Improved Region-Growing and Combinatorial Algorithms for k -Route Cut Problems
IImproved Region-Growing and Combinatorial Algorithms for k -Route Cut Problems Guru Guruganesh ∗ Laura Sanita † Chaitanya Swamy ‡ Abstract
We study the k -route generalizations of various cut problems, the most general of which is k -routemulticut ( k - MC ) problem, wherein we have r source-sink pairs and the goal is to delete a minimum-costset of edges to reduce the edge-connectivity of every source-sink pair to below k . The k -route extensionsof multiway cut ( k - MWC ), and the minimum s - t cut problem ( k - ( s, t ) - Cut ), are similarly defined. Wepresent various approximation and hardness results for k - MC , k - MWC , and k - ( s, t ) - Cut that improvethe state-of-the-art for these problems in several cases. Our contributions are threefold. • For k -route multiway cut , we devise simple, but surprisingly effective, combinatorial algorithms thatyield bicriteria approximation guarantees that markedly improve upon the previous-best guarantees. • For k -route multicut , we design algorithms that improve upon the previous-best approximation fac-tors by roughly an O ( √ log r ) -factor, when k = 2 , and for general k and unit costs and any fixedviolation of the connectivity threshold k . The main technical innovation is the definition of a new,powerful region growing lemma that allows us to perform region-growing in a recursive fashion eventhough the LP solution yields a different metric for each source-sink pair, and without incurring an O (log r ) blow-up in the cost that is inherent in some previous applications of region growing to k -route cuts. We obtain the same benefits as [15] do in their divide-and-conquer algorithms, andthereby obtain an O (ln r ln ln r ) -approximation to the cost. We also obtain some extensions to k -route node-multicut problems. • We complement these results by showing that the k -route s - t cut problem is at least as hard toapproximate as the densest- k -subgraph ( D k S ) problem on uniform hypergraphs. In particular, thisimplies that one cannot avoid a poly( k ) -factor if one seeks a unicriterion approximation, withoutimproving the state-of-the-art for D k S on graphs, and proving the existence of a family of one-wayfunctions. Previously, only NP -hardness of k - ( s, t ) - Cut was known.
The problem of finding minimum size cuts for a given graph has a rich history in the field of combinatorialoptimization, with a wide range of applications in logistics, transportation and telecommunication systems.One key problem of interest is that of disconnecting a given set of node pairs in a network by removingedges at minimum cost. Formally, in the multicut problem, we are given an undirected graph G = ( V, E ) with nonnegative edge costs { c e } e ∈ E and pairs of nodes ( s , t ) , . . . , ( s r , t r ) called source-sink pairs orcommodities, and we seek a minimum-cost set of edges whose removal disconnects every s i - t i pair. Twospecial cases of this problem have by themselves attracted widespread attention: (i) the celebrated minimum ∗ [email protected] . Department of Computer Science, Carnegie Mellon University, Pittsburgh, USA. Work donewhile the author was an undergraduate research assistant at the University of Waterloo under Prof. Sanita. † [email protected] . Dept. of Combinatorics and Optimization, Univ. Waterloo, Waterloo, ON N2L 3G1. ‡ [email protected] . Dept. of Combinatorics and Optimization, Univ. Waterloo, Waterloo, ON N2L 3G1. Sup-ported in part by NSERC grant 327620-09, an NSERC Discovery Accelerator Supplement Award and an Ontario Early ResearcherAward. a r X i v : . [ c s . D S ] O c t - t cut problem, which is the special case when r = 1 ; and (ii) the multiway cut problem [12], whereevery pair of nodes from a given set T ⊆ V of terminals forms a commodity. These cut problems andtheir variants have been widely studied in terms of hardness and approximation (see, e.g., [36, 37]), havenumerous direct applications (e.g., identifying bottlenecks in a network), and algorithms for them serve asimportant primitives in the design of divide-and-conquer algorithms (see, e.g., [15, 34]) and find applicationin diverse settings such as image segmentation, VLSI design and network routing (see, e.g., [27, 29, 6, 32]).We study a natural generalization of the above cut problems motivated by the fact that in various settings,we are not interested in a complete disconnection of our terminals but rather in reducing their connectivitybelow a certain threshold. Specifically, in the k -route multicut ( k - MC ) problem, the input is a multicutinstance and an integer k ≥ ; the goal is to find a minimum-cost set F ⊆ E of edges so that there are atmost ( k − edge-disjoint s i - t i paths in ( V, E \ F ) for all i = 1 , . . . , r . We define the k -route multiway cut ( k - MWC ), and the k -route ( s, t ) -cut ( k - ( s, t ) - Cut ) problems analogously.The study of k -route cut problems can be motivated from various perspectives. One motivation comesfrom the fact that k -route cuts are dual objects to k -route flows [21], which can be seen as a robust or fault-tolerant version of flows where we seek to send traffic along tuples of k edge-disjoint paths. A k -route cutestablishes an upper bound on the value (suitably defined) of the maximum k -route flow, and can thus be seenas identifying the bottleneck in a network when we seek a certain level of robustness. k -route cut problemscan also be directly motivated as abstracting the problem of an attacker who seeks to reduce connectivityin a given network while incurring minimum cost. Viewed from this perspective, k -route cut problems areclosely related to network interdiction problems, which typically consider the complementary objective ofminimizing source-sink connectivity subject to a budget constraint on the edge-removal cost [30, 38, 39].The k -route cut problems are at least as hard as their 1-route counterparts. Multicut and multiway cutare APX -hard [12], with the former not admitting any constant-factor approximation assuming the unique-games conjecture [9], and k - ( s, t ) - Cut is NP -hard; hence, we focus on approximation algorithms. Moreover,as highlighted in [10, 5, 23, 11, 22], k -route cut problems turn out to be much more challenging than their1-route counterparts, especially for non-constant k , so (as in [11]) we consider bicriteria approximationguarantees. (This is further justified by our hardness result for k - ( s, t ) - Cut in Section 4.) We say that asolution F ⊆ E is an ( α, β ) -approximation for the given k - MC instance if (cid:80) e ∈ F c e is at most β times theoptimal value, and ( V, E \ F ) contains at most α ( k − edge-disjoint s i - t i paths for all i = 1 , . . . , r . Our results.
We develop various approximation and hardness results for k - MC , k - MWC and k - ( s, t ) - Cut that improve upon the current-best approximation and hardness results in several cases.In Section 2, we consider the k -route multiway cut problem. We devise an (cid:0) O (1) , O (1) (cid:1) -approximationfor k - MWC with unit costs (Theorem 2.2), and an (cid:0) O (1) , O (log r ) (cid:1) -approximation with general costs (The-orem 2.4), where r = | T | . The previous-best guarantees for k - MWC (for general k ) are those that followfrom the results of Chuzhoy et al. [11] for k - MC , namely, an (cid:0) O (1) , O (log . r ) (cid:1) -approximation for unitcosts and an (cid:0) O (log r ) , O (log r ) (cid:1) -approximation for general costs. Thus, our guarantees constitute a sig-nificant improvement in the state-of-the-art for k - MWC . We also show that the special case where T = V ,which we call k -route all-pairs cut , is APX -hard for k ≥ (Appendix A). (For k = 1 , , it is easy to see thatall-pairs k -route cut is polytime solvable.)In Section 3, we design algorithms for the k -route multicut problem. We achieve approximation ratiosof O (ln r ln ln r ) for 2- MC , and (cid:0) γ, O ( γ ( √ γ − ln r ln ln r ) (cid:1) for k - MC with unit costs. In contrast, Chuzhoyet al. [11] obtain approximation ratios of O (log . r ) for 2- MC , and (cid:0) γ, O ( log . r min { ,γ − } ) (cid:1) for k - MC withunit costs. Thus, for any fixed γ (i.e., independent of k and r ), our results improve upon the previous-bestguarantees for these cases in [11] by roughly an O ( √ log r ) -factor. (Setting γ = kk − , our guarantee andthe one in [11] become unicriterion approximations that are incomparable.) In contrast to the algorithmsin [11], which rely on approximations to suitable variants of sparsest cut, we devise rounding algorithms fora natural LP-relaxation for k - MC , and our guarantees therefore also translate to integrality-gap results. In2ection 5, we consider some extensions to k -route node-multicut problems.Complementing the above results, we show in Section 4 that k - ( s, t ) - Cut is at least as hard as the densest k -subgraph ( D k S ) problem: a ρ -approximation for k - ( s, t ) - Cut yields a (2 ρ λ ) -approximation for D k S on λ -uniform hypergraphs (Theorem 4.2). The latter problem is hard to approximate within an n (cid:15) -factor,for some constant (cid:15) , for all λ ≥ , unless a certain family of one-way functions exists [3]. This impliesthat obtaining a unicriterion O (cid:0) k (cid:15) polylog( n ) (cid:1) -approximation (even) for k - ( s, t ) - Cut for some constant (cid:15) would improve the state-of-the-art for the notoriously hard densest k -subgraph problem on graphs, andimply the existence of certain one-way functions. Previously, only NP -hardness of k - ( s, t ) - Cut was known,as a consequence of the fact that certain NP -hard unbalanced graph partitioning problems [20, 28] can becast as special cases of k - ( s, t ) - Cut . Our techniques.
Our algorithms for k - MWC are combinatorial, and rely on the following simple, butquite useful observation: if F ⊆ E is feasible, then G = ( V, E \ F ) has a multiway cut with at most ( k − r − edges (Claim 2.1). Using this, we show that we can identify a terminal t i ∈ T , a t i -isolatingcut, and a set of edges of cost O (cid:0) OPT | T | (cid:1) whose removal causes the t i -isolating cut to have O ( k − edges.We include these edges, drop t i from T , and repeat, which naturally yields an O (log r ) -approximation in thecost. The improvement for unit costs stems from the stronger property that either the minimum multiway-cut in G has cost O ( OPT ) , or there is some t i -isolating cut of value O ( k − ; thus, we may now dropterminals incurring zero cost , which results in an improved O (1) cost-approximation.Interestingly, [5] use a similar approach to obtain an (cid:0) O (1) , O (1) (cid:1) -approximation for single-source k - MC with unit costs and they remark that such an approach is unlikely to work for k - MWC because thereare examples where every pair of terminals is k − -edge connected but the optimal multiway cut valueis Ω( r ) · OPT . Thus, a useful insight to emerge from our work is that whereas a -factor violation in thepairwise terminal connectivity does not ensure that the multiway cut value is O ( OPT ) , a (2 + ε ) -factorviolation in connectivity does, for any ε > .Our algorithms for k - MC are based on rounding an optimal solution to a natural LP-relaxation of theproblem. This is technically the most sophisticated part of the paper. The main technique that we use is region growing . The idea is to view the LP solution as a metric, grow a suitable ball in this metric andprove a region-growing lemma showing that the cost of the ball-boundary edges can be charged to the ball-volume, where volume measures the contribution to the LP objective from the edges inside the ball. Thiswas introduced by [27, 19] in the context of the sparsest cut and multicut problems, and Even et al. [15],building upon the work of Seymour [33], extended the technique to obtain improved guarantees for variousdivide-and-conquer algorithms that involve recursive applications of region growing. However, in contrastwith various applications of region growing considered in [27, 19, 16, 15], the difficulty in the k -routemulticut problem stems from the fact that an LP-solution yields a different metric for each source-sink pair instead of a single common metric that can be applied in the region-growing process. (In particular, k - MC does not fall into the divide-and-conquer framework of Even et al. [15].) Although [5, 23, 22] adaptedthe region-growing lemma in [27, 19] to the 2-route, 3-route, and the k -route single-source settings, theirapproach seems incapable of obtaining any thing better than an O (log r ) -approximation—one loses one log -factor due to region growing and another due to recursion—which is worse than the guarantees in [11].(In fact, [11] abandoned the region-growing approach and used a greedy set-cover strategy to obtain theirimprovements over [5, 23, 22].)Our chief technical novelty is to prove a region-growing lemma (see Lemmas 3.1 and 3.3) applicableto settings with different metrics, that is inspired by, but more general, than the analogous lemma in [15],and much more sophisticated than the one used in [5, 23, 22]. This lemma, coupled with a subtle insightabout the metrics derived from the LP solution, allows us to obtain the same kind of savings in our recursiveregion-growing algorithm that Even et al. [15] obtain (via their region-growing lemma) in their divide-and-conquer algorithms; this yields our improved approximation guarantees. We believe that our region-growing3emma and its application in the context of different metrics are tools of independent interest that will findfurther application in the study of cut problems.The hardness proof for k - ( s, t ) - Cut dovetails the hardness proof in [11] for the vertex-connectivity ver-sion of k - ( s, t ) - Cut (where we want to decrease the s - t vertex connectivity to below k ), who reduce from the small-set vertex expansion ( SSVE ) problem, which they show is D k S -hard. We observe that this reductionimmediately implies the same hardness for k - ( s, t ) - Cut on a directed graph , and combine this with a usefultrick from [8] that allows us to move from digraphs to undirected graphs. The idea is to take the digraphused in the hardness proof, remove edge directions, and add some extra nodes and expensive edges so thatthe residual digraph obtained after sending a partial s - t flow along the expensive edges essentially coincideswith the digraph used in the hardness proof. Related work.
Standard (i.e., 1-route) cut problems have been extensively studied; we refer the reader tothe textbooks [2, 36, 37] for more information.The study of k -route flow and k -route cut problems was initiated by Kishimoto [21], and has sincereceived much attention in the theoretical Computer Science community [7, 10, 5, 23, 22, 11]. Bruhn etal. [7] gave a k − -approximation for single-source k - MC with unit costs, whereas [10, 5, 23] obtainedefficient polylogarithmic approximation results for k - MC with small values of k . Subsequently, Chuzhoy etal. [11] obtained the first non-trivial results for k - MC with arbitrary k in the form of bicriteria approximationguarantees. Independently, Kolman and Scheideler [22] obtained an O (cid:0) exp( k ) polylog( r ) (cid:1) -approximationfor single-source k - MC (with general costs). As shown by our hardness result for k - ( s, t ) - Cut in Section 4,the move to bicriteria approximations is necessary unless one incurs a poly( k ) -factor in the approximation.As noted earlier, k -route cut problems and network interdiction problems (see, e.g., [30, 38, 39, 13]and the references therein) can be viewed as complementary problems. For instance, in the maximum-flowinterdiction problem ( MFIP ) we are given edge capacities in addition to edge costs, and the goal is tominimize the maximum s - t flow in the graph remaining after removing edges of total cost at most a givenbudget. MFIP with unit capacities is thus complementary to k - ( s, t ) - Cut , and bicriteria guarantees for onetranslate to the other. Unit-capacity
MFIP is known to be polytime solvable for planar graphs [30, 39]. Dinitzand Gupta [13] propose a general framework for attacking packing interdiction problems. However, theirresults do not quite apply to
MFIP (since phrasing max-flow in terms of edge-flows destroys the packingproperty, and phrasing it in terms of path-flows yields an interdiction problem where one removes paths). k -route multiway cut Recall that in the k -route multiway cut ( k - MWC ) problem, we are given a set T = { t , . . . , t r } ⊆ V ofterminals and we seek to remove a minimum-cost set of edges so that the edge-connectivity between any twoterminals is less than k . The case k = 1 is the multiway cut problem, which is known to be APX -hard [12]even with unit edge costs. We devise an (cid:0) O (1) , O (1) (cid:1) -approximation for k - MWC with unit costs, and an (cid:0) O (1) , O (log r ) (cid:1) -approximation with general costs. These improve upon the previous-best guarantees (forgeneral k ) of (cid:0) O (1) , O (log . r ) (cid:1) for unit costs, and O (cid:0) O (log r ) , O (log r ) (cid:1) for general costs due to [11].Remark 2.5 shows that our guarantees also translate to integrality-gap bounds for a suitable LP-relaxation.Let O ∗ denote the optimal set of edges, and let G = ( V, E \ O ∗ ) be the remainder graph. Let k (cid:48) = k − .Our algorithms are quite easy to describe and analyze. We first prove a simple claim about G . Claim 2.1
There is a set E of edges of G with (cid:12)(cid:12) E (cid:12)(cid:12) ≤ k (cid:48) ( r − such that O ∗ ∪ E is a multiway cut in G . Proof :
Compute a minimum t - t cut F in G , where F is a set of edges. By the definition of G , | F | ≤ k (cid:48) .Removing F from G creates at least two components. We can now recurse in each connected component,and after computing at most r − min cuts, each terminal will be in a different connected component.4he idea behind the algorithm for unit costs is the following. Claim 2.1 shows that the optimal multiwaycut in G would be a good approximation to k - MWC if | E | = O (cid:0) | O ∗ | (cid:1) . Otherwise, there is a multiway cutin G of cost O ( k (cid:48) r ) , and so there is some terminal (in fact Ω( r ) terminals) that has an isolating cut in G ofsize O ( k (cid:48) ) ; we simply remove this terminal from T and repeat this process. Theorem 2.2
There is a (cid:0) γ, γγ − (cid:1) -approximation algorithm for k - MWC with unit costs for any γ > . Proof :
For all i , compute a minimum t i -isolating cut F i . It is well known that, even with non-unit costs, (cid:80) ri =1 c ( F i ) is a 2-approximation to the minimum multiway cut [12]. In particular, C = (cid:80) ri =1 | F i | ≤ | O ∗ | + 2 | E | (by Claim 2.1). If C ≥ γk (cid:48) r , then we have | O ∗ | ≥ C − k (cid:48) r ≥ C (cid:0) − γ (cid:1) , so taking the unionof the F i s yields a γγ − -approximation. Otherwise, there is some t i such that | F i | < γk (cid:48) , so we can simplyremove t i from T and decrease r , and repeat.We remark that the number of iterations can be reduced to log r at the expense of increasing the con-nectivity to γk (cid:48) , since must be at least r/ terminals such that | F i | ≤ γk (cid:48) . Remark 2.3
The condition γ > above is tight. This follows from an example in [5] where every pair ofterminals is k (cid:48) -edge connected but the minimum multiway cut yields an Ω( r ) -approximation.To generalize this algorithm to general edge costs, assume for now that we know OPT = c ( O ∗ ) . Unlikein the unit edge-cost case where we could make progress by dropping terminals while incurring zero cost,here we will need to incur cost O (cid:0) OPT r (cid:1) to drop a terminal (or incur cost O ( OPT ) to drop r/ terminals).This naturally leads to an O (log r ) -approximation in the cost. Let H r := 1 + + . . . + r = O (log r ) . Theorem 2.4
There is a (cid:0) γ, γγ − H r (cid:1) -approximation algorithm for k - MWC with general edge costs, for any γ > . Proof :
Let T (cid:48) initialized to T denote the current terminal set and r (cid:48) ← r . Let F initialized to ∅ denote theset of edges removed. Let α = γ − . While | T (cid:48) | > , we do the following. Set c (cid:48) e = min { c e , α OPT k (cid:48) r (cid:48) } forevery edge e . Note that the c (cid:48) -cost of the minimum multiway cut is at most c (cid:48) ( O ∗ ∪ E ) ≤ OPT + k (cid:48) r (cid:48) · α OPT k (cid:48) r (cid:48) .For every terminal t ∈ T (cid:48) , compute a minimum c (cid:48) -cost t -isolating cut F t . Then, we have (cid:80) t ∈ T (cid:48) c (cid:48) ( F t ) ≤ α ) OPT . So there is some t ∈ T (cid:48) such that c (cid:48) ( F t ) ≤ α ) OPT r (cid:48) . The number of edges in F t with c e > α OPT k (cid:48) r (cid:48) is less than α ) α k (cid:48) = γk (cid:48) . We add edges in F t with c e ≤ α OPT k (cid:48) r (cid:48) to F . This incurs cost atmost c (cid:48) ( F t ) ≤ α ) OPT r (cid:48) , and ensures that t is less than γk (cid:48) connected to every other terminal in T (cid:48) in theremaining graph. We now set T (cid:48) ← T (cid:48) \ { t } , r (cid:48) ← r (cid:48) − , and repeat the above process.Clearly, every pair of terminals is at most γk (cid:48) connected in ( V, E \ F ) . Also, c ( F ) ≤ (cid:80) r (cid:48) = r α ) OPT r (cid:48) = OPT · γγ − · H r .Finally, we can eliminate the need for knowing OPT as follows. Given a guess C of OPT , if at someiteration we have (cid:80) t ∈ T (cid:48) c (cid:48) ( F t ) > α ) C then we know that C <
OPT ; otherwise, we obtain a solutionof cost at most C · γγ − · H r . So we can try powers of (1 + (cid:15) ) to find the smallest C such that the latter casehappens; this blows up the approximation in cost by at most a (1 + (cid:15) ) -factor. Remark 2.5 (LP-relative bounds)
The guarantees in Theorems 2.2 and 2.4 also translate to integrality-gapbounds for the following LP-relaxation of k - MWC . Let P ij be the collection of all t i - t j paths. min c T x s.t. (cid:88) e ∈ P ( x e + y e ) ≥ ∀ t i , t j ∈ T, P ∈ P ij ; (cid:88) e y e ≤ k (cid:48) ( r − x, y ≥ . (P’)5laim 2.1 implies that (P’) is indeed a valid relaxation of k - MWC . Let ( x, y ) be an optimal solution to (P’)and OPT P’ be its value. Then, for any λ ≥ , for the cost function c (cid:48) e = min { c e , λ } , there is a fractionalmultiway-cut of c (cid:48) -cost at most OPT P’ + λ (cid:80) e y e . Also, if F i is a minimum c (cid:48) -cost t i -isolating cut then wehave (cid:80) i c (cid:48) ( F i ) ≤ (cid:0) OPT P’ + λ (cid:80) e y e (cid:1) . (This follows since an optimal solution to the multiway-cut LP isknown to be half-integral (see, e.g., [36]); this implies that 2(cost of an optimal solution) is at least (cid:80) i (costof a minimum t i -isolating cut).) This implies that we can replace | O ∗ | and | E | in the proof of Theorem 2.2by OPT P’ and (cid:80) e y e , and OPT in the proof of Theorem 2.4 by
OPT P’ , and all the arguments go through. The all-pairs case.
This is the special case of k - MWC where T = V . To our knowledge, this k -routeall-pairs cut problem has not been explicitly studied before. When k = 1 , the all-pairs problem is trivial asthe remainder graph cannot contain any edge. When k = 2 , this problem is still in P as the remainder graphis a maximum-cost spanning forest. We prove that the problem is APX -hard for all k ≥ (see Appendix A),thus resolving the complexity (with respect to polytime solvability) of k -route all-pairs cut. The all-pairsproblem can also be stated in terms of properties required of the remainder graph. For example, in -routeall-pairs cut, we seek a minimum-cost edge set such that the remainder graph does not contain a diamond asa minor. Interestingly, this is equivalent to requiring that the remainder graph be a maximum-weight cactus ,which is a graph where every edge lies in at most one cycle. As noted above, this problem is APX -hard.But we observe that this problem admits an O (1) -approximation as a consequence of the results of Fioriniet al. [17]; see Appendix A. k -route multicut We now consider general k -route multicut ( k - MC ): given source-sink pairs/commodities ( s , t ) , . . . , ( s r , t r ) ,we want to find a minimum-cost set of edges whose removal reduces the s i - t i edge connectivity to less than k for all i = 1 , . . . , r . We consider the following LP-relaxation of the problem, which was also consideredby Barman and Chawla [5]. Throughout e indexes the edges in E , and i indexes the commodities. Let P i denote the collection of all (simple) s i - t i paths in G . min (cid:88) e c e x e s.t. (cid:88) e ∈ P ( x e + y ie ) ≥ ∀ i, ∀ P ∈ P i ; (cid:88) e y ie ≤ k − ∀ i ; x, y ≥ . (P)Let (cid:0) x, { y i } (cid:1) denote an optimal solution to (P), and OPT be its value. We show that (cid:0) x, { y i } (cid:1) can berounded to yield an O (ln r ln ln r ) -approximation when k = 2 , and a bicriteria (cid:0) γ, O (cid:0) γ ( √ γ − ln r ln ln r (cid:1)(cid:1) -approximation for k - MC with unit edge costs, for any γ > . Notably, our cost-approximation is with respectto OPT , and so they translate to integrality-gap upper bounds for (P). Our results improve upon (for anyfixed γ ) the previous-best guarantees for these cases in [11] by roughly a √ log r -factor. The main tool that we leverage is region growing [27, 19, 15]. The idea is to view the LP solution as a metric,grow a suitable ball in this metric and prove that the cost of the ball-boundary edges can be charged to theball-volume, where volume measures the contribution to the LP objective from the edges inside the ball. Themain difficulty in applying this idea to (P) is that, unlike most applications of region growing [27, 19, 16, 15],the LP solution yields a different metric for each commodity. The key technical ingredient and novelty isa new region-growing lemma (see Lemmas 3.1 and 3.3) that is analogous to, but more general, than theone in [15], and much more sophisticated than the one used in [5, 23, 22]. Roughly speaking, we prove thatgiven a current set S of nodes, one can construct a ball around any s i in the ( x + y i ) -metric such that the costof the “boundary x -edges” can be charged to ( x -volume of S ) · ln (cid:0) x -volume of S/x -volume of the ball (cid:1) · n ln r (Lemma 3.3). A subtle insight that helps deal with the complication that different applications ofregion growing involve different commodities and therefore different metrics is the following. Since the x -contribution is common to all ( x + y i ) -metrics, even though we consider different commodity-metrics wecan leverage the above guarantee and obtain the same kind of savings that [15] obtain in their divide-and-conquer algorithms (see Lemma 3.5); this leads to our improved approximation guarantees.We now state our region-growing lemmas in a general form and then apply these to the optimal solution (cid:0) x, { y i } (cid:1) to obtain various useful corollaries. In Section 5, we extend our arguments to prove region-growing lemmas in settings that involve both edge and node lengths. Let n = | V | , m = | E | . Let (cid:96) : V × V (cid:55)→ R ≥ be a metric on V × V . Our algorithm will iteratively focus on certain regions of the graph G .Let S ⊆ V , which is intended to represent the node-set of the current region. Let F ⊆ E , which is intendedto represent the edges that contribute to the volume, and whose cost we care about. Let β ≥ . Let z ∈ V ,and ρ ≥ . • Define B (cid:96) ( z, ρ ) := { v ∈ V : (cid:96) zv ≤ ρ } to be the ball of radius ρ around z . • Let B S(cid:96) ( z, ρ ) := B (cid:96) ( z, ρ ) ∩ S and B S(cid:96) ( z, ρ ) := S \ B (cid:96) ( z, ρ ) . • Define the following volumes: V S,F(cid:96) ( β ; z, ρ ) := β + (cid:88) ( u,v ) ∈ F : u,v ∈ B S(cid:96) ( z,ρ ) c uv (cid:96) uv + (cid:88) ( u,v ) ∈ F : u ∈ B S(cid:96) ( z,ρ ) v ∈ B S(cid:96) ( z,ρ ) c uv (cid:0) ρ − (cid:96) zu (cid:1) V S,F(cid:96) ( β ; z, ρ ) := β + (cid:88) ( u,v ) ∈ F : u,v ∈ B S(cid:96) ( z,ρ ) c uv (cid:96) uv + (cid:88) ( u,v ) ∈ F : u ∈ B S(cid:96) ( z,ρ ) v ∈ B S(cid:96) ( z,ρ ) c uv (cid:0) (cid:96) zv − ρ (cid:1) • For a subset T ⊆ S , let δ SF ( T ) denote { ( u, v ) ∈ F : u ∈ T, v ∈ S \ T } . • Define ∂ S,F(cid:96) ( z, ρ ) := δ SF (cid:0) B S(cid:96) ( z, ρ ) (cid:1) = δ SF (cid:0) B S(cid:96) ( z, ρ ) (cid:1) .When F = E , we drop F from the above pieces of notation (e.g., δ SE ( T ) is shortened to δ S ( T ) ). For H ⊆ E , we use (cid:96) ( H ) to denote (cid:80) e ∈ H (cid:96) e . Lemma 3.1 (Region-growing lemma)
Let F ⊆ E , S ⊆ V , z ∈ V , and ≤ a < b . Let ρ be chosenuniformly at random from [ a, b ) . Then, E ρ c (cid:0) ∂ S,F(cid:96) ( z, ρ ) (cid:1) V S,F(cid:96) ( β ; z, ρ ) ln (cid:16) e V S,F(cid:96) ( β ; z,b ) V S,F(cid:96) ( β ; z,ρ ) (cid:17) ≤ b − a · ln ln (cid:18) e V S,F(cid:96) ( β ; z, b ) V S,F(cid:96) ( β ; z, a ) (cid:19) and (1) E ρ c (cid:0) ∂ S,F(cid:96) ( z, ρ ) (cid:1) V S,F(cid:96) ( β ; z, ρ ) ln (cid:16) e V S,F(cid:96) ( β ; z,a ) V S,F(cid:96) ( β ; z,ρ ) (cid:17) ≤ b − a · ln ln (cid:18) e V S,F(cid:96) ( β ; z, a ) V S,F(cid:96) ( β ; z, b ) (cid:19) (2) Proof :
We abbreviate c (cid:0) ∂ S,F(cid:96) ( z, ρ ) (cid:1) to c ( ρ ) , V S,F(cid:96) ( β ; z, ρ ) to V ( ρ ) and V S,F(cid:96) ( β ; z, ρ ) to V ( ρ ) . Let x − be avalue infinitesimally smaller than x . Let I = { (cid:96) ( s i , v ) : v ∈ V } . Note that V ( ρ ) and V ( ρ ) are differentiableat all ρ ∈ [ a, b ) \ I and for each such ρ , we have d V ( ρ ) dρ = c ( ρ ) and d V ( ρ ) dρ = − c ( ρ ) . Let a = a , a k = b , and7 a , . . . , a k − } = ( a, b ) ∩ I with a < . . . < a k − . Then, ( b − a ) · E ρ c ( ρ ) V ( ρ ) ln (cid:0) e V ( b ) V ( ρ ) (cid:1) = k (cid:88) i =1 (cid:90) a − i a i − d V ( ρ ) V ( ρ ) ln (cid:0) e V ( b ) V ( ρ ) (cid:1) = k (cid:88) i =1 (cid:90) a − i a i − d (ln V ( ρ ))ln (cid:0) e V ( b ) (cid:1) − ln( V ( ρ ))= k (cid:88) i =1 − ln ln (cid:16) e V ( b ) V ( ρ ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) a − i a i − = k (cid:88) i =1 (cid:20) ln ln (cid:16) e V ( b ) V ( a i − ) (cid:17) − ln ln (cid:16) e V ( b ) V ( a − i ) (cid:17)(cid:21) ≤ ln ln (cid:16) e V ( b ) V ( a ) (cid:17) . The final inequality follows since V ( ρ ) increases with ρ , and ln ln (cid:0) e V ( b ) V ( ρ ) (cid:1) decreases with ρ . Similarly, weobtain that ( b − a ) · E ρ (cid:34) c ( ρ ) V ( ρ ) ln (cid:0) e V ( a ) V ( ρ ) (cid:1) (cid:35) = (cid:80) ki =1 ln ln (cid:16) e V ( a ) V ( ρ ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) a − i a i − ≤ ln ln (cid:16) e V ( a ) V ( b ) (cid:17) . Corollary 3.2
Let
F, H ⊆ E , S ⊆ V , z ∈ V , and ≤ a < b . For any α ∈ (0 , , we can efficiently find aradius ρ ∈ [ a, b ) such that c (cid:0) ∂ S,F(cid:96) ( z, ρ ) (cid:1) ≤ − α )( b − a ) · V S,F(cid:96) ( β ; z, ρ ) · ln (cid:16) e V S,F(cid:96) ( β ; z,b ) V S,F(cid:96) ( β ; z,ρ ) (cid:17) · ln ln (cid:16) e V S,F(cid:96) ( β ; z,b ) V S,F(cid:96) ( β ; z,a ) (cid:17) (3) c (cid:0) ∂ S,F(cid:96) ( z, ρ ) (cid:1) ≤ − α )( b − a ) · V S,F(cid:96) ( β ; z, ρ ) · ln (cid:16) e V S,F(cid:96) ( β ; z,a ) V S,F(cid:96) ( β ; z,ρ ) (cid:17) · ln ln (cid:16) e V S,F(cid:96) ( β ; z,a ) V S,F(cid:96) ( β ; z,b ) (cid:17) (4) (cid:12)(cid:12) ∂ S,H(cid:96) ( z, ρ ) (cid:12)(cid:12) < (cid:96) ( H ) α ( b − a ) . (5) Proof :
Suppose we pick ρ uniformly at random from [ a, b ) . Define the following events. Ω := (cid:110) ρ ∈ [ a, b ) : (cid:12)(cid:12) ∂ S,H(cid:96) ( z, ρ ) (cid:12)(cid:12) ≥ (cid:96) ( H ) α ( b − a ) (cid:111) Ω := (cid:26) ρ ∈ [ a, b ) : c (cid:0) ∂ S,F(cid:96) ( z, ρ ) (cid:1) > − α )( b − a ) · V S,F(cid:96) ( β ; z, ρ ) ln (cid:16) e V S,F(cid:96) ( β ; z,b ) V S,F(cid:96) ( β ; z,ρ ) (cid:17) · ln ln (cid:16) e V S,F(cid:96) ( β ; z,b ) V S,F(cid:96) ( β ; z,a ) (cid:17)(cid:27) Ω := (cid:26) ρ ∈ [ a, b ) : c (cid:0) ∂ S,F(cid:96) ( z, ρ ) (cid:1) > − α )( b − a ) · V S,F(cid:96) ( β ; z, ρ ) ln (cid:16) e V S,F(cid:96) ( β ; z,a ) V S,F(cid:96) ( β ; z,ρ ) (cid:17) · ln ln (cid:16) e V S,F(cid:96) ( β ; z,a ) V S,F(cid:96) ( β ; z,b ) (cid:17)(cid:27) For an edge ( u, v ) ∈ E , Pr (cid:2) ( u, v ) ∈ ∂ S,H(cid:96) ( z, ρ )] ≤ (cid:96) uv b − a . Hence, E ρ (cid:104)(cid:12)(cid:12) ∂ S,H(cid:96) ( z, ρ ) (cid:12)(cid:12)(cid:105) ≤ (cid:96) ( H ) b − a , andtherefore Pr[Ω] ≤ α . By Lemma 3.1 and Markov’s inequality, we have that Pr[Ω ] , Pr[Ω ] < (1 − α ) / .Conditioning on Ω c := [ a, b ) \ Ω increases the probability of an event by at most a factor − Pr[Ω] ≤ − α ,so Pr[Ω | Ω c ] , Pr[Ω | Ω c ] < / . Thus, Pr[Ω c ∩ Ω c ∩ Ω c ] > .We argue that Ω c , Ω c , and Ω c are all unions of at most n subintervals of [ a, b ) , and we can find theseefficiently. Since Pr[Ω c ∩ Ω c ∩ Ω c ] > , we can then efficiently find an interval contained in Ω c ∩ Ω c ∩ Ω c (in fact, a non-singleton interval), and hence, find ρ ∈ [ a, b ] satisfying (3)–(5) (in fact, there are infinitelymany such ρ ).There are at most n distinct sets B S(cid:96) ( z, ρ ) that one may encounter as ρ varies in [ a, b ) . For each suchset A , there is an interval [ lb , ub ) such that A = B S(cid:96) ( z, ρ ) for all ρ ∈ [ lb , ub ) . Note that the right-hand-sides (RHSs) of (3) and (4) are continuous and differentiable in ( lb , ub ) , and are monotonic (increasing anddecreasing, respectively) functions of ρ . We call [ lb , ub ) a smooth subinterval of [ a, b ) . By definition, theleft-hand-sides (LHSs) of (3)–(5) are invariant over a smooth subinterval. Hence, Ω c is the union of somesmooth subintervals. Consider a smooth subinterval [ lb , ub ) . By continuity, if some ρ ∈ [ lb , ub ) satisfies (3)or (4), then we can efficiently find the maximal subinterval of [ lb , ub ) (which may be a singleton interval)8uch that all ρ in the subinterval satisfy the given bound. Hence, both Ω c and Ω c are the union of at most n subintervals of [ a, b ) . By trying out the at most n possible subintervals of Ω c ∪ Ω c ∪ Ω c , we can find someinterval contained in Ω c ∩ Ω c ! ∩ Ω c and hence obtain ρ satisfying (3)–(5). Applications of the region-growing lemmas.
To apply the above results to the metrics obtained from (cid:0) x, { y i } (cid:1) , it will be convenient to modify G by subdividing every edge e into r + 1 edges e , e , . . . , e r ,and setting x f = x e for f = e and 0 otherwise, and y if = y ie if f = e i and 0 otherwise. We call e an x -edge, and we call e i , a y i -edge for i = 1 , . . . , r . Clearly, any solution in G yields a solution in thesubdivided graph of the same cost and vice versa, and this holds even for fractional solutions to (P). In thesequel, we work with the subdivided graph. To keep notation simple, we continue to use G = ( V, E ) todenote the subdivided graph, and (cid:0) x, { y i } (cid:1) to denote the above solution in the subdivided graph. Let F bethe set of all x -edges, and H i be the set of all y i -edges for all i = 1 , . . . , r . Consider a commodity i . Let (cid:96) i denote the shortest-path metric of G (i.e., the subdivided graph) induced by the { x e + y ie } edge lengths. Set β = OPT /r . To avoid cumbersome notation, we shorten: B S(cid:96) i ( z, ρ ) and B S(cid:96) i ( z, ρ ) to B Si ( z, ρ ) and B Si ( z, ρ ) respectively V S,F(cid:96) i ( β ; z, ρ ) and V S,F(cid:96) i ( β ; z, ρ ) to V S,xi ( z, ρ ) and V S,xi ( z, ρ ) respectively ∂ S,F(cid:96) i ( z, ρ ) , ∂ S,H i (cid:96) i ( z, ρ ) , and ∂ S,E(cid:96) i ( z, ρ ) to ∂ S,xi ( z, ρ ) , ∂ S,yi ( z, ρ ) , and ∂ Si ( z, ρ ) respectivelyAlso, define V x ( S ) := β + (cid:80) e ∈ E ( S ) c e x e , where E ( S ) is the set of edges having both endpoints in S .Finally, for an integer q ≥ and a set A of edges let c q ( A ) be the cost of all but the q − most expensiveedges of A (so c q ( A ) = 0 if | A | < q ). Lemma 3.3
Let S ⊆ V , z ∈ V , and i be some commodity. Let α ∈ (0 , and q = (cid:6) k − α (cid:7) . We canefficiently find ρ ∈ [0 , such that c q (cid:0) ∂ Si ( z, ρ ) (cid:1) ≤ − α · V S,xi ( z, ρ ) ln (cid:16) e V x ( S ) V S,xi ( z, ρ ) (cid:17) ln ln (cid:0) e ( r + 1) (cid:1) (6) c q (cid:0) ∂ Si ( z, ρ ) (cid:1) ≤ − α · V S,xi ( z, ρ ) ln (cid:16) e V x ( S ) V S,xi ( z, ρ ) (cid:17) ln ln (cid:0) e ( r + 1) (cid:1) . (7) Proof :
We apply Corollary 3.2 taking (cid:96) = (cid:96) i , H = H i , and [ a, b ) = [0 , (and S , z , α as given by thestatement of the lemma, and F to be the set of x -edges). Note that V S,xi ( z, ≤ V x ( S ) , V S,xi ( z, ≤ V x ( S ) , V S,xi ( z, V S,xi ( z, ≤ r + 1 , V S,xi ( z, V S,xi ( z, ≤ r + 1 . Thus, we obtain ≤ ρ < such that c (cid:0) ∂ S,xi ( z, ρ ) (cid:1) satisfies the bounds given by the RHS of (6) and(7) respectively. Moreover, since (cid:96) i ( H i ) ≤ k − , we have that (cid:12)(cid:12) ∂ S,yi ( z, ρ ) (cid:12)(cid:12) < k − α due to (5), and so (cid:12)(cid:12) ∂ S,yi ( z, ρ ) (cid:12)(cid:12) < q . Finally, since edges not in F ∪ H i have zero (cid:96) i -length, ∂ S,xi ( z, ρ ) and ∂ S,yi ( z, ρ ) partition ∂ Si ( z, ρ ) for all ρ . Therefore, c (cid:0) ∂ S,xi ( z, ρ ) (cid:1) ≥ c q (cid:0) ∂ Si ( z, ρ ) (cid:1) , and the lemma follows. Corollary 3.4
Let S ⊆ V . Suppose that s i , t i ∈ S and there are γ ( k − edge-disjoint s i - t i paths internalto S , where γ > . Suppose that c e = 1 for all edges e . We can efficiently find ρ ∈ [0 , such that c (cid:0) ∂ Si ( s i , ρ ) (cid:1) ≤ γ ( √ γ − · V S,xi ( s i , ρ ) ln (cid:16) e V x ( S ) V S,xi ( s i , ρ ) (cid:17) ln ln (cid:0) e ( r + 1) (cid:1) c (cid:0) ∂ Si ( s i , ρ ) (cid:1) ≤ γ ( √ γ − · V S,xi ( s i , ρ ) ln (cid:16) e V x ( S ) V S,xi ( s i , ρ ) (cid:17) ln ln (cid:0) e ( r + 1) (cid:1) roof : We apply Lemma 3.3 with α ∈ (0 , , whose value we will fix later, to obtain ρ ∈ [0 , . Note that B Si ( s i , ρ ) is an s i - t i cut. Let q = (cid:6) k − α (cid:7) . For any s i - t i cut A ⊆ S , we know that c (cid:0) δ S ( A ) (cid:1) ≥ γ ( k − .Therefore, since we have unit edge costs, c (cid:0) δ S ( A ) (cid:1) ≤ c q (cid:0) δ S ( A ) (cid:1) + q − ≤ c q (cid:0) δ S ( A ) (cid:1) · γγ − /α . Plugging inthe bounds for c q ( . ) from (6), (7), we see that the constant factor multiplying the volume terms on the RHSis γ (1 − α )( γ − /α ) . This factor is minimized by setting α = γ − / , which yields the constant factor γ ( √ γ − and completes the proof. The case k = 2 . The algorithm for k = 2 follows a similar template as the algorithm in [5] for 2- MC . However, its analysis resulting in our improved guarantee relies crucially on Lemma 3.3 which isderived from our stronger region-growing lemma. The algorithm proceeds as follows. Given a current set U of nodes and a current set of N source-sink pairs, we repeatedly use Lemma 3.3 to “carve out” disjointregions A , . . . , A h ⊆ U and build a set Z of edges until there is no -edge-connected source-sink pair in U \ ( A ∪ . . . ∪ A h ) . Given A , . . . , A p − , we obtain A p as follows. We choose an s i - t i pair that is at least -edge connected in S = U \ (cid:83) p − q =1 A q , and use Lemma 3.3 with center s i , α = 0 . and set S . Note that k −
1) = 2 = k . We set A p to be B Si ( s i , ρ ) or B Si ( s i , ρ ) , so as to ensure there are at most N/ source-sink pairs inside A p . We add the edges corresponding to c (cid:0) δ S ( A p ) (cid:1) to Z ; Lemma 3.3 ensures that thecost of these edges can be bounded in terms of the volume contained in A p . Having obtained A , . . . , A h this way, we now recurse on each set A p and the source-sink pairs contained in A p , to obtain edge-sets Z , . . . , Z h . The solution we return is Z ∪ ( Z ∪ . . . ∪ Z h ) . A more formal description follows. Algorithm - MCAlg ( U, T = { ( s , t ) , . . . , ( s N , t N ) } ) Input:
A subset U ⊆ V , and a collection T = { ( s i , t i ) } Ni =1 of source-sink pairs, where s i , t i ∈ U for all i = 1 , . . . , N . Output:
A set Z ⊆ E ( U ) such that s i and t i are at most 1-edge-connected in ( U, E ( U ) \ Z ) for all i = 1 , . . . , N .A1. Set S = U , Z = ∅ , S = ∅ , and T (cid:48) = (cid:8) ( s i , t i ) ∈ T : s i and t i are at least -edge-connected in ( S, E ( S )) (cid:9) .A2. If T (cid:48) = ∅ , return Z .A3. While T (cid:48) (cid:54) = ∅ , we do the following.A3.1 Pick some ( s i , t i ) ∈ T (cid:48) .A3.2 Apply Lemma 3.3 with z = s i , α = 0 . and the set S to find a radius ≤ ρ < satisfying (6), (7).A3.3 If B Si ( s i , ρ ) contains at most N/ pairs from T then set A = B Si ( s i , ρ ) , else set A = B Si ( s i , ρ ) .A3.4 Set S ← S ∪ { A } . Add the edges contributing to c (cid:0) δ S ( A ) (cid:1) (i.e., all edges of δ S ( A ) except the most-expensive one) to Z .A3.5 Set S ← S \ A . Update T (cid:48) to be the s i - t i pairs from T that are at least -edge-connected in ( S, E ( S )) .A4. For every set A ∈ S , set Z ← Z ∪ - MCAlg (cid:0) A, { ( s i , t i ) ∈ T : s i , t i ∈ A } (cid:1) .A5. Return Z .The initial call to - MCAlg , which computes the solution we return, is - MCAlg (cid:0) V, { ( s , t ) , . . . , ( s r , t r ) } (cid:1) . Let Z := 2 - MCAlg (cid:0) V, { ( s , t ) , . . . , ( s r , t r ) } (cid:1) . The feasibility of Z follows from the same argumentsas in [5]; Lemma 3.7 gives a self-contained proof. We focus on showing that c ( Z ) ≤ O (ln r ln ln r ) · OPT .Consider the recursion tree associated with the execution of - MCAlg , where each node is labeled witharguments passed in the current invocation of - MCAlg . Define the depth of a subtree of this recursion treeto be the maximum number of edges on a root to leaf path of the subtree. Recall that β = OPT r . Lemma 3.5
Let d be the depth of a subtree of the recursion tree rooted at (cid:0) U ⊆ V, T ⊆ { ( s , t ) , . . . , ( s r , t r ) } (cid:1) .Let Z U = 2 - MCAlg ( U, T ) . We have c ( Z U ) ≤ (cid:0) β |T | + V x ( U ) (cid:1) ln (cid:0) e d r V x ( U ) OPT (cid:1) ln ln (cid:0) e ( r + 1) (cid:1) . roof : The proof is by induction on d . If d = 0 , there is no recursive call; so Z U = ∅ , which satisfies thestated bound. Otherwise, suppose that we make the recursive calls - MCAlg ( A , T ) , . . . , - MCAlg ( A h , T h ) in step A4 to obtain edge-sets Z , . . . , Z h respectively. For p = 1 , . . . , h , let S p be the current set S when set A p was added to S in step A3.4 (so A p ⊆ S p ), and let E p be the edge-set added to Z in this step. Then, Z U = (cid:83) hp =1 ( E p ∪ Z p ) . By the induction hypothesis, c ( Z p ) ≤ (cid:16) β |T p | + V x ( A p ) (cid:17) ln (cid:16) e d − r V x ( A p ) OPT (cid:17) ln ln (cid:0) e ( r +1) (cid:1) .Let vol p = V S p ,xi ( s i , ρ ) if A p = B S p i ( s i , ρ ) and vol p = V S p ,xi ( s i , ρ ) if A p = B S p i ( s i , ρ ) . Note that V x ( A p ) ≤ vol p ≤ V x ( S p ) ≤ V x ( U ) . By Lemma 3.3 and the above upper bounds, we have c ( E p ) = c (cid:0) δ S p ( A p ) (cid:1) ≤ p ln (cid:16) e V x ( U ) V x ( A p ) (cid:17) ln ln (cid:0) e ( r + 1) (cid:1) , and therefore c ( E p ) + c ( Z p ) ≤ (cid:16) β |T p | + vol p (cid:17) ln (cid:16) e d r V x ( U ) OPT (cid:17) ln ln (cid:0) e ( r + 1) (cid:1) . (8)Note that (cid:80) hp =1 vol p ≤ V x ( U ) + β ( h − and (cid:80) hp =1 |T p | ≤ N − h (since each time we create a child of ( U, T ) we remove at least one new ( s i , t i ) pair from T ). So adding (8) over all p = 1 , . . . , h , we obtain that c ( Z U ) ≤ (cid:0) β |T | + V x ( U ) (cid:1) ln (cid:0) e d r V x ( U ) OPT (cid:1) ln ln (cid:0) e ( r + 1) (cid:1) . Theorem 3.6
Algorithm - MCAlg returns a feasible solution of cost at most O (ln r ln ln r ) · OPT . Proof :
Feasibility of Z is shown in Lemma 3.7. Each time we make a recursive call to - MCAlg , thenumber of source-sink pairs involved decreases by at least a factor of 2, so the depth d of the overallrecursion tree is O (log r ) . Since βr + V x ( V ) = (cid:0) r (cid:1) OPT and r V x ( V ) OPT = r + 1 , by Lemma 3.5,this implies that c ( Z ) ≤ O ( OPT ) · (cid:0) ln( r + 1) + O (log r ) (cid:1) ln ln (cid:0) e ( r + 1) (cid:1) . Lemma 3.7
The solution Z returned by - MCAlg is feasible.
Proof :
Suppose for a contradiction that there is some s i - t i pair such that there are (at least) 2 edge-disjoint s i - t i paths P , P in ( V, E \ Z ) . Consider the recursion tree of - MCAlg , and let ( Y, T Y ) be the node furthestfrom the root such that P , P ⊆ E ( Y ) . (Such a node must exist since the root satisfies this property.)Note that there is at least one child ( X, . ) of ( Y, T Y ) such that δ YP ∪ P ( X ) (cid:54) = ∅ . If not and both P and P are contained in some child of ( Y, T Y ) then this contradicts the definition of ( Y, T Y ) . Otherwise, we have P , P ⊆ E ( A ) , where A = Y \ (cid:83) children ( X, . ) of ( Y, T Y ) X . But then we would have processed A in step A3and created at least one child ( A (cid:48) , . ) for some A (cid:48) ⊆ A .We claim that if δ YP ∪ P ( X ) (cid:54) = ∅ for a child ( X, . ) of ( Y, T Y ) , then | δ YP ∪ P ( X ) | ≥ . This is true ifboth s i and t i are in X or if neither of them are in X since then a path crossing X must cross it at leasttwice. Otherwise, X is an s i - t i cut, and since P and P are edge-disjoint s i - t i paths in E ( Y ) , we againhave | δ YP ∪ P ( X ) | ≥ . Among all the children ( X, . ) of ( Y, T Y ) such that δ YP ∪ P ( X ) (cid:54) = ∅ , let ( W, . ) be thechild that was added to S earliest in step A3.4 of - MCAlg ( Y, T Y ) ; let S (cid:48) ⊆ Y be the current set S when W was added. Then, P ∪ P ⊆ E ( S (cid:48) ) , and so | δ S (cid:48) E \ Z ( W ) | ≥ | δ S (cid:48) P ∪ P ( W (cid:48) ) | ≥ . But this is a contradiction,since we include in Z all but at most one edge of δ S (cid:48) ( W ) . General k and unit costs. The algorithm, which we denote by k - MCAlg , leading to our bicriteria guar-antee is quite similar to - MCAlg . The only changes are the following: • In steps A1 and A3.5, we set T (cid:48) to be the s i - t i pairs from T that are at least γ ( k − -edge-connectedin ( S, E ( S )) . 11 In step A3.2, we apply Corollary 3.4 with the set S to find the radius ρ ∈ [0 , . • In step A3.4, we add all edges of δ S ( A ) to Z . (Unlike 2- MC , if we only include the edges contributingto c q (cid:0) δ S ( A ) (cid:1) for some suitable q , then we cannot necessarily argue that the final solution satisfies thestated connectivity guarantee.) • Of course, in step A4, we now recursively call k - MCAlg (with the same arguments).
Theorem 3.8
For any γ > , algorithm k - MCAlg returns a solution Z such that c ( Z ) ≤ O (cid:0) γ ( √ γ − ln r ln ln r (cid:1) · OPT and every s i - t i pair is less than γ ( k − -edge-connected in ( V, E \ Z ) .Thus, taking γ = kk − , we obtain a feasible solution of cost at most O (cid:0) ( k − ln r ln ln r (cid:1) · OPT . Proof :
Let Z be the output of k - MCAlg (cid:0) V, { ( s , t ) , . . . , ( s r , t r ) } (cid:1) . It is clear that Z is feasible: every s i - t i pair that is at least γ ( k − -edge-connected in ( U, E ( U )) , where ( U, . ) is a node of the recursion treeis either taken care of (i.e., rendered less than γ ( k − -edge-connected) by the edges added to Z in step A3,or, by induction, is taken care of by a recursive call.Mimicking the proof of Lemma 3.5, and using Corollary 3.4 in place of Lemma 3.3 in the proof, onecan easily show that if d is the depth of the recursion tree rooted ( U, T ) and Z U = k - MCAlg ( U, T ) , then c ( Z U ) ≤ γ ( √ γ − (cid:16) β |T | + V x ( U ) (cid:17) ln (cid:16) e d r V x ( U ) OPT (cid:17) ln ln (cid:0) e ( r + 1) (cid:1) . Since the depth of the overall recursion tree is O (log r ) , as argued in the proof of Theorem 3.6, we obtainthat c ( Z ) ≤ O (cid:0) γ ( √ γ − ln r ln ln r (cid:1) · OPT . k - ( s, t ) - Cut
Theorems 4.1 and 4.2 together prove that k - ( s, t ) - Cut is at least as hard as the densest- k -subgraph problem( D k S ) problem. In D k S on hypergraphs, we seek a set of k nodes containing the maximum the number ofhyperedges. Our hardness result implies that obtaining a unicriterion O (cid:0) k (cid:15) polylog( n ) (cid:1) -approximation forsome constant (cid:15) would improve the current-best guarantee for D k S on graphs, and imply the existence ofa certain family of one-way functions. Our reduction is from small set vertex expansion ( SSVE ), whereinwe have a bipartite graph G = ( U ∪ V, E ) and a parameter < α ≤ , and we seek a subset S ⊆ U with | S | ≥ α | U | that minimizes the number of neighbors, Γ( S ) . Chuzoy et al. [11] show that SSVE reduces tothe minimization version of D k S , MinD k S , wherein we seek a minimum number of nodes that contain atleast k hyperedges. They also show that a ρ -approximation for MinD k S on λ -uniform hypergraphs yields a (2 ρ λ ) -approximation for D k S on λ -uniform hypergraphs.For a graph H = ( V H , E H ) , subset S ⊆ V H , and v ∈ V H \ S , we use δ H ( S, v ) = δ H ( v, S ) to denotethe edges between v and S in H , and Γ H ( S ) to denote the set of neighbors of S in H . As is standard, weabbreviate δ H ( { v } , V H \ { v } ) to δ H ( v ) . Theorem 4.1 ([11])
For any λ ≥ , there is a polytime approximation-preserving reduction that given a MinD k S -instance on a λ -uniform hypergraph with n nodes and m edges, creates an SSVE -instance with m + n nodes and λm edges. Hence, a ρ ( m, n ) -approximation for SSVE yields, for λ -uniform hypergraphs,a ρ ( λm, m + n ) -approximation for MinD k S and a (cid:0) (cid:0) ρ ( λm, m + n ) (cid:1) λ (cid:1) -approximation for D k S . Theorem 4.2
There is a polytime approximation-preserving reduction that given an
SSVE -instance with n nodes and m edges, creates a k - ( s, t ) - Cut -instance with O ( n ) nodes, O ( mn + n ) edges, and k = O ( mn ) .Hence, a ρ ( k, m, n ) -approximation for k - ( s, t ) - Cut yields a ρ (cid:0) O ( mn ) , O ( mn + n ) , O ( n ) (cid:1) -approximationfor SSVE . u u v v v v G=(U U V) u u u s t b a K(v ) K(v ) K(v ) K(v ) … … … … … … … … … … … … … … Figure 4.1: To the left, a graph of a given SSVE instance. To the right, the graph for the k - ( s, t ) - Cut instancecreated by our reduction. The edges incident into t have unit cost, while all other edges have infinite cost.Each K ( v i ) is a clique with N = 25 vertices. Dashed edges have unit capacity. The other edges havethe following capacities: edge { s, a } and edge { b, t } have capacity 150. Each edge { b, u i } ( i = 1 , , )has capacity 50. Each edge { a, x } for x ∈ K ( v ) and x ∈ K ( v ) has capacity 2. Each edge { a, x } for x ∈ K ( v ) and x ∈ K ( v ) has capacity 1. Proof :
Given an instance (cid:0) G = ( U ∪ V, E ) , α (cid:1) of SSVE , we construct the following instance of k - ( s, t ) - Cut ; see Fig. 4.1. Let N = 2 | U || V | + 1 . Below, an infinite-cost edge ( u, v ) of capacity b uv is simplya shorthand for b uv parallel infinite-cost edges. Also, unless otherwise specified, an edge has unit capacity.(i) We replace each vertex v ∈ V with a clique K ( v ) of size N . All edges in the clique have infinite cost.For each edge ( u, v ) ∈ E , we add an edge between u and every vertex in K ( v ) .(ii) We add the source s and connect it to all vertices in U ; we add the sink t and connect it to all verticesin K ( v ) for every v ∈ V .(iii) Finally, we add a vertex b , an edge ( b, t ) with capacity | E |· N , and edges ( b, u ) with capacity | δ G ( u ) |· N for all u ∈ U . We also add a vertex a , an edge ( s, a ) with capacity | E | · N , and for all v ∈ V , we addedges ( a, x ) for all x ∈ K ( v ) with capacity | δ G ( v ) | .All edges have infinite cost except for the edges between (cid:83) v ∈ V K ( v ) and t , which have unit cost. We set k = | U | (1 − α ) + N | E | + 1 .We claim that there exists a solution of value at most C for the SSVE instance iff there is a solutionof value at most CN for the k - ( s, t ) - Cut instance. Note that a solution F consisting of unit-cost edges isfeasible if the maximum s - t flow in the capacitated remainder graph G (cid:48) \ F has value at most k − . Theintuition is that if we send N | E | units of flow along the paths s − a − x − u − b − t for all ( u, v ) ∈ E, x ∈ K ( v ) ,then in the residual digraph, all arcs between U and (cid:83) v ∈ V K ( v ) leave U . Given this, one can mimic thearguments in [11] to show the desired claim.Suppose the SSVE instance has a solution S ⊆ U of value at most C . Construct a k - ( s, t ) - Cut -solutionby removing the ( v, t ) edges for all v ∈ Γ G ( S ) . Clearly the cost of this set is at most CN . We now argue13easibility. Consider the s - t cut induced by A = { s, a } ∪ S ∪ (cid:83) v ∈ Γ G ( S ) K ( v ) . Then | δ G (cid:48) ( A ) | is equal to | U \ S | (cid:124) (cid:123)(cid:122) (cid:125) edges between s and U \ S + (cid:88) u ∈ S N | δ G ( u ) | (cid:124) (cid:123)(cid:122) (cid:125) edges between b and S + (cid:88) v ∈ V \ Γ G ( S ) N | δ G ( v ) | (cid:124) (cid:123)(cid:122) (cid:125) edges between a and (cid:83) v ∈ V \ Γ G ( s ) K ( v ) + N (cid:0) G between U \ S and Γ G ( S ) (cid:1)(cid:124) (cid:123)(cid:122) (cid:125) edges between U \ S and (cid:83) v ∈ Γ G ( S ) K ( v ) . The sum of the last two terms is (cid:80) u ∈ U \ S N | δ G ( u ) | and | U \ S | ≤ | U | (1 − α ) , so the size of the s - t cut isat most | U | (1 − α ) + (cid:80) u ∈ U N | δ G ( u ) | ≤ | U | (1 − α ) + N | E | ≤ k − .For the other direction, suppose G (cid:48) has a solution F of value at most CN . Clearly, F can consist of onlyunit-cost edges (incident to t ). We first argue that we may convert F into a structured feasible solution F (cid:48) of cost at most CN where | F (cid:48) ∩ δ G (cid:48) ( K ( v )) | ∈ { , N } for all v ∈ V .Fix v ∈ V . If | δ G (cid:48) ( K ( v ) , t ) \ F | ≤ k (cid:48) := | U | (1 − α ) , then we add all edges of δ G (cid:48) ( K ( v ) , t ) \ F to F .Now suppose | δ G (cid:48) ( K ( v ) , t ) \ F | > k (cid:48) and let w , . . . , w k (cid:48) +1 be vertices in K ( v ) such that ( w i , t ) / ∈ F forall i = 1 , . . . , k (cid:48) + 1 . We claim that F \ δ G (cid:48) ( K ( v ) , t ) is also feasible. Suppose to the contrary that we nowhave k (cid:48) + 1 edge-disjoint s - t -paths. We may assume that each such path contains at most one vertex from K ( v ) since we can always shortcut the path to t . Consider a path P that contains a vertex w ∈ K ( v ) where w / ∈ { w , . . . , w k (cid:48) +1 } . Then we can construct another path P (cid:48) by switching w with a distinct vertex w j forsome j ∈ { . . . , k (cid:48) + 1 } . Note that P (cid:48) is an s - t path that avoids edges in F . If we repeat this argument forall such paths, we obtain k (cid:48) + 1 edge-disjoint s - t -paths in G (cid:48) \ F , contradicting the feasibility of F .If we perform the above transformation for all v ∈ V , then we obtain a feasible solution F (cid:48) of cost atmost | F | + | V | k (cid:48) < ( C + 1) N . But by construction | F (cid:48) | must be an integer multiple of N , so | F (cid:48) | ≤ CN .Consider the residual network ˜ G that is obtained from G (cid:48) \ F (cid:48) as follows. We first bidirect the edges of G (cid:48) \ F (cid:48) , giving each resulting arc the same capacity as that of the corresponding edge of G (cid:48) . ˜ G is the residualnetwork obtained after we send one unit of flow along the path s - a - x - u - b - t for every edge ( u, v ) ∈ E andevery x ∈ K ( v ) . Note that these paths are edge disjoint, so we send N | E | units of flow. By flow theory [2],we know that the value of the maximum s - t -flow in G (cid:48) \ F (cid:48) is at most k (cid:48) := | U | (1 − α ) iff the value ofmaximum s - t -flow in ˜ G , which equals the capacity of the minimum s - t cut in ˜ G , is at most k (cid:48) . It followsthere is an s - t cut in ˜ G of capacity at most k (cid:48) .Let A be the vertices that are on the s -side of this cut. Let S := U ∩ A . Then | S | ≥ α | U | , otherwise thecut would have capacity more than | U | (1 − α ) due to the arcs ( s, u ) for u ∈ U \ S . Consider v ∈ Γ G ( S ) . Wemust have K ( v ) ⊆ A : if K ( v ) ∩ A = ∅ , then considering u ∈ S such that ( u, v ) ∈ E , the cut has capacityat least N > k (cid:48) due to the edges between u and K ( v ) ; otherwise, since K ( v ) is split between the s - and t -sides, the cut has capacity at least N − > k (cid:48) . Finally, δ G (cid:48) ( K ( v ) , t ) ⊆ F (cid:48) , otherwise δ G (cid:48) ( K ( v ) , t ) ∩ F (cid:48) = ∅ ,and again the cut has capacity at least N > k (cid:48) . Thus, | Γ G ( S ) | ≤ C , so S is an SSVE -solution. k - MC We now consider variants of k - MC where we seek to delete edges or nodes so as to reduce the node connec-tivity of each s i - t i pair to at most k − . Formally, as before, we are given an undirected graph G = ( V, E ) , r source-sink pairs ( s , t ) , . . . , ( s r , t r ) , and an integer k ≥ . In the edge-deletion k -route node-multicut ( ED - k - NMC ) problem, we have nonnnegative edge-costs { c e } e ∈ E and we seek a minimum-cost set F ⊆ E of edges such that the remainder graph G = ( V, E \ F ) contains at most k − node-disjoint s i - t i paths forall i = 1 , . . . , r . In the node-deletion k -route node-multicut ( ND - k - NMC ) problem, we have nonnegativenode costs { c v } v ∈ V and we seek a minimum-cost set A ⊆ V \ { s , t , . . . , s r , t r } of nodes such that theremainder graph G = (cid:0) V \ A, E ( V \ A ) (cid:1) contains at most k − node-disjoint s i - t i paths for all i = 1 , . . . , r .The LP-relaxations of these k -route node-multicut problems induce both edge and node lengths, soto round these we develop region-growing lemmas that also incorporate node lengths. To keep notation14imple, instead of proving an overly- general region-growing lemma and obtaining the lemmas required for ED - k - NMC and ND - k - NMC as corollaries, we specifically focus on ED - k - NMC (Section 5.1) and ND - k - NMC (Section 5.2) and prove suitable region-growing lemas. We use these to obtain an O (ln r ln ln r ) -approximation for ED - k - NMC with k = 2 , and a bicriteria (cid:0) γ, O (cid:0) γ ( √ γ − ln r ln ln r (cid:1)(cid:1) -approximation for ND - k - NMC with general k and unit node costs. k -route node-multicut ( ED - k - NMC ) with k = 2 The LP-relaxation for ED - k - NMC is as follows. min (cid:88) e c e x e (P2)s.t. (cid:88) e ∈ E ( P ) x e + (cid:88) v ∈ V ( P ) y iv ≥ ∀ i, ∀ P ∈ P i (cid:88) v y iv ≤ k − , y is i = y it i = 0 ∀ ix, y ≥ . Region-growing lemma.
Let (cid:0) x, { y i } (cid:1) be an optimal solution to (P2), and OPT be its value. Let S ⊆ V represent the node-set of the current region. For T ⊆ S ⊆ V , recall that E ( S ) is the set of edges with bothendpoints in S and δ S ( T ) is the set of boundary edges of T in E ( S ) . Set β = OPT /r . As before, define V x ( S ) := β + (cid:80) e ∈ E ( S ) c e x e . Let ρ ≥ . Let z ∈ V . Fix a commodity i . • Define (cid:96) i ( u ; v ) = min P : P is a u - v path (cid:0)(cid:80) e ∈ E ( P ) x e + (cid:80) w ∈ V ( P ): w (cid:54) = u y iw (cid:1) , where E ( P ) and V ( P ) denoterespectively the set of edges and nodes of P . Note that (cid:96) i defines an asymmetric metric on V × V . • Define B i ( z, ρ ) := { v ∈ V : (cid:96) i ( z ; v ) ≤ ρ } to be the ball of radius ρ around z . Let B Si ( z, ρ ) := B i ( z, ρ ) ∩ S . • Define the edge-boundary ∂ S,xi , and node-boundary Γ S,yi , of B Si ( z, ρ ) in S as follows. ∂ S,xi ( z, ρ ) := { ( u, v ) ∈ E : u, v ∈ S, (cid:96) i ( z ; u ) ≤ ρ, (cid:96) i ( z ; v ) − y iv > ρ } Γ S,yi ( z, ρ ) := { v ∈ S : ρ < (cid:96) i ( z ; v ) ≤ ρ + y iv } . Let B Si ( z, ρ ) := S \ (cid:0) B Si ( z, ρ ) ∪ Γ S,yi ( z, ρ ) (cid:1) . • Define the following volumes: V S,xi ( z, ρ ) := β + (cid:88) ( u,v ) ∈ E : u ∈ B Si ( z,ρ ) v ∈ B Si ( z,ρ ) ∪ Γ S,yi ( z,ρ ) c uv x uv + (cid:88) ( u,v ) ∈ ∂ S,xi ( z,ρ ): u ∈ B Si ( z,ρ ) c uv (cid:0) ρ − (cid:96) i ( z ; u ) (cid:1) V S,xi ( z, ρ ) := β + (cid:88) ( u,v ) ∈ E : u ∈ B Si ( z,ρ ) v ∈ B Si ( z,ρ ) ∪ Γ S,yi ( z,ρ ) c uv x uv + (cid:88) ( u,v ) ∈ ∂ S,xi ( z,ρ ): u ∈ B Si ( z,ρ ) c uv (cid:0) (cid:96) i ( z ; v ) − ρ − y iv (cid:1) emma 5.1 Let S ⊆ V , z ∈ V , i be some commodity, and ≤ a < b . Let ρ be chosen uniformly at randomfrom [ a, b ) . Then, E ρ c (cid:0) ∂ S,xi ( z, ρ ) (cid:1) V S,xi ( z, ρ ) ln (cid:16) e V S,xi ( z,b ) V S,xi ( z,ρ ) (cid:17) ≤ b − a · ln ln (cid:18) e V S,xi ( z, b ) V S,xi ( z, a ) (cid:19) and (9) E ρ c (cid:0) ∂ S,xi ( z, ρ ) (cid:1) V S,xi ( z, ρ ) ln (cid:16) e V S,xi ( z,a ) V S,xi ( z,ρ ) (cid:17) ≤ b − a · ln ln (cid:18) e V S,xi ( z, a ) V S,xi ( z, b ) (cid:19) (10) Proof :
We abbreviate c (cid:0) ∂ S,xi ( z, ρ ) (cid:1) to c ( ρ ) , V S,xi ( z, ρ ) to V ( ρ ) and V S,xi ( z, ρ ) to V ( ρ ) . Let I = { (cid:96) i ( z ; v ) , (cid:96) i ( z ; v ) − y iv : v ∈ V } . Note that V ( ρ ) and V ( ρ ) are differentiable at all ρ ∈ [ a, b ) \ I andfor each such ρ , we have d V ( ρ ) dρ = c ( ρ ) and d V ( ρ ) dρ = − c ( ρ ) . The proof now follows from exactly the samearguments as in the proof of Lemma 3.1. Corollary 5.2
Let S ⊆ V , z ∈ V , and i be some commodity. Let α ∈ (0 , and q = (cid:6) k − α (cid:7) . We canefficiently find ρ ∈ [0 , such that c (cid:0) ∂ S,xi ( z, ρ ) (cid:1) ≤ − α · V S,xi ( z, ρ ) ln (cid:16) e V x ( S ) V S,xi ( z, ρ ) (cid:17) ln ln (cid:0) e ( r + 1) (cid:1) c (cid:0) ∂ S,xi ( z, ρ ) (cid:1) ≤ − α · V S,xi ( z, ρ ) ln (cid:16) e V x ( S ) V S,xi ( z, ρ ) (cid:17) ln ln (cid:0) e ( r + 1) (cid:1)(cid:12)(cid:12) Γ S,yi ( z, ρ ) (cid:12)(cid:12) < q. Proof :
If we choose ρ uniformly at random from [0 , then E ρ (cid:104)(cid:12)(cid:12) Γ S,yi ( z, ρ ) (cid:12)(cid:12)(cid:105) ≤ (cid:80) i y iv ≤ k − . Taking [ a, b ) = [0 , , the arguments in Corollary 3.2 readily generalize to show that we can efficiently find ρ ∈ [0 , such that c (cid:0) ∂ S,xi ( z, ρ ) (cid:1) / V S,xi ( z, ρ ) ln (cid:0) e V S,xi ( z, V S,xi ( z,ρ ) (cid:1) , and c (cid:0) ∂ S,xi ( z, ρ ) (cid:1) / V S,xi ( z, ρ ) ln (cid:0) e V S,xi ( z, V S,xi ( z,ρ ) (cid:1) areat most − α ) times the right-hand-sides of (9) and (10) respectively, and (cid:12)(cid:12) Γ S,yi ( z, ρ ) (cid:12)(cid:12) < k − α . The lemmanow follows by noting that V S,xi ( z, ≤ V x ( S ) , V S,xi ( z, ≤ V x ( S ) , V S,xi ( z, V S,xi ( z, ≤ r + 1 , V S,xi ( z, V S,xi ( z, ≤ r + 1 . Algorithm and analysis.
The algorithm and analysis dovetail the one in Section 3.2 for k = 2 . Algorithm ED - - NMCAlg ( U, T = { ( s , t ) , . . . , ( s N , t N ) } ) Input:
A subset U ⊆ V , and a collection T = { ( s i , t i ) } Ni =1 of source-sink pairs, where s i , t i ∈ U for all i = 1 , . . . , N . Output:
A set Z ⊆ E ( U ) such that s i and t i are at most 1-node-connected in ( U, E ( U ) \ Z ) for all i = 1 , . . . , N .B1. Set S = U , Z = ∅ , S = ∅ , and T (cid:48) = (cid:8) ( s i , t i ) ∈ T : s i and t i are at least -node-connected in ( S, E ( S )) (cid:9) .B2. If T (cid:48) = ∅ , return Z .B3. While T (cid:48) (cid:54) = ∅ , we do the following. ( s i , t i ) ∈ T (cid:48) .A3.2 Apply Corollary 5.2 with z = s i , α = 0 . and the set S (and k = 2 ) to find a radius ≤ ρ < .A3.3 If B Si ( s i , ρ ) ∪ Γ S,yi ( s i , ρ ) contains at most N/ pairs from T then set A = B Si ( s i , ρ ) , else set A = B Si ( s i , ρ ) .A3.4 Set S ← S ∪ { A ∪ Γ S,yi ( s i , ρ ) } . Add the edges in ∂ S,xi ( s i , ρ ) to Z .A3.5 Set S ← S \ A . Update T (cid:48) to be the s i - t i pairs from T that are at least -node-connected in ( S, E ( S )) .B4. For every set A ∈ S , set Z ← Z ∪ ED - - NMCAlg (cid:0) A, { ( s i , t i ) ∈ T : s i , t i ∈ A } (cid:1) .B5. Return Z .The initial call to ED - - NMCAlg , which computes the solution we return, is ED - - NMCAlg (cid:0) V, { ( s , t ) , . . . , ( s r , t r ) } (cid:1) . Let Z := ED - - NMCAlg (cid:0) V, { ( s , t ) , . . . , ( s r , t r ) } (cid:1) . Define the depth of a subtree of the recursion treecorresponding to the execution of ED - - NMCAlg to be the maximum number of edges on a root to leaf pathof the subtree. The following claim will be useful to prove feasibility of Z . Claim 5.3
Let
S, T ⊆ V with | S ∩ T | ≤ . Let E S ⊆ E ( S ) and E T ⊆ E ( T ) . Let u, v ∈ S be suchthat u and v are at most 1-node-connected in ( S, E S ) . Then, u and v are at most 1-node-connected in ( S ∪ T, E S ∪ E T ) . Proof : If S ∩ T = ∅ , this clearly holds. So assume otherwise. Suppose P , P are two simple node-disjoint u - v paths in G (cid:48) = ( S ∪ T, E S ∪ E T ) . At least one of P and P does not lie completely in ( S, E S ) ; suppose P is this path. But since all edges of δ G (cid:48) ( S ) are incident to a single node, and P both exits and leaves S , P contains a repeated node, which is a contradiction. Lemma 5.4
The solution Z returned is feasible. Proof :
Suppose for a contradiction that there is some s i - t i pair that is (at least) 2-node-connected in ( V, E \ Z ) . Consider the recursion tree of ED - - NMCAlg , and let ( Y, T Y ) be the node furthest from the rootsuch that s i and t i are at least 2-node-connected in the subgraph ( Y, E ( Y )) induced by Y . Suppose that theloop in step B3 of ED - - NMCAlg ( Y, T Y ) runs for h iterations. Note that h ≥ since s i and t i are at least2-node-connected in ( Y, E ( Y )) . Let X p be the set added to S in step B3.4 in the p -th iteration of the loop.Let X h +1 ⊆ Y be the set S at the termination of the loop. Let S p = (cid:83) h +1 q = p X q (so S = Y ). Notice that | X p ∩ S p +1 | ≤ , since X p ∩ S p +1 ⊆ Γ S p ,yp ( s p , ρ p ) , where s p - t p is the source-sink pair and ρ p is the radiuschosen in the p -th iteration, and | Γ S p ,yp ( s p , ρ p ) | < by Lemma 5.2.Let p be the highest index such that s i and t i are at least 2-node-connected in ( S p , E ( S p )) . Note that p ≤ h , otherwise the loop in step B3 would not have terminated with S = X h +1 . If s i , t i ∈ S p +1 , they are at most1-node-connected in ( S p +1 , E ( S p +1 )) . Since | X p ∩ S p +1 | ≤ , we have E ( S p ) = E ( X p ) ∪ E ( S p +1 ) , andby Claim 5.3 it follows that s i and t i are at most 1-node-connected in ( S p , E ( S p )) , which is a contradiction.If s i , t i ∈ X p , they are at most 1-node-connected in ( X p , E ( X p )) due to the definition of ( Y, T Y ) , and sowe arrive at the same contradiction. So it must be that (cid:12)(cid:12) { s i , t i } ∩ ( X p \ S p +1 ) (cid:12)(cid:12) = 1 . But then all s i - t i paths in ( S p , E ( S p )) contain the singleton node in X p ∩ S p +1 . So s i and t i are at most 1-node-connected in ( S p , E ( S p )) , and we have the same contradiction. Lemma 5.5
Let d be the depth of a subtree of the recursion tree rooted at (cid:0) U ⊆ V, T ⊆ { ( s , t ) , . . . , ( s r , t r ) } (cid:1) .Let Z U = ED - - NMCAlg ( U, T ) . Then c ( Z U ) ≤ (cid:0) β |T | + V x ( U ) (cid:1) ln (cid:0) e d r V x ( U ) OPT (cid:1) ln ln (cid:0) e ( r + 1) (cid:1) . roof : When d = 0 , we have Z U = ∅ , so the statement holds. Suppose in step B3 of ED - - NMCAlg ( U, T ) ,we add sets A , . . . , A h to S (where h ≥ ), in that order. For p = 1 , . . . , h , let S p be the current set S when A p was added to S in step B3.4, and let E p be the edge-set added to Z in this step. Let Z , . . . , Z h bethe edge-sets returned by the recursive calls to ED - - NMCAlg ( A , T ) , . . . , ED - - NMCAlg ( A h , T h ) in stepB4. Let vol p = V S p ,xi ( s i , ρ ) if A p = B S p i ( s i , ρ ) ∪ Γ S p ,yi ( s i , ρ ) and vol p = V S p ,xi ( s i , ρ ) if A p = B S p i ( s i , ρ ) ∪ Γ S p ,yi ( s i , ρ ) . The key thing to note is that we still have V x ( A p ) ≤ vol p ≤ V x ( S p ) ≤ V x ( U ) and (cid:80) hp =1 vol p ≤ V x ( U ) + β ( h − . The latter follows since an easy induction argument shows that (cid:80) hp = q vol p ≤ V x ( S q ) + β ( h − q ) for all q = 1 , . . . , h . Given this, the rest of the proof is identical to that ofLemma 3.5.Each recursive call to ED - - NMCAlg , reduces the number of source-sink pairs involved by a factor ofat least 2, so the depth d of the entire recursion tree is O (log r ) . So we have shown the following. Theorem 5.6
Algorithm ED - - NMCAlg returns a feasible solution of cost at most O (ln r ln ln r ) · OPT . k -route node-multicut ( ND - k - NMC ) with unit costs
The LP-relaxation for ND - k - NMC is as follows. min (cid:88) v c v x v (P3)s.t. (cid:88) v ∈ V ( P ) ( x v + y iv ) ≥ ∀ i, ∀ P ∈ P i (cid:88) v y iv ≤ k − , y is i = y it i = 0 ∀ ix, y ≥ , x v = 0 ∀ v ∈ { s , t , . . . , s r , t r } . Region-growing lemma.
Let (cid:0) x, { y i } (cid:1) be an optimal solution to (P3), and OPT be its value. As before,let S ⊆ V represent the node-set of the current region. Set β = OPT /r . Let z ∈ V and ρ ≥ . Fix acommodity i . • Define (cid:96) i ( u ; v ) = min P : P is a u - v path (cid:80) w ∈ V ( P ): w (cid:54) = u ( x w + y iw ) , where V ( P ) is the set of nodes of P . Asbefore, (cid:96) i defines an asymmetric metric on V × V . • Define B i ( z, ρ ) := { v ∈ V : (cid:96) i ( z ; v ) ≤ ρ } , and B Si ( z, ρ ) := B i ( z, ρ ) ∩ S . • Define the x -boundary of B Si ( z, ρ ) in S to be Γ S,xi ( z, ρ ) := { v ∈ S : ρ + y iv < (cid:96) i ( z ; v ) ≤ ρ + x v + y iv } .Define the y -boundary of B Si ( z, ρ ) in S to be Γ S,yi ( z, ρ ) := { v ∈ S : ρ < (cid:96) i ( z ; v ) ≤ ρ + y iv } . Note that Γ S,xi ( z, ρ ) and Γ S,yi ( z, ρ ) partition Γ Si ( z, ρ ) := { v ∈ S \ B Si ( z, ρ ) : ∃ u ∈ B si ( z, ρ ) s.t. ( u, v ) ∈ E } . Let B Si ( z, ρ ) := S \ (cid:0) B Si ( z, ρ ) ∪ Γ Si ( z, ρ ) (cid:1) . • Define the following volumes: V S,xi ( z, ρ ) := β + (cid:88) u ∈ B Si ( z,ρ ) ∪ Γ S,yi ( z,ρ ) c u x u + (cid:88) u ∈ Γ S,xi ( z,ρ ) c u (cid:0) ρ − ( (cid:96) i ( z ; u ) − x u − y iu ) (cid:1) V S,xi ( z, ρ ) := β + (cid:88) u ∈ B Si ( z,ρ ) ∪ Γ S,yi ( z,ρ ) c u x u + (cid:88) u ∈ Γ S,xi ( z,ρ ) c u ( (cid:96) i ( z ; u ) − y iu − ρ ) Lemma 5.7
Let S ⊆ V , z ∈ V , i be some commodity, and ≤ a < b . Let ρ be chosen uniformly at randomfrom [ a, b ) . Then, E ρ c (cid:0) Γ S,xi ( z, ρ ) (cid:1) V S,xi ( z, ρ ) ln (cid:16) e V S,xi ( z,b ) V S,xi ( z,ρ ) (cid:17) ≤ b − a · ln ln (cid:18) e V S,xi ( z, b ) V S,xi ( z, a ) (cid:19) and E ρ c (cid:0) Γ S,xi ( z, ρ ) (cid:1) V S,xi ( z, ρ ) ln (cid:16) e V S,xi ( z,a ) V S,xi ( z,ρ ) (cid:17) ≤ b − a · ln ln (cid:18) e V S,xi ( z, a ) V S,xi ( z, b ) (cid:19) Corollary 5.8
Let S ⊆ V . Suppose that s i , t i ∈ S and there are γ ( k − node-disjoint s i - t i paths internalto S , where γ > . Suppose that c v = 1 for all nodes v . We can efficiently find ρ ∈ [0 , such that c (cid:0) Γ Si ( s i , ρ ) (cid:1) ≤ γ ( √ γ − · V S,xi ( s i , ρ ) ln (cid:16) e V x ( S ) V S,xi ( s i , ρ ) (cid:17) ln ln (cid:0) e ( r + 1) (cid:1) c (cid:0) Γ Si ( s i , ρ ) (cid:1) ≤ γ ( √ γ − · V S,xi ( s i , ρ ) ln (cid:16) e V x ( S ) V S,xi ( s i , ρ ) (cid:17) ln ln (cid:0) e ( r + 1) (cid:1) Proof :
Let α ∈ (0 , , whose value we will fix later. Take [ a, b ) = [0 , . We have V S,xi ( z, ≤ V x ( S ) , V S,xi ( z, ≤ V x ( S ) , V S,xi ( z, V S,xi ( z, ≤ r + 1 , V S,xi ( z, V S,xi ( z, ≤ r + 1 . Given this, the arguments in Corollary 3.2 readily generalize to show that we can efficiently find ρ ∈ [0 , such that c (cid:0) Γ S,xi ( s i , ρ ) (cid:1) ≤ − α · V S,xi ( z, ρ ) ln (cid:16) e V x ( S ) V S,xi ( z, ρ ) (cid:17) ln ln (cid:0) e ( r + 1) (cid:1) (11) c (cid:0) Γ S,xi ( s i , ρ ) (cid:1) ≤ − α · V S,xi ( z, ρ ) ln (cid:16) e V x ( S ) V S,xi ( z, ρ ) (cid:17) ln ln (cid:0) e ( r + 1) (cid:1) (12) (cid:12)(cid:12) Γ S,yi ( z, ρ ) (cid:12)(cid:12) < k − α . (13)Note that t i / ∈ Γ Si ( s i , ρ ) since ρ < . So removing Γ Si ( s i , ρ ) disconnects s i and t i , and hence, | Γ Si ( s i , ρ ) | ≥ γ ( k − . Therefore, since we have unit node costs and Γ S,xi ( s i , ρ ) and Γ S,yi ( s i , ρ ) partition Γ Si ( s i , ρ ) , we have c (cid:0) Γ Si ( s i , ρ ) (cid:1) ≤ c (cid:0) Γ S,xi ( s i , ρ ) (cid:1) · γγ − /α . Plugging in the bounds from (11), (12), wesee that the constant factor multiplying the volume terms on the RHS is minimized by setting α = γ − / ,which yields the constant factor γ ( √ γ − . Algorithm and analysis.
The algorithm, ND - k - NMCAlg , for ND - k - NMC is quite similar to ED - - NMCAlg .The only changes are the following: • In steps B1 and B3.5, we set T (cid:48) to be the s i - t i pairs from T that are at least γ ( k − -node-connected in ( S, E ( S )) . 19 In step B3.2, we apply Corollary 5.8 with the set S to find the radius ρ ∈ [0 , . • In step B3.4, we add A to S , and add all nodes of Γ Si ( s i , ρ ) to Z . • Of course, in step B4, we now recursively call ND - k - NMCAlg (with the same arguments).
Theorem 5.9
For any γ > , algorithm ND - k - NMCAlg returns a solution Z such that c ( Z ) ≤ O (cid:0) γ ( √ γ − ln r ln ln r (cid:1) · OPT and every s i - t i pair is less than γ ( k − -node-connected in ( V \ Z, E ( V \ Z )) .Thus, taking γ = kk − , we obtain a feasible solution of cost at most O (cid:0) ( k − ln r ln ln r (cid:1) · OPT . Proof :
Let Z be the output of ND - k - NMCAlg (cid:0) V, { ( s , t ) , . . . , ( s r , t r ) } (cid:1) . It is clear that Z is feasible.Mimicking the proof of Lemma 3.5, and using Corollary 5.8 in place of Lemma 3.3 in the proof, one caneasily show that if d is the depth of the recursion tree rooted ( U, T ) and Z U = ND - k - NMCAlg ( U, T ) , then c ( Z U ) ≤ γ ( √ γ − (cid:16) β |T | + V x ( U ) (cid:17) ln (cid:16) e d r V x ( U ) OPT (cid:17) ln ln (cid:0) e ( r + 1) (cid:1) . Since the depth of the recursion tree is O (log r ) , we obtain that c ( Z ) ≤ O (cid:0) γ ( √ γ − ln r ln ln r (cid:1) · OPT . References [1] S. Arora and C. Lund. Hardness of approximations. In D. Hochbaum, editor,
Approximation Algo-rithms for NP-hard Problems , PWS Publishing co., Boston, MA, USA, 1997.[2] R. Ahuja, T. Magnanti, J. Orlin, Network Flows: Theory, Algorithms, and Applications.
Prentice-Hall,Inc.
Electronic Colloquium on Computational Complexity (ECCC), 18:7, 2011.[4] S. Arora, L. Babai, J. Stern, and Z. Swedyk, The hardness of approximate optima in lattices, codes,and systems of linear equations.
Journal of Computer System Science , 54:317331, 1997[5] S. Barman and S. Chawla, Region growing for multi-route cuts. In
Proceedings of the 21st SODA ,pages 404–418, 2010.[6] S. Bhatt, and F. T. Leighton, A framework for solving VLSI graph layout problems.
Journal ofComputer and System Sciences
Theory of Computing , 4(1):1–20, 2008.[8] D. Chakrabarty, R. Krishnaswamy, S. Li, and S. Narayanan, Capacitated Network Design on Undi-rected Graphs. In
Proceedings of the 16th APPROX , pages 71–80, 2013.[9] S. Chawla, A. Gupta, and H. R¨acke. Embeddings of negative-type metrics and an improved approxi-mation to generalized sparsest cut.
ACM Transactions on Algorithms , 4(2), 2008.[10] C. Chekuri and S. Khanna, Algorithms for 2-Route Cut Problems.
Automata, Languages and Pro-gramming , 472-484, 2008.[11] J. Chuzhoy, Y. Makarychev, A. Viajayaraghavan, and Y. Zhou, Approximation algorithms and hardnessof the k -route cut problem. In Proceedings of the 23rd SODA , pages 780–799, 2012.2012] D. Dahlhaus, D. S. Johnson, C. H. Papadimitriou, P. D. Seymour, and M. Yannakakis, The complexityof multiway cuts.
Proceedings of the 24th STOC , 241-251, 1992[13] M. Dinitz and A. Gupta. Packing interdiction and partial covering problems. In
Proceedings of the16th IPCO , pages 157–168, 2013.[14] M. Dinitz, G. Kortsarz, and R. Raz. Label cover instances with large girth and the hardness of approx-imating basic k-spanner,
Proceedings of the 39th international colloquium conference on Automata,Languages, and Programming
Journal of the ACM , 47(4):585–616, 2000.[16] G. Even, J. Naor, B. Schieber, and M. Sudan, Approximating minimum feedback sets and multicuts indirected graphs.
Algorithmica , 20:151–174, 1998.[17] S. Fiorini, G. Joret, U. Pietropaoli, Hitting Diamonds and Growing Cacti. , pages 191-204, 2010.[18] F. Fomin, D. Lokshtanov, N. Misra, and S. Saurabh, Planar F -Deletion: Approximation, Kernelizationand Optimal FPT Algorithms. In Proceedings of the 53rd FOCS , pages 470–479, 2012.[19] N. Garg, V. Vazirani, and M. Yannakakis, Approximate max-flow min-(multi)-cut theorems and theirapplications.
SIAM Journal on Computing , 25:235–251, 1996.[20] A. Hayrapetyan, D. Kempe, M. Pl, and Z. Svitkina, Unbalanced graph cuts.
In Proceedings of the 13thESA , pages 191–202, 2005.[21] W. Kishimoto, A method for obtaining the maximum multi-route flow in a network.
Networks ,27(4):279-291, 1996.[22] P. Kolman, and C. Scheideler, Approximate Duality of Multicommodity Multiroute Flows and Cuts:Single Source Case. In
Proceedings of the 23rd SODA , pages 800–810, 2012.[23] P. Kolman, and C. Scheideler, Towards Duality of Multicommodity Multiroute Cuts and Flows: Mul-tilevel Ball-Growing. In
Proceedings of the 28th STACS , pages 129–140, 2011.[24] G. Kortsarz, On the Hardness of Approximating Spanners.
Algorithmica , 30, 1999.[25] B. Laekhanukit,
Personal communication , October 2013.[26] B. Laekhanukit, Parameters of two-prover-one-round game and the hardness of connectivity problems.In
Proceedings of the 25th SODA , pages 1626–1643, 2014.[27] F. Leighton and S. Rao., Multicommodity max-flow min-cut theorems and their use in designingapproximation algorithms.
Journal of the ACM , 46:787-832, 1999.[28] A. Li, and P. Zhang, Unbalanced graph partitioning.
Theory of Computing Systems , 53:454-466, 2013.[29] J. Shi, and J. Malik, Normalized cuts and image segmentation.
IEEE Transactions on Pattern Analysisand Machine Intelligence , 888–905, 2000.[30] C. A. Phillips. The network inhibition problem. In
Proceedings of the 25th STOC , pages 776–785,1993. 2131] C. Papadimitriou and M. Yannakakis. Optimization, approximation, and complexity classes.
Journalof Computer and System Sciences , 43:425–440, 1991.[32] H. R¨acke. Minimizing congestion in general networks. In
Proceedings of the 43rd FOCS , pages 43–52,2002.[33] P. Seymour. Packing directed circuits fractionally.
Combinatorica , 15:281–288, 1995.[34] D. Shmoys Cut problems and their application to divide-and-conquer. In D. Hochbaum, editor,
Ap-proximation Algorithms for NP-hard Problems , PWS Publishing co., Boston, MA, USA, 1997.[35] C. Tovey, A simplified NP-complete satisfiability problem.
Discrete Applied Mathematics
Approximation Algorithms . Springer Verlag, 2001.[37] D. Williamson and D. Shmoys.
The Design of Approximation Algorithms , Cambridge University Press,2010.[38] R. K. Wood. Deterministic network interdiction.
Mathematical and Computer Modeling , 17(2):1–18,1993.[39] R. Zenklusen. Network flow interdiction on planar graphs.
Discrete Applied Mathematics , 158:1441–1455, 2010.
A The k -route all-pairs cut problem Theorem A.1
The -route all-pairs cut problem is APX-hard. Proof :
We give an L -reduction from vertex cover on bounded-degree graphs, which is known to be APX -hard [31]. Given a vertex-cover instance ˆ G = ( ˆ V , ˆ E ) , where ˆ G has maximum degree α = O (1) , weconstruct an instance G = ( V, E ) of the -route all-pairs cut problem. In the following, to avoid confusion,we will refer to the elements ( ˆ V , ˆ E ) of the vertex-cover instance ˆ G as vertices and edges , and to the elements ( V, E ) of the constructed -route all-pairs-cut instance as nodes and links .Let the vertices in ˆ V be numbered , , . . . , | ˆ V | . For every vertex v ∈ ˆ V , we introduce a path P v in G that contains as many links as the degree of v . That is, P v has one link f e v for every edge e ∈ ˆ E incidentto v in ˆ G . We give infinite cost to such links. Let a v be the first node of the path P v and b v be the last. Weadd a link ( a v , b v ) of unit cost in G . Note that P v and ( a v , b v ) yields a cycle in G for every v ∈ ˆ V . Wealso connect a v to a v +1 through a cycle formed by 3 links with infinite cost. That is, we introduce | ˆ V | − triangles connecting all paths. For each edge e = ( u, v ) ∈ ˆ E we introduce a node σ e and we connect σ e tothe endpoints of f e u and to the endpoints of f e v , with links of unit cost. We let G = ( V, E ) be the resultinggraph for our -route all-pairs cut instance (see Fig.A.2).Let p ∗ and c ∗ be the cost of an optimal solution for the vertex-cover instance and the cost of an optimalsolution for the -route all-pairs cut instance, respectively. We claim that:(i) If there exists a vertex cover in ˆ G of size p , then there is a solution for the -route all-pairs cut instanceof cost at most | ˆ E | + p . Note that this implies that c ∗ ≤ | ˆ E | + p ∗ ≤ (2 α + 1) p ∗ .(ii) For any feasible solution for the -route all-pairs cut instance of cost at most | ˆ E | + p we can constructa cover of ˆ G of size at most p . 22 … σ e b a b a b a q b q … σ e’ Figure A.2: The instance created by our reduction from a vertex-cover instance ˆ G on q vertices. Black edgeshave infinite cost and grey edges have unit cost. Node σ e represents the edge (1 , and node σ e (cid:48) representsthe edge (3 , q ) . Vertex has degree 4 in ˆ G and vertex has degree 5 in ˆ G .This implies that we have an L -reduction, and shows that if we have a β -approximation for -route all-pairscut, then we can obtain a vertex-cover solution of size at most βc ∗ − | ˆ E | = O ( β ) p ∗ , yielding the theorem.In proving this, a useful observation is that a set F of edges is feasible for -route all-pairs cut problemiff the remainder graph G = ( V, E \ F ) has the property that every two simple cycles meet at most at onevertex. Such a graph is called a cactus graph.For (i), suppose there exists a vertex cover of size p . For every v in the cover, we add ( a v , b v ) in F .Furthermore, for each edge e = ( u, w ) ∈ ˆ E , we select one vertex between u and w that is in the cover(at least one of them is in the cover by definition), say u , and we add to F the links connecting σ e to theendpoints of f e w for the other vertex w . It is not difficult to check that F yields a feasible solution for the -route all-pairs cut instance (using the relationship to cactus graphs) of the claimed cost.For (ii), suppose we have a feasible solution F for the -route all-pairs cut instance, and consider theremainder graph obtained by removing F . Clearly, all links of infinite cost are still present. Note that eachnode σ e can have at most two links incident to it in the remainder graph, and both these links must beincident to two nodes of the same path P v for some v . If not, then we would have an infinite-cost link ofsome triangle that connects the vertices { a v } v ∈ ˆ V that is contained into another cycle other than the triangle,contradicting feasibility of our solution. We first argue that we may assume that each σ e has exactly two linksincident to it in the remainder graph (and hence, also in F ). Let e = ( u, w ) be the edge in ˆ E correspondingto σ e . As argued above, F contains at least one pair of links that connect σ e to the nodes of a path, say P u . Suppose that F also contains some links connecting σ e to P w . If we remove such links from F , wecreate one additional cycle, containing the link f e w , in the remainder graph. Thus, the new remainder graphis not a cactus iff f e w is already contained in some cycle in ( V, E \ F ) . But there is only one possiblecycle in ( V, E \ F ) containing f e w , namely the cycle formed by P w and ( a w , b w ) . This observation impliesthat F ∪ { ( a w , b w ) } \ { the two links connecting σ e to P w } is a feasible solution to our -route all-pairs cutinstance. Furthermore, this solution has no greater cost since we are adding at most one link of unit cost,and we removing at least one link of the same cost.Since the cost of our solution is at most | ˆ E | + p , it follows that there are at most p links in F of theform ( a v , b v ) . We claim that these vertices v form a cover in ˆ G . Suppose not. Then there is at least one edge e ∈ ˆ E that is not covered by these vertices. We know that the node σ e is connected in G to the endpointsof the link f e u for one of the endpoints, say u , of e . The link f e u is therefore contained in the cycle formedby P u and ( a u , b u ) , since ( a u , b u ) is not in F , and is also contained in the triangle with the node σ e , whichcontradicts feasibility of F . 23 orollary A.2 The k -route all-pairs cut problem is APX-hard for all k ≥ . Proof :
The reduction is very similar to the one in the proof of Theorem A.1. The only change is that inthe graph G created from the given vertex-cover instance, we now have: (a) k − parallel links betweenevery pair of consecutive nodes of every path P v ; and (b) k − parallel links between a j , a j +1 for all j = 1 , . . . , | ˆ V | − .Suppose there exists a vertex cover of size p . As before, for each node σ e we remove exactly one pairof links incident to σ e , and in particular we choose the pair of links that connect σ e to the endpoints of theedge f e u if e = ( u, v ) and v is in the cover, and the other pair otherwise. We also remove all edges of theform ( a v , b v ) for v in the cover. We remove | ˆ E | + p edges in total.We claim that for every pair of nodes z, w of the remainder graph, we have at most k − edge-disjointpaths. Every node that is not a node of a path P v has maximum degree two, and therefore this is clear. If z and w belongs to two different paths P v and P u with u > v , then every path connecting them must use eitherthe link ( a v , a v +1 ) (and there are at most k − such links) or the two upper links of the triangle formedwith a v and a v +1 . Therefore, we can have at most k − edge-disjoint such paths. Finally, if z and w belongto the same path P u , we have k − paths given by the infinite cost links, plus at most one additional paththat uses either the edge ( a u , b u ) or a sequence of pairs of links incident into the nodes { σ e } for the edges e that have u as an endpoint in ˆ G . Note that, by construction, if ( a u , b u ) is still in the graph, then all thepairs of links incident into the nodes { σ e } for the edges e that have u as an endpoint have been removed,and therefore, once again we get at most k − edge-disjoint paths.For the other direction, suppose we have a feasible solution F for the k -route all-pairs cut instance, andconsider the remainder graph obtained by removing such set of links. Clearly, all links of infinite cost arestill present. Once again, each node σ e corresponding to an edge e = ( u, v ) can have at most two linksincident to it in the remainder graph, and both these links must be incident to two nodes of the same path.If not, then we would have a pair of nodes ( a v and a u ) that are connected by k edge-disjoint paths: k − given by the infinite-cost links not in the paths P u and P v , and one which uses the links in the paths P u and P v and two links incident to σ e . Also, as before, we may assume that each σ e has exactly two linksincident into it in the remainder graph (and hence, in F ), because otherwise for one endpoint u of e we getthat F ∪ { ( a u , b u ) } \ { the two links connecting σ e to P u } is a feasible solution to our k -route all-pairs cutinstance of no larger cost. So if the cost of F is at most | ˆ E | + p , it follows that we have at most p links in F of the form ( a v , b v ) . We claim that these vertices v form a cover in ˆ G .Suppose not. Then there is at least one edge e ∈ ˆ E that is not covered by such vertices. We know thatthe node σ e is connected in G to the endpoints of the link f e u for one of the endpoints, say u , of e . Theendpoints of f e u are therefore connected by k − parallel paths using one single link, one path formed bythe edge ( a u , b u ) and edges of P u \ { f e u } , and one path contained in the triangle with the node σ e . Thiscontradicts feasibility of F .On the positive side, the -route all-pairs cut problem admits an O (1) -approximation. This followsfrom: (1) the equivalence of -route all-pairs cut and the problem of removing a min-cost set of edges sothat the remainder graph is a cactus; (2) the results of Fiorini et al. [17], who gave an O (1) -approximationfor the problem of removing a minimum-weight node set so that the remaining graph is a cactus; and (3) theedge-removal version easily reduces to the node-removal version by subdividing edges, and setting the costof the original vertices to ∞ and the cost of each vertex corresponding to an edge to be the cost of the edge.Recently, Fomin et al. [18] developed an O (1) -approximation algorithm for the problem of removingthe fewest number of nodes so that the remaining graph excludes a minor from a given list of graphs, at leastone of which should be planar. While k -route all-pairs cut can be stated as excluding the planar graph with k parallel edges as a minor of the remainder graph, the result of [18] does not directly apply here. This isbecause our transformation of an edge-weighted instance to a node weighted one introduces non-uniformnode weights, whereas the algorithm in [18] is for uniform node weights.24 Hardness of the edge-deletion k -route node-multicut problem Recall that in the edge-deletion k -route node-multicut ( ED - k - NMC ) problem, we have an undirected graph G = ( V, E ) with nonnegative edge costs { c e } e ∈ E , and r source-sink pairs ( s , t ) , . . . , ( s r , t r ) and aninteger k ≥ . We seek a minimum-cost set F ⊆ E of edges such that the remainder graph G = ( V, E \ F ) contains at most k − node-disjoint s i - t i paths for all i = 1 , . . . , r .Chuzhoy et al. [11] show that ED - k - NMC is hard to approximate within a factor Ω( k (cid:15) ) . They presenta reduction from -SAT (5) , which is the variant of -SAT where each variable occurs in at most clauses,coupled with the parallel-repetition theorem, which is essentially a reduction from (the minimization ver-sion) of label cover . However, Laekhanukit [26] pointed out some subtle (but fixable) errors in their proofand proposed a correction, but his reduction also suffers from some subtle (again fixable) errors [25]. Wegive a correct proof below via a somewhat simpler reduction than the ones in [11, 26].Label cover was first introduced by Arora et al. [4] and has been subsequently used as a basis for manyhardness reductions (see, e.g., [1]). Kortsarz [24] presented a minimization version of label cover (some-times known as MinRep ) with the same hardness guarantee, that has since found use in various network-design applications (see, e.g., [14]).In the
MinRep problem, we are given a bipartite graph H = ( U ∪ W, F ) , two sets of labels L (forvertices in U ) and L (for vertices in W ), and a constraint function for each edge e defined as π e : L → L .A labeling is given by specifying a set of labels f ( u ) ⊆ L for every vertex u ∈ U and a set of labels f ( w ) ⊆ L for every vertex w ∈ W . We say that a labeling covers an edge e = uw ∈ F if there exists a ∈ f ( u ) and b ∈ f ( w ) such that π e ( a ) = b . Min-Rep asks for a labeling that covers all the edges whileminimizing (cid:80) u ∈ U | f ( u ) | + (cid:80) w ∈ W | f ( w ) | . Theorem B.1 (see, e.g., [37])
There are constants (cid:15) , δ > such that there is no polytime algorithm for MinRep with approximation factor:– O (cid:0) q (cid:15) (cid:1) unless P=NP, where q = | L | + | L | is the size of the label set;– O (cid:0) ∆ δ (cid:1) unless P=NP, where ∆ is the maximum degree of the underlying graph;– log − (cid:15) m for any constant (cid:15) , unless NP is contained in deterministic quasipolynomial time, where m isthe number of edges. Theorem B.2
There is a polytime approximation-preserving reduction that given a
MinRep -instance ( H, π, L , L ) with label-size q = | L | + | L | and maximum-degree ∆ , constructs an ED - k - NMC -instance (cid:0) G, { c e } , { s , t , . . . , s r , t r } , k (cid:1) with k = O (∆ q ) , r = | E H | , and | E G | = O ( | E H | q ) .Hence, there is no O (cid:0) k (cid:15) (cid:1) -approximation for ED - k - NMC , for some constant (cid:15) > , unless P=NP, and no log − (cid:15) | E G | -approximation for any (cid:15) > unless NP is contained in deterministic quasipolynomial time. Proof :
We first describe the construction and then argue the approximation-preservation property.
The construction.
For each vertex u ∈ U and for each label in a ∈ L we introduce two vertices a u , ¯ a u in G connected by an edge of unit cost. Intuitively, if we select this edge in an ED - k - NMC solution for theinstance we construct, this implies that we are selecting label a for u . Similarly, for each w ∈ W and foreach label b ∈ L we introduce two vertices b w , ¯ b w connected by an edge of unit cost. The above edges willbe the only ones having unit cost. All subsequent edges added to this construction will have infinite cost.Consider an edge e = ( u, w ) of H . For each such edge, we construct the following gadget. We introducetwo nodes s e , t e that will form a terminal pair in our new instance. s e and t e are connected as follows. Foreach b ∈ L , let L eb ⊆ L be the labels of L such that a ∈ L eb implies π e ( a ) = b . Clearly, the sets L eb ,25 e t e a u b u c u d u f w g w P ef P eg a -u b -u c -u d -u f -w g -w Figure B.3: The gadget ( N e , E ( N e )) introduced for an edge e = ( u, w ) . Here L = { a, b, c, d } , L = { f, g } , L ef = { a, b, c } , L eg = { d } . Each black edge has unit cost, while all other edges have ∞ cost. Blue(grey) rectangles indicate the nodes in V ( C ef ) ∪ V ( C eg ) .for all b ∈ L , form a partition of the labels L . For each non empty set L eb we add a path P eb of length2 with the middle vertex connected to t e . We then add edges to form a cycle C eb starting and ending at s e ,containing all the edges in L eb , the edges in the path P eb and the edge b w ¯ b w . Finally we split each of theseadded edges into 2 by introducing a middle vertex. We let V ( C eb ) we the the set of new vertices introducedby this operation, and let N e be all the vertices participating in this gadget; see Fig. B.3.Let G (cid:48) = (cid:0)(cid:83) e (cid:48) ∈ E H N e (cid:48) , (cid:83) e (cid:48) ∈ E H E ( N e (cid:48) ) (cid:1) be the graph formed by the vertices and edges of all the edgegadgets. For all e , we are going to add other edges forming paths (of length 2) between s e and t e . Weadd edges ( s e , v ) , ( v, t e ) for all vertices v ∈ Γ G (cid:48) ( N e ) , that is, for all v / ∈ N e that are adjacent in someedge-gadget to some node in N e . Note that N e ∩ N e (cid:48) = ∅ unless e and e (cid:48) share an endpoint in H , say u , inwhich case, the two gadgets share the vertices { a u , ¯ a u } for all labels a of u . Thus, | Γ G (cid:48) ( N e ) | = O (∆ q ) .Define k e := |{ b ∈ L : L eb (cid:54) = ∅}| + | Γ G (cid:48) ( N e ) | . Finally, set k := max e k e . For all edges e with k e < k ,we add k − k e new vertices and connect these to s e and t e (via ∞ -cost edges). Let G be the resulting graph.This concludes our construction. Approximation preservation.
We now argue that any feasible solution to the ED - k - NMC -instance offinite cost yields a feasible solution to the
MinRep -instance of no greater cost, and vice versa. This willcomplete the proof.( ⇒ ) Let Z be a solution of finite cost for our ED - k - NMC instance. Consider a node u ∈ V H and let L u ∈ { L , L } be the label-set of u . Set f ( u ) := { a ∈ L u : ( a u , ¯ a u ) ∈ Z } . Clearly, the cost of the twosolutions are the same. We now claim that the resulting labeling is feasible for the label-cover instance.Suppose not, then there is an edge e ∈ E H that is not covered. By our construction, this means that for each b ∈ L with L eb (cid:54) = ∅ , the subgraph of the remainder subgraph G = ( V G , E G \ Z ) induced by the nodes of thecycle C eb and t e is connected. Each such cycle C eb , yields therefore one vertex-disjoint path in G between s e and t e . Also, all edges incident to s e and t e are present in G (since they have ∞ cost), so all length-2 pathsin G between s e and t e are still present in G . It follows that the vertex connectivity of s e and t e is at least k ,a contradiction. 26 ⇐ ) For the other direction, given a labeling for the label-cover instance, we construct Z := { ( a u , ¯ a u ) : u ∈ V H , a ∈ f ( u ) } . Clearly, the cost of the two solutions is the same. We claim that Z is a feasible solutionto our ED - k - NMC instance. Suppose not. Then, for some e = ( u, w ) ∈ E H , the s e - t e vertex connectivityin the remainder graph G = ( V G , E G \ Z ) is at least k . Therefore we can find a set of vertex-disjointpaths P between these vertices of size |P| ≥ k . Without loss of generality, we may assume that all the k − |{ b ∈ L : L eb (cid:54) = ∅}| length-2 paths between s e , t e are in P . If we remove the internal nodes on theselength-2 paths from G , the connected component containing s e in the remaining portion of G is a subgraphof the gadget ( N e , E ( N e )) for edge e (shown in Fig. B.3). This means that this subgraph contains at least |{ b ∈ L : L eb (cid:54) = ∅}| s e - t e vertex-disjoint paths. Clearly, this is only possible if, for every label b such that L eb (cid:54) = ∅ , either ( b w , ¯ b w ) / ∈ Z or ( a u , ¯ a u ) / ∈ Z for every a ∈ L eb . But that means that the edge ee