = D(S, 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 q(s) < 2 x. Since S S q(s) =, thee is at least one index x, fo which S S q(s) 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 D(U, 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 q(s) < 2 x. Theefoe, U P D(U, U) S S q(s) S x 2 2 x 1 = S x 2 x 2. On the othe hand, S S q(s) x 2 log, and so S x 2 x +1 log. We conclude that U P D(U) 8 log. Let P be the set family computed by Lemma 4.2. Clealy, fo each U P, w (2k 1) (U, U) = Φ (2k 1) (U) D(U, U) Φ (2k 1) (G) D(U, U). (2) On the othe hand, since E H (U, U) 2k 2, w(e E G (U, U)) w (2k 1) (U, U) must hold. Theefoe, U P w (2k 1) (U, U) U P Combining Equations (2) and (3), we get that: 2OPT U P Theefoe, OPT = Ω ( log ) Φ(2k 1) (G). w(e E G (U, U)) 2OPT. (3) w (2k 1) (U, U) Φ (2k 1) (G) U P D(U, U) Φ (2k 1) (G) 8 log In ode to bound the cost w(e ) of the solution, we note that D(S, S) D 0, and so w(e 0 ) = w (2k 1) (S, S) = Φ (2k 1) (S) D(S, S) α ALN () Φ (2k 1) (G) D 0 = O(α ALN () log ) D 0 OPT We can now use Theoem 2.5 with α = O(α ALN () log ) to conclude that w(e ) = O(α ALN () log 2 ) = O(log 2.5 log log ). 17

19 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 O(k). 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, Φ (k ) (S) = O(log ) Φ (2k 1), whee k = C(2k 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 w(e 0 ) Φ (C log (2k 1)) (S) D(S, S) O(log ) Φ (C log (2k 1)) (G) D 0 O ) OPT by Theoem 4.1. Theefoe, ) OPT D 0. Using ( log 2 Theoem 2.5 with α = O(log 2 ), we get that the algoithm etuns a bi-citeia (O(log ), O(log 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 sepaato fo (s, t) iff fo each subset sepaating s fom t,. Given any pai S, T V of vetex subsets, let E(S, 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 E(S, 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 E(S, 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 2.3 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 Minimum 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 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. 18

20 Poof. We stat by consideing a non-degeneate case, whee evey subset S V of vetices has a distinct cost. Fix some 1 i. Let i be the unique minimum-cost sepaato fo (s i, t i ). Conside the connected components of G \ i that contain s i and t i. Let S i be the component of the smalle cost. We then set T i = V \ (S i i ). This finishes the definition of the cuts (S i, i, T i ). Clealy, these cuts satisfy popety 1. We claim that sets {S i } i=1 fom a lamina family. Assume fo contadiction that this is not the case, so thee ae two sets S i, S j whose intesection is non-empty, but neithe of them is a subset of the othe. Assume w.l.o.g. that these two sets ae S 1 and S 2. Notice that c( 1 ) c( 2 ) must hold: othewise, if c( 1 ) = c( 2 ), then 1 = 2 must hold, and thus S 1 and S 2 ae some connected components of G \ 1. But in that case, eithe S 1 = S 2 o S 1 S 2 = must hold, a contadiction. Without loss of geneality, we assume that c( 1 ) > c( 2 ). Since 1 is the minimum cost sepaato fo s 1 and t 1, and c( 2 ) < c( 1 ), set 2 cannot be a sepaato fo s 1 and t 1, and so both these vetices must eithe belong to S 2 o to T 2. Let X 2 {S 2, T 2 } be the set that contains neithe s 1 no t 1, and let Y 2 be the othe set (ecall that by ou assumption sets 1 and 2 contain no teminals). Recall that X 2 contains eithe s 2 o t 2 but not both of them. Let us assume, without loss of geneality, that s 2 X 2 and t 2 Y 2. Let X 1 {S 1, T 1 } be set containing s 2, and let Y 1 be the othe set. Assume without loss of geneality that s 1 S 1 and t 1 T 1. Figue 3 shows which teminals lie in which sets. X 2 2 Y 2 X 1 s 2 s 1, (t 2 ) 1 Y 1 t 1, (t 2 ) Figue 3: The figue shows the intesections of the sets S 1, 1, T 1 with the sets S 2, 2, T 2. Thee ae edges only between sets located in hoizontally, vetically, o diagonally adjacent cells. The figue also shows how teminals s 1, t 1, s 2, and t 2 ae distibuted among the sets, with the two possible locations of t 2 appeaing in paentheses. We need the following claim. Claim Y 1 2 = X 2 1 = Poof. Let A = (Y 1 2 ) (Y 2 1 ) ( 1 2 ), and let B = (X 1 2 ) (X 2 1 ) ( 1 2 ) (see Figue 4). Notice that A is a sepaato fo s 1 and t 1, and B is a sepaato fo s 2 and t 2 By ou definition of cuts (S i, i, T i ): eithe c( 1 ) < c(a) o 1 = A; and eithe c( 2 ) < c(b) o 2 = B. 19

21 X 2 2 Y 2 X 1 s 2 s 1, (t 2 ) 1 Y 1 t 1, (t 2 ) X 2 2 Y 2 X 1 s 2 s 1, (t 2 ) 1 Y 1 t 1, (t 2 ) X 2 2 Y 2 X 1 s 2 s 1, (t 2 ) 1 Y 1 t 1, (t 2 ) Set A Set B Sets Y 1 2 and X 2 1 Figue 4: Illustation fo Claim 5.2 Howeve, c(a) + c(b) = c( 1 ) + c( 2 ). Y 1 2 = X 2 1 =. Theefoe, 1 = A and 2 = B must hold, and so X 2 2 Y 2 X 1 X Y 1 Y 1 X 2 Y 1 Y 2 Thus sets Y 1 X 2, Y 1 Y 2 and X 1 ae all disconnected in G\ 1. That is, 1 is a sepaato fo each pai of these sets. We claim that S 1 = X 1 must hold. Indeed, since neithe s 1 no t 1 lie in Y 1 X 2, S 1 Y 1 X 2. It is also impossible that S 1 = Y 1 Y 2, since then eithe S 1 S 2, o S 1 S 2 =, contadicting ou initial assumption. Similaly, sets Y 1 X 2, X 1 X 2, and Y 2 ae disconnected in G \ 2. X 1 X 1 X 2 1 X 2 2 Y 2 Y 1 Y 1 X 2 2 Y 2 We claim that S 2 = Y 2 must hold. Indeed, Y 1 X 2 does not contain s 2 o t 2, so S 2 Y 1 X 2. It is also impossible that S 2 = X 1 X 2, since then S 2 X 1 S 1 must hold, contadicting ou initial assumption. To summaize, we have shown that S 1 = X 1, and S 2 = Y 2 must hold. But then, by the definition of the sets S i, c(y 1 Y 2 ) > c(s 1 ) = c(x 1 ), and c(x 1 X 2 ) > c(s 2 ) = c(y 2 ). Theefoe, c(y 1 Y 2 ) + c(x 1 X 2 ) > c(x 1 ) + c(y 2 ), which is impossible. Finally, we show that we can petub the costs of the vetices so that all subsets have diffeent costs. Let δ = min c(a) c(b). A,B V : c(a) c(b)>0 We assign a new cost c u to evey vetex u unifomly at andom fom the inteval [c u, c u + δ/(2 V )]. Note that with pobability 1, the costs of evey two distinct vetex subsets will be diffeent. We find a family of vetex cuts (S i, i, T i ) w..t. the costs c v. We veify that i is a minimum cost 20

22 sepaato fo s i and t i with espect to the oiginal costs c v. Assume fo contadiction that thee is a sepaato fo s i and t i, with c( ) < c( i ). Then c( i ) c( i ) c( ) + δ > c( ) + δ c( ), 2 V which contadicts to the fact that i is the minimum cost sepaato fo s i and t i w..t. costs c. Poof of Theoem 1.3 In this section, we pove Theoem 1.3, by showing a andomized (2, O(log 5/2 log log ) dk) biciteia 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 O(k). 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 D(S, 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 E(S,V \(S )) w e, whee E(S, 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 and the k-oute vetex spasity of the gaph G is: { } Υ ( ) (S), D(S, V \ (S )) { Ψ (k) (G) = min (S)} Ψ(k) S V 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 O(k) to within a facto of α ALN (), as we show in the next theoem. Theoem 5.3. Thee is a andomized algoithm that finds, in time n O(k), disjoint subsets S, V of vetices, with k 1 such that Υ ( ) (S) α ALN () Ψ (k) (G) D(S, 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 5.3 to find an appoximate k-oute vetex spasest cut. The algoithm is summaized in Figue 5. 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 21

23 Input: A weighted gaph G(V, 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 ( ) (S) α ALN () Ψ (2k 1) (G) D(S, V \(S )) using Theoem Let E 0 = E(U, 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 5: Bi-citeia appoximation algoithm fo VC-kRC in time n O(k). (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 5.4. ( ) 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 5.5. 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 22

24 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 c(v) = 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 5.5 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 pove a countepat of Lemma 4.2. Lemma 5.6. 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 D(U j, R j ) 8 log. Poof. The poof of closely follows the poof of Lemma 4.2. Let S = {S 1,..., S } be the family of vetex subsets fom Lemma 5.5, and assume that the vetex cut coesponding to set S i S is (S i, i, T i ). Family P will contain two type of vetex subsets. Subset U j is a subset of the fist type iff U j = S i fo some S i S. In this case, we set Λ j = i, and the coesponding cut (U j, Λ j, R j ) = (S i, i, T i ). It is easy to see that the fist condition of the lemma will hold fo vetex subsets of this type. Subset U j of vetices is a subset of the second type iff U j = S i \ (S i i ) fo some S i, S i S, whee S i S i. In this case, we set Λ j = i i, and R j = V \ (U j Λ j ). Notice that if e = (u, v) E(H) has u U j, v U j, then v Λ j must hold. Indeed, if v S i, then since (S i, i, T i ) is a valid vetex cut, v i must hold. Othewise, if v S i, but v i, then v S i must hold, and since (S i, i, T i ) is a valid vetex cut, u i must hold, which is impossible. Theefoe, (U j, Λ j, R j ) is a valid vetex cut. Moeove, Λ j = i i 2(k 1), and so the fist condition of the lemma holds. We now show how to define the family P, so that the second condition of the lemma is satisfied as well. We assume w.l.o.g. that S does not contain two copies of the same set: othewise, we simply emove copies of the same set, until just one copy emains in S. Fo evey set S i S, let D (S i ) = S j S: D(S j, T j ) D, and let q(s i ) = (D(S i, T i ) S j S i D ) \ D (S i ). As befoe, we patition the set S as follows: fo x = 1,..., log 2 + 1, let S x = { S i S 2 x 1 q(s i ) < 2 x}. S i S x q(s i ) =, we can choose an index x, such that S i S x q(s i) Since log 2 +1 x=1 2 log. We say that a set S i is good if it belongs to S x. Conside the decomposition foest F fo the good sets S i : the nodes of the foest ae the sets of S x, and S i is the paent of S j iff S j S i, and thee is no othe set S l S x with S j S l S i. Let S be the subset of nodes of F with at 23

25 most one child. Note that S S x /2 S i S x q(s i) 2 2 x On the othe hand, since q(s i ) 2 x 1 fo S i S, 2 x+1 (2 log ). S i S q(s i ) 8 log (4) Fo evey set S i S, we let U i = S i and Λ i = i if S i is a leaf of F; we let U i = S i \ (S j j ) and Λ i = i j if S i has a unique child S j in F. Let R i = V \ (U i Λ i ). If set U i is of the fist type, then D(U i, R i ) = D(S i, T i ), and so D(U i, R i ) D q(s i ). Othewise, if U i = S i \ S j, then D(U i, R i ) D contains all demand pais in D(S i, T i ) D, except fo pais (x, y) with x S j j, y T i. But since j does not contain any teminals paticipating in pais in D, x S j, y T j and (x, y) D(S j, T j ) must hold. Theefoe, D(U i, R i ) D q(s i ), and so U i P D(U i, R i ) 8 log. Conside the family P = {U 1,..., U p } and the coesponding cuts (U i, Λ i, R i ) as in Lemma 5.6. 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 Υ(Λj) ( ) (U j ) log p j=1 D(U j, R j ) O OPT O Theefoe, thee is an index 1 j p, such that ( dk log Υ (Λ j) (U j ) D(U j, R j ) O ( dk log ( dk log ) OPT. ) 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 OPT. In ode to complete the poof of Theoem 1.3, obseve that w(e 0 ) = Υ ( ) (U), and by Theoem 5.4, ( ) dk log w(e 0 ) α ALN ()Ψ (2k 1) (G) D(U, V \ (U )) O OPT D(U, V \ (U )). Note that we emove all demand pais in D(U, V \ (U )) in step 4 of the algoithm. We can now use Theoem 2.5 with α = O(log α ALN () dk) to conclude that the cost of the solution etuned by the algoithm is bounded by O(log 5/2 log log ) dk OPT. 24

26 6 Algoithms fo 2-oute cuts In this section we pove Theoem 1.5. Since we pove in Section A 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 O(log 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) Υ ( ) (S) (S) = min, V \S: min {D(S), D(V \ (S ))} 1 and the unifom vetex 2-oute spasity of the gaph G is: { Ψ (2) (G) = min (S)} Ψ(2) 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 andomized polynomial time algoithm that finds disjoint subsets S V and V of vetices, with 1 and 0 < D(S), such that Υ ( ) (S) α ARV () Ψ (2) (G) D(S). 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 6. In ode to analyze the algoithm, we stat by showing that it is guaanteed to poduce a feasible solution. Claim 6.2. 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 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. 25

27 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 < D(S), such that Υ ( ) (S) α ARV () Ψ (2) (G) D(S), using Theoem 6.1. Let T = V \ (S ). 3. Let E 0 = E(S, 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 6: Appoximation algoithm fo VC-kRC, k = 2 (weighted case). Theoem 6.3. 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. Poof. We will assume that G is connected: if G is not connected and all teminals lie in one connected component then we just let G to be this connected component; othewise, if some teminals lie in one connected component and othes lie in anothe connected component then Ψ (2) (G) = 0 and we ae done. Conside the gaph H = G \ E. By the optimality of E, gaph H is connected. Since E is a solution to the VC-kRC poblem with k = 2, fo evey demand pai (s i, t i ), thee is a one-vetex subset i of vetices sepaating s i fom t i in H. Conside the block decomposition of H. Recall that a block B of H is a maximal 2-vetex connected subgaph of H. Evey pai (B 1, B 2 ) of distinct blocks shae eithe no vetices o only one vetex, and in the latte case, this vetex is a cut vetex. Let B be the set of all blocks, and let S be the set of all cut vetices of H. The block tee B(H) of H is a bipatite gaph, with pats B and S, in which a block B is connected to a cut vetex u iff u lies in B. We assign costs c(u) to each node u of the tee B(H) as follows. If u is a cut vetex of H, let c(u) = D u (the numbe of demand pais in which u paticipates); if B is a block, then c(b) = D(V (B) \ S). Then each vetex u S contibutes D u to the cost of u in B(H); each vetex u V \ S contibutes D u to the cost of the block that contains u. Thus the total cost of all vetices in B(H) is exactly 2. Note that the cost c(u) of a cut vetex u is at most (since thee ae demand pais); the cost c(u) of a block B is also at most since fo evey i [] the block B contains at most one of the teminals s i and t i. We find a node x in the tee B(X) with the following popety: if we oot the tee B(H) at 26

28 x, then fo each child x of x, the cost of the subtee ooted at x is at most (half of the cost of B(H)). In ode to find such a node x, stat with an abitay node x 0 = x, and oot the tee B(H) at x. As long as x has a child x, such that the cost of the sub-tee ooted at x is moe than, we set x = x and continue. Since duing this pocess we always move down the tee ooted at x 0, it is guaanteed to teminate. It is easy to veify that the node x at which the pocess teminates has the desied popeties. Suppose fist that x is a cut vetex of H. Let T 1,..., T p be the set of the subtees of B(H) ooted at the childen nodes of x, soted by thei cost, with T 1 being the most expensive subtee and T p the cheapest one. Note that p i=1 c(t i) = 2 c(x). Let j be the maximal index j such that j i=1 c(t i) < /2. By ou choice of x, we have 1 j < p. Then j i=1 c(t i ) j j +1 j + 1 i=1 c(t i ) 4. Let S be the set of all vetices of H contained in all blocks B V (T i ), fo all 1 i j, and let T = V \ (S {x}). Then D(T ) = 2 D(S) D x /2. Theefoe, Ψ (2) (G) Ψ (2) (S) Υ ({x}) (S) min{d(s),d(t )} OPT /4. Assume now that x is a block, and denote it by B. Let u 1,..., u p be the neighbos of B in tee B(H) (ecall that they must be the cut points of H that lie in B). Fo 1 i p, let T i be the subtee of B(H) ooted at u i, and let S i = V (B ) \ {u i }; T i = V \ (S i {u i }) block B is a node of T i Note that u i is a sepaato vetex fo S i and T i in H. By ou choice of x, we have D(S i ) = c(t i ) c(u i ), and D(T i ) 2 c(t i ) D(S i ). On the othe hand, p i=1 D(S i) = 2 D(B). Fo each 1 i p, thee ae no edges between S i and V \ (S i {u i }) in H, thus we have w(e E G (S i, T i )) = w(e G (S i, T i )) Ψ (2) (G) min {D(S i ), D(T i )} Ψ (2) (G) D(S i ). Summing this inequality ove all 1 i p, we get 2OPT p w(e E G (S i, T i )) i=1 p Ψ (2) (G) d(s i ) Ψ (2) (G). i=1 We conclude that Ψ (2) (G) 4OPT/. 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 6.3, w(e 0 ) α ARV () Ψ (2) (G) D(S) 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 2.4: 27

29 w(e ) w(e 1) + w(e 2) min { a, b} OPT + 2α Solving this ecuence as in Theoem 2.4, we get that w(e ) O(log 3/2 )OPT. 7 A facto k ɛ -hadness fo k-vc-krc In this section we pove Theoem 1.4. We pefom a eduction fom the 3SAT(5) poblem. In this poblem we ae given a 3SAT fomula ϕ on n vaiables and 5n/3 clauses. Each clause contains 3 distinct liteals and each vaiable paticipates in exactly 5 diffeent clauses. We say that ϕ is a Yes-Instance if it is satisfiable. We say that ϕ is a No-Instance with espect to some paamete ɛ, iff no assignment satisfies moe than ɛ-faction of clauses. The following well-known theoem follows fom the PCP theoem [AS98, ALM + 98]. Theoem 7.1. Thee is a constant ɛ : 0 < ɛ < 1, such that it is NP-had to distinguish between Yes-Instances and No-Instances (defined with espect to ɛ) of the 3SAT(5) poblem. We use the Raz veifie fo 3SAT(5) with l paallel epetitions. This is an inteactive poof system, in which two poves ty to convince the veifie that the input 3SAT(5) fomula ϕ is satisfiable. 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, i.e., fo each i : 1 i l, the assignment to x i, etuned by the fist pove, is identical to the assignment to x i, obtained by pojecting the assignment to the vaiables of C i, etuned by the second pove, onto x i. (We assume that the answes sent by the second pove always satisfy the clauses appeaing in its quey). The following theoem is obtained by combining the PCP theoem with the paallel epetition theoem [Raz98]. Theoem 7.2 ( [AS98, ALM + 98, Raz98]). Thee exists 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 A(q) be the collection of all the possible answes to q (if q is a quey of the second pove, then A(q) only contains answes that satisfy all the clauses of the quey). Let A = 2 l, A = 7 l. Then fo each q Q 1, A(q) = A, and fo each q Q 2, A(q ) = A. Given a andom sting R, let q 1 (), q 2 () be the 28

30 ... queies sent to the fist and the second pove espectively, when the veifie chooses. Fo each q Q 1, let N(q) = {q Q 2 R : q 1 () = q, q 2 () = q }, and fo each q Q 2, let N(q ) = {q Q 1 R : q 1 () = q, q 2 () = q }. Notice that fo all q Q 1, N(q) = 5 l, and fo all q Q 2, N(q ) = 3 l. Constuction: We now tun to descibe ou eduction. Fo each quey q Q 1 of the fist pove, fo each answe a A(q), we have an edge e(q, a), whose endpoints ae denoted by v(q, a), u(q, a), and whose cost is (5/3) l. We will think of v(q, a) as the fist endpoint of e(q, a) and of u(q, 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 A(q), thee is an edge e(q, a) = (v(q, a), u(q, a)), of cost 1. As befoe, v(q, a) is called the fist endpoint and u(q, a) the second endpoint of e(q, a). Let E 0 be the set of all esulting edges. Fo each q Q, let V (q) = {v(q, a), u(q, a) a A(q)}. 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 A(q) 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 e(q, b j (a i )) to the fist endpoint of edge e(q, b j+1 (a i )). We will efe to v(q, b 1 (a i )) as the fist vetex on path P i, and to u(q, b zi (a i )) as the last vetex. Next, we connect the souce s() to the fist vetex of e(q, a 1 ) and the fist vetex of P 1. We also connect the second vetex of e(q, 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 e(q, a i+1 ) and P i+1, and the second vetex of e(q, a i ) to the fist vetices of e(q, 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: {e(q, a) a A(q)} {e(q, a ) a A(q )}. Then G() is an almost layeed gaph, whee fo each 1 i A, laye i consists of the edge e(q 1 (), a i ) and of the path P i (see Figue 7). 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 e(q, a), e(q, a ) of edges, whee a and a ae matching answes to queies q and q, espectively. P i b 1 (a i ) b 2 (a i ) b zi (a i ) a 1 a 2 a A s() t() P 1 P 2 P A Figue 7: 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() = N(q 1 ()) N(q 2 ()), and let U() = q N() V (q). Notice that U() = N(q 1()) 7 l + N(q 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 29