Results 1  10
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25
Fixedparameter tractability of multicut parameterized by the size of the cutset
, 2011
"... Given an undirected graph G, a collection {(s1, t1),...,(sk, tk)} of pairs of vertices, and an integer p, the EDGE MULTICUT problem ask if there is a set S of at most p edges such that the removal of S disconnects every si from the corresponding ti. VERTEX MULTICUT is the analogous problem where S i ..."
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Cited by 33 (5 self)
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Given an undirected graph G, a collection {(s1, t1),...,(sk, tk)} of pairs of vertices, and an integer p, the EDGE MULTICUT problem ask if there is a set S of at most p edges such that the removal of S disconnects every si from the corresponding ti. VERTEX MULTICUT is the analogous problem where S is a set of at most p vertices. Our main result is that both problems can be solved in time 2O(p3) · nO(1), i.e., fixedparameter tractable parameterized by the size p of the cutset in the solution. By contrast, it is unlikely that an algorithm with running time of the form f (p) · nO(1) exists for the directed version of the problem, as we show it to be W[1]hard parameterized by the size of the cutset.
Improved approximation for directed cut problems
, 2007
"... We present improved approximation algorithms for directed multicut and directed sparsest cut. The current best known approximation ratio for these problems is O(n 1/2). We obtain an Õ(n11/23)approximation. Our algorithm works with the natural LP relaxation used in prior work. We use a randomized ro ..."
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Cited by 21 (0 self)
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We present improved approximation algorithms for directed multicut and directed sparsest cut. The current best known approximation ratio for these problems is O(n 1/2). We obtain an Õ(n11/23)approximation. Our algorithm works with the natural LP relaxation used in prior work. We use a randomized rounding algorithm with a more sophisticated charging scheme and analysis to obtain our improvement. This also implies a Õ(n11/23) upper bound on the ratio between the maximum multicommodity flow and minimum multicut in directed graphs.
On the unique games conjecture
 In FOCS
, 2005
"... This article surveys recently discovered connections between the Unique Games Conjecture and computational complexity, algorithms, discrete Fourier analysis, and geometry. 1 ..."
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Cited by 15 (1 self)
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This article surveys recently discovered connections between the Unique Games Conjecture and computational complexity, algorithms, discrete Fourier analysis, and geometry. 1
A (log n)Ω(1) integrality gap for the Sparsest Cut SDP
 In Proceedings of 50th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2009
, 2009
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A (log n) Ω(1) integrality gap for the Sparsest Cut SDP
 IN PROCEEDINGS OF 50TH ANNUAL IEEE SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE (FOCS 2009
, 2009
"... We show that the GoemansLinial semidefinite relaxation of the Sparsest Cut problem with general demands has integrality gap (log n) Ω(1). This is achieved by exhibiting npoint metric spaces of negative type whose L1 distortion is (log n) Ω(1). Our result is based on quantitative bounds on the ra ..."
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Cited by 11 (3 self)
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We show that the GoemansLinial semidefinite relaxation of the Sparsest Cut problem with general demands has integrality gap (log n) Ω(1). This is achieved by exhibiting npoint metric spaces of negative type whose L1 distortion is (log n) Ω(1). Our result is based on quantitative bounds on the rate of degeneration of Lipschitz maps from the Heisenberg group to L1 when restricted to cosets of the center.
Approximating Minimum Multicuts by Evolutionary MultiObjective Algorithms
"... It has been shown that simple evolutionary algorithms are able to solve the minimum cut problem in expected polynomial time when using a multiobjective model of the problem. In this paper, we generalize these ideas to the NPhard minimum multicut problem. Given a set of k terminal pairs, we prove t ..."
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Cited by 11 (4 self)
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It has been shown that simple evolutionary algorithms are able to solve the minimum cut problem in expected polynomial time when using a multiobjective model of the problem. In this paper, we generalize these ideas to the NPhard minimum multicut problem. Given a set of k terminal pairs, we prove that evolutionary algorithms in combination with a multiobjective model of the problem are able to obtain a kapproximation for this problem in expected polynomial time.
Multicommodity Flows and Cuts in Polymatroidal Networks
, 2011
"... We consider multicommodity flow and cut problems in polymatroidal networks where there are submodular capacity constraints on the edges incident to a node. Polymatroidal networks were introduced by Lawler and Martel [24] and Hassin [18] in the singlecommodity setting and are closely related to the ..."
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Cited by 6 (3 self)
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We consider multicommodity flow and cut problems in polymatroidal networks where there are submodular capacity constraints on the edges incident to a node. Polymatroidal networks were introduced by Lawler and Martel [24] and Hassin [18] in the singlecommodity setting and are closely related to the submodular flow model of Edmonds and Giles [10]; the wellknown maxflowmincut theorem generalizes to this more general setting. Polymatroidal networks for the multicommodity case have not, as far as the authors are aware, been previously explored. Our work is primarily motivated by applications to information flow in wireless networks. We also consider the notion of undirected polymatroidal networks and observe that they provide a natural way to generalize flows and cuts in edge and node capacitated undirected networks. We establish polylogarithmic flowcut gap results in several scenarios that have been previously considered in the standard network flow models where capacities are on the edges or nodes [25, 26, 14, 23, 13]. Our results from a preliminary version have already found applications in wireless network information flow [20, 21] and we anticipate more in the future. On the technical side our key tools are the formulation and analysis of the dual of the flow relaxations via continuous extensions of submodular functions, in particular the Lovász extension. For directed graphs we rely on a simple yet useful reduction from
Sparsest cut on bounded treewidth graphs: Algorithms and hardness results
 In 45th Annual ACM Symposium on Symposium on Theory of Computing
, 2013
"... We give a 2approximation algorithm for NonUniform Sparsest Cut that runs in time nO(k), where k is the treewidth of the graph. This improves on the previous 22 kapproximation in time poly(n)2O(k) due to Chlamtác ̌ et al. [CKR10]. To complement this algorithm, we show the following hardness resul ..."
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Cited by 4 (0 self)
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We give a 2approximation algorithm for NonUniform Sparsest Cut that runs in time nO(k), where k is the treewidth of the graph. This improves on the previous 22 kapproximation in time poly(n)2O(k) due to Chlamtác ̌ et al. [CKR10]. To complement this algorithm, we show the following hardness results: If the NonUniform Sparsest Cut problem has a ρapproximation for seriesparallel graphs (where ρ ≥ 1), then the MaxCut problem has an algorithm with approximation factor arbitrarily close to 1/ρ. Hence, even for such restricted graphs (which have treewidth 2), the Sparsest Cut problem is NPhard to approximate better than 17/16 − ε for ε> 0; assuming the Unique Games Conjecture the hardness becomes 1/αGW − ε. For graphs with large (but constant) treewidth, we show a hardness result of 2 − ε assuming the Unique Games Conjecture. Our algorithm rounds a linear program based on (a subset of) the SheraliAdams lift of the standard Sparsest Cut LP. We show that even for treewidth2 graphs, the LP has an integrality gap close to 2 even after polynomially many rounds of SheraliAdams. Hence our approach cannot be improved even on such restricted graphs without using a stronger relaxation. 1
An informationtheoretic metatheorem on edgecut bounds
 IN PROCEEDINGS IEEE INTERNATIONAL SYMPOSIUM ON INFORMATION THEORY (ISIT
, 2012
"... We consider the problem of multiple unicast in wireline networks. Edgecut based bounds which are simple bounds on the rates achievable by routing flow are not in general, fundamental, i.e. they are not outer bounds on the capacity region. It has been observed that when the problem has some kind of ..."
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Cited by 3 (1 self)
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We consider the problem of multiple unicast in wireline networks. Edgecut based bounds which are simple bounds on the rates achievable by routing flow are not in general, fundamental, i.e. they are not outer bounds on the capacity region. It has been observed that when the problem has some kind of symmetry involved, then flows and edgecut based bounds are ‘close’, i.e. within a constant or polylogarithmic factor of each other. In this paper, we make the observation that in these very cases, such edgecut based bounds are actually ‘close ’ to fundamental yielding an approximate characterization of the capacity region for these problems. We demonstrate this in the case of kunicast in undirected networks, kpair unicast in directed networks with symmetric demands i.e. for every source communicating to a destination at a certain rate, the destination communicates an independent message back to the source at the same rate, and sumrate of kgroupcast in directed networks, i.e. a group of nodes, each of which has an independent message for every other node in the group. We place our work in context of existing results to suggest a metatheorem: if there is inherent symmetry either in the network connectivity or in the traffic pattern, then edgecut bounds are nearfundamental and flows approximately achieve capacity.