1,337 research outputs found

    Rounding Algorithms for a Geometric Embedding of Minimum Multiway Cut

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    The multiway-cut problem is, given a weighted graph and k >= 2 terminal nodes, to find a minimum-weight set of edges whose removal separates all the terminals. The problem is NP-hard, and even NP-hard to approximate within 1+delta for some small delta > 0. Calinescu, Karloff, and Rabani (1998) gave an algorithm with performance guarantee 3/2-1/k, based on a geometric relaxation of the problem. In this paper, we give improved randomized rounding schemes for their relaxation, yielding a 12/11-approximation algorithm for k=3 and a 1.3438-approximation algorithm in general. Our approach hinges on the observation that the problem of designing a randomized rounding scheme for a geometric relaxation is itself a linear programming problem. The paper explores computational solutions to this problem, and gives a proof that for a general class of geometric relaxations, there are always randomized rounding schemes that match the integrality gap.Comment: Conference version in ACM Symposium on Theory of Computing (1999). To appear in Mathematics of Operations Researc

    Multiway Cut, Pairwise Realizable Distributions, and Descending Thresholds

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    We design new approximation algorithms for the Multiway Cut problem, improving the previously known factor of 1.32388 [Buchbinder et al., 2013]. We proceed in three steps. First, we analyze the rounding scheme of Buchbinder et al., 2013 and design a modification that improves the approximation to (3+sqrt(5))/4 (approximately 1.309017). We also present a tight example showing that this is the best approximation one can achieve with the type of cuts considered by Buchbinder et al., 2013: (1) partitioning by exponential clocks, and (2) single-coordinate cuts with equal thresholds. Then, we prove that this factor can be improved by introducing a new rounding scheme: (3) single-coordinate cuts with descending thresholds. By combining these three schemes, we design an algorithm that achieves a factor of (10 + 4 sqrt(3))/13 (approximately 1.30217). This is the best approximation factor that we are able to verify by hand. Finally, we show that by combining these three rounding schemes with the scheme of independent thresholds from Karger et al., 2004, the approximation factor can be further improved to 1.2965. This approximation factor has been verified only by computer.Comment: This is an updated version and is the full version of STOC 2014 pape

    Directed Multicut with linearly ordered terminals

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    Motivated by an application in network security, we investigate the following "linear" case of Directed Mutlicut. Let GG be a directed graph which includes some distinguished vertices t1,,tkt_1, \ldots, t_k. What is the size of the smallest edge cut which eliminates all paths from tit_i to tjt_j for all i<ji < j? We show that this problem is fixed-parameter tractable when parametrized in the cutset size pp via an algorithm running in O(4ppn4)O(4^p p n^4) time.Comment: 12 pages, 1 figur

    Improved Hardness for Cut, Interdiction, and Firefighter Problems

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    We study variants of the classic s-t cut problem and prove the following improved hardness results assuming the Unique Games Conjecture (UGC). * For Length-Bounded Cut and Shortest Path Interdiction, we show that both problems are hard to approximate within any constant factor, even if we allow bicriteria approximation. If we want to cut vertices or the graph is directed, our hardness ratio for Length-Bounded Cut matches the best approximation ratio up to a constant. Previously, the best hardness ratio was 1.1377 for Length-Bounded Cut and 2 for Shortest Path Interdiction. * For any constant k >= 2 and epsilon > 0, we show that Directed Multicut with k source-sink pairs is hard to approximate within a factor k - epsilon. This matches the trivial k-approximation algorithm. By a simple reduction, our result for k = 2 implies that Directed Multiway Cut with two terminals (also known as s-t Bicut} is hard to approximate within a factor 2 - epsilon, matching the trivial 2-approximation algorithm. * Assuming a variant of the UGC (implied by another variant of Bansal and Khot), we prove that it is hard to approximate Resource Minimization Fire Containment within any constant factor. Previously, the best hardness ratio was 2. For directed layered graphs with b layers, our hardness ratio Omega(log b) matches the best approximation algorithm. Our results are based on a general method of converting an integrality gap instance to a length-control dictatorship test for variants of the s-t cut problem, which may be useful for other problems

    Bilu-Linial Stable Instances of Max Cut and Minimum Multiway Cut

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    We investigate the notion of stability proposed by Bilu and Linial. We obtain an exact polynomial-time algorithm for γ\gamma-stable Max Cut instances with γclognloglogn\gamma \geq c\sqrt{\log n}\log\log n for some absolute constant c>0c > 0. Our algorithm is robust: it never returns an incorrect answer; if the instance is γ\gamma-stable, it finds the maximum cut, otherwise, it either finds the maximum cut or certifies that the instance is not γ\gamma-stable. We prove that there is no robust polynomial-time algorithm for γ\gamma-stable instances of Max Cut when γ<αSC(n/2)\gamma < \alpha_{SC}(n/2), where αSC\alpha_{SC} is the best approximation factor for Sparsest Cut with non-uniform demands. Our algorithm is based on semidefinite programming. We show that the standard SDP relaxation for Max Cut (with 22\ell_2^2 triangle inequalities) is integral if γD221(n)\gamma \geq D_{\ell_2^2\to \ell_1}(n), where D221(n)D_{\ell_2^2\to \ell_1}(n) is the least distortion with which every nn point metric space of negative type embeds into 1\ell_1. On the negative side, we show that the SDP relaxation is not integral when γ<D221(n/2)\gamma < D_{\ell_2^2\to \ell_1}(n/2). Moreover, there is no tractable convex relaxation for γ\gamma-stable instances of Max Cut when γ<αSC(n/2)\gamma < \alpha_{SC}(n/2). That suggests that solving γ\gamma-stable instances with γ=o(logn)\gamma =o(\sqrt{\log n}) might be difficult or impossible. Our results significantly improve previously known results. The best previously known algorithm for γ\gamma-stable instances of Max Cut required that γcn\gamma \geq c\sqrt{n} (for some c>0c > 0) [Bilu, Daniely, Linial, and Saks]. No hardness results were known for the problem. Additionally, we present an algorithm for 4-stable instances of Minimum Multiway Cut. We also study a relaxed notion of weak stability.Comment: 24 page

    Fixed-Parameter Tractability of Directed Multiway Cut Parameterized by the Size of the Cutset

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    Given a directed graph GG, a set of kk terminals and an integer pp, the \textsc{Directed Vertex Multiway Cut} problem asks if there is a set SS of at most pp (nonterminal) vertices whose removal disconnects each terminal from all other terminals. \textsc{Directed Edge Multiway Cut} is the analogous problem where SS is a set of at most pp edges. These two problems indeed are known to be equivalent. A natural generalization of the multiway cut is the \emph{multicut} problem, in which we want to disconnect only a set of kk given pairs instead of all pairs. Marx (Theor. Comp. Sci. 2006) showed that in undirected graphs multiway cut is fixed-parameter tractable (FPT) parameterized by pp. Marx and Razgon (STOC 2011) showed that undirected multicut is FPT and directed multicut is W[1]-hard parameterized by pp. We complete the picture here by our main result which is that both \textsc{Directed Vertex Multiway Cut} and \textsc{Directed Edge Multiway Cut} can be solved in time 22O(p)nO(1)2^{2^{O(p)}}n^{O(1)}, i.e., FPT parameterized by size pp of the cutset of the solution. This answers an open question raised by Marx (Theor. Comp. Sci. 2006) and Marx and Razgon (STOC 2011). It follows from our result that \textsc{Directed Multicut} is FPT for the case of k=2k=2 terminal pairs, which answers another open problem raised in Marx and Razgon (STOC 2011)
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