Approximation Algorithms and Hardness of the <i>k</i>-Route Cut Problem

Chuzhoy, Julia; Makarychev, Yury; Vijayaraghavan, Aravindan; Zhou, Yuan

doi:10.1137/1.9781611973099.63

Cited by 7 publications

(8 citation statements)

References 33 publications

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“…All the reductions are known and were previously presented in [7,3,11]. We present these results here for two reasons.…”

Section: Hardness Constructions Of Connectivity Problemsmentioning

confidence: 90%

“…We will give the hardness construction of this problem based on the construction in [11] 3 . Take an instance (G = (U, W, E), {π e : e ∈ E}, L 1 , L 2 , c 1 , c 2 ) of the minimum-cost label cove problem.…”

Section: Theorem 69 ([11])mentioning

confidence: 99%

“…These problems have hardness of the form k σ , where σ is a small constant that has not been calculated. (See [24,7,3,11]). The common source of hardness of these problems is the label cover problem (a.k.a., 2P1R) with parallel repetition.…”

Section: Introductionmentioning

confidence: 99%

“…, (s D , t D )} ⊆ V × V and a connectivity parameter k. The goal is to find a minimumcost subset E ⊆ E of edges such that G−E has no k openly disjoint s i , t i -paths for every source-sink pairs s i , t i . The best known approximation guarantee for this problem is O(D · k) due to the work of Chuzhoy, Makarychev, Vijayaraghavan and Zhou [11], and the best known hardness is O(k σ ), for some (very) small constant σ > 0. The approximation ratio is slightly better when we turn to a bi-criteria approximation algorithm.…”

Section: Introductionmentioning

confidence: 99%

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Parameters of Two-Prover-One-Round Game and The Hardness of Connectivity Problems

Laekhanukit

2013

Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms

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Optimizing parameters of Two-Prover-One-Round Game (2P1R) is an important task in PCPs literature as it would imply a smaller PCP with the same or stronger soundness. While this is a basic question in PCPs community, the connection between the parameters of PCPs and hardness of approximations is sometimes obscure to approximation algorithm community. In this paper, we investigate the connection between the parameters of 2P1R and the hardness of approximating the class of so-called connectivity problems, which includes as subclasses the survivable network design and (multi)cut problems. Based on recent development on 2P1R by Chan (STOC 2013) and several techniques in PCPs literature, we improve hardness results of some connectivity problems that are in the form k σ , for some (very) small constant σ > 0, to hardness results of the form k c for some explicit constant c, where k is a connectivity parameter. In addition, we show how to convert these hardness into hardness results of the form D c , where D is the number of demand pairs (or the number of terminals). Our results are as follows.

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“…All the reductions are known and were previously presented in [7,3,11]. We present these results here for two reasons.…”

Section: Hardness Constructions Of Connectivity Problemsmentioning

confidence: 90%

Section: Theorem 69 ([11])mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Parameters of Two-Prover-One-Round Game and The Hardness of Connectivity Problems

Laekhanukit

2013

Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…Chuzhoy et al [10] study the edge-connectivity k-route cut problem. In this problem an undirected edge-weighted graph, a set of demands consisting of pairs of vertices, and a number k are given.…”

Section: Subsequent Workmentioning

confidence: 99%

Vertex Sparsifiers: New Results from Old Techniques

Englert¹,

Gupta²,

Krauthgamer³

et al. 2014

SIAM J. Comput.

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Abstract. Given a capacitated graph G = (V, E) and a set of terminals K ⊆ V , how should we produce a graph H only on the terminals K so that every (multicommodity) ), where k = |K|; (b) a convex combination of trees over the terminals K that maintains congestion up to a factor of O(log k); (c) for a planar graph G, a convex combination of planar graphs that maintains congestion up to a constant factor. This requires us to give a new algorithm for the 0-extension problem, the first one in which the preimages of each terminal are connected in G. Moreover, this result extends to minor-closed families of graphs. Our bounds immediately imply improved approximation guarantees for several terminal-based cut and ordering problems. 1. Introduction. Given an undirected capacitated graph G = (V, E) and a set of terminal nodes K ⊆ V , we consider the question of producing a graph H only on the terminals K so that the congestion incurred on G and H for any multicommodity flow routed between terminal nodes is similar. Often, we will want the graph H to be structurally "simpler" than G as well. Such a graph H will be called a flow sparsifier for G; the loss (also known as quality) of the flow sparsifier is the factor by which the congestions in the graphs G and H differ. For instance, when K = V , the results of Räcke [31] give a convex combination of trees H with a loss of O(log n). We call this a tree-based

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Hardness and approximation for network flow interdiction

Chestnut

Zenklusen

2017

Networks

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In the Network Flow Interdiction problem, an adversary attacks a network in order to minimize the maximum s‐t‐flow. Very little is known about the approximatibility of this problem, despite decades of interest in it. It is, surprisingly, nontrivial to obtain in polynomial time an approximation guarantee that is independent of the number of edges in the graph and their capacities. We present the first such approximation algorithm, which has approximation ratio at most 2(n−1) for any graph with n vertices. We complement the algorithm with a hardness theorem. Past work has shown that Network Flow Interdiction cannot be much easier to approximate than Densest k‐Subgraph. We show that any nϵ‐approximation algorithm for Network Flow Interdiction, or one of several variants, would imply a 2n4ϵ‐approximation algorithm for Densest k‐Subgraph, which is an improvement over past work in terms of the polynomial factor. We also show that Network Flow Interdiction is essentially the same as the Budgeted Minimum s‐t‐Cut problem, and transferring our results gives hardness and an approximation algorithm for that problem, as well. © 2017 Wiley Periodicals, Inc. NETWORKS, Vol. 69(4), 378–387 2017

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Approximation Algorithms and Hardness of the k-Route Cut Problem

Cited by 7 publications

References 33 publications

Parameters of Two-Prover-One-Round Game and The Hardness of Connectivity Problems

Parameters of Two-Prover-One-Round Game and The Hardness of Connectivity Problems

Vertex Sparsifiers: New Results from Old Techniques

Hardness and approximation for network flow interdiction

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