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13
Edge Disjoint Paths Revisited
 In Proceedings of the 14th ACMSIAM Symposium on Discrete Algorithms
, 2003
"... The approximability of the maximum edge disjoint paths problem (EDP) in directed graphs was seemingly settled by the )hardness result of Guruswami et al. [10] and the O( m) approximation achievable via both the natural LP relaxation [19] and the greedy algorithm [11, 12]. Here m is the numb ..."
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Cited by 40 (4 self)
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The approximability of the maximum edge disjoint paths problem (EDP) in directed graphs was seemingly settled by the )hardness result of Guruswami et al. [10] and the O( m) approximation achievable via both the natural LP relaxation [19] and the greedy algorithm [11, 12]. Here m is the number of edges in the graph. However, we observe that the hardness of approximation shown in [10] applies to sparse graphs and hence when expressed as a function of n, the number of vertices, only an \Omega\Gamma n )hardness follows. On the other hand, the O( m)approximation algorithms do not guarantee a sublinear (in terms of n) approximation algorithm for dense graphs. We note that a similar gap exists in the known results on the integrality gap of the natural LP relaxation: an \Omega\Gamma n) lower bound and an O( m) upper bound. Motivated by this discrepancy in the upper and lower bounds we study algorithms for the EDP in directed and undirected graphs obtaining improved approximation ratios. We show that the greedy algorithm has an approximation ratio of O(min(n m)) in undirected graphs and a ratio of O(min(n m)) in directed graphs. For ayclic graphs we give an O( n log n) approximation via LP rounding. These are the first sublinear approximation ratios for EDP. Our results also extend to EDP with weights and to the unsplittable flow problem with uniform edge capacities.
Polynomial Flowcut Gaps and Hardness of Directed Cut Problems
 In Proc. of STOC, 2007
"... We study the multicut and the sparsest cut problems in directed graphs. In the multicut problem, we are a given an nvertex graph G along with k sourcesink pairs, and the goal is to find the minimum cardinality subset of edges whose removal separates all sourcesink pairs. The sparsest cut problem ..."
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Cited by 25 (0 self)
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We study the multicut and the sparsest cut problems in directed graphs. In the multicut problem, we are a given an nvertex graph G along with k sourcesink pairs, and the goal is to find the minimum cardinality subset of edges whose removal separates all sourcesink pairs. The sparsest cut problem has the same input, but the goal is to find a subset of edges to delete so as to minimize the ratio of the number of deleted edges to the number of sourcesink pairs that are separated by this deletion. The natural linear programming relaxation for multicut corresponds, by LPduality, to the wellstudied maximum (fractional) multicommodity flow problem, while the standard LPrelaxation for sparsest cut corresponds to maximum concurrent flow. Therefore, the integrality gap of the linear programming relaxation for multicut/sparsest cut is also the flowcut gap: the largest gap, achievable for any graph, between the maximum flow value and the minimum cost solution for the corresponding cut problem. Our first result is that the flowcut gap between maximum multicommodity flow and minimum multicut is ˜ Ω(n 1/7) in directed graphs. We show a similar result for the gap between maximum concurrent flow and sparsest cut in directed graphs. These results improve upon a
Approximating minimum multicuts by evolutionary multiobjective algorithms
 In Proc. of Parallel Problem Solving from Nature (PPSN’08
, 2008
"... Abstract. 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, ..."
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Abstract. 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.
On the MaxFlow MinCut Ratio for Directed Multicommodity Flows
 Theor. Comput. Sci
, 2003
"... We give a pure combinatorial problem whose solution determines maxflow mincut ratio for directed multicommodity flows. In addition, this combinatorial problem has applications in improving the approximation factor of Greedy algorithm for maximum edge disjoint path problem. More precisely, our u ..."
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Cited by 7 (1 self)
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We give a pure combinatorial problem whose solution determines maxflow mincut ratio for directed multicommodity flows. In addition, this combinatorial problem has applications in improving the approximation factor of Greedy algorithm for maximum edge disjoint path problem. More precisely, our upper bound improves the approximation factor for this problem to O(n ). Finally, we demonstrate how even for very simple graphs the aforementioned ratio might be very large.
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 5 (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
The Generalized Deadlock Resolution Problem
"... Abstract. In this paper we initiate the study of the ANDOR directed feedback vertex set problem from the viewpoint of approximation algorithms. This ANDOR feedback vertex set problem is motivated by a practical deadlock resolution problem that appears in the development of distributed database sys ..."
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Abstract. In this paper we initiate the study of the ANDOR directed feedback vertex set problem from the viewpoint of approximation algorithms. This ANDOR feedback vertex set problem is motivated by a practical deadlock resolution problem that appears in the development of distributed database systems 1.This problem also turns out be a natural generalization of the directed feedback vertex set problem. Awerbuch and Micali [1] gave a polynomial time algorithm to find a minimal solution for this problem. Unfortunately, a minimal solution can be arbitrarily more expensive than the minimum cost solution. We show that finding the minimum cost solution is as hard as the directed Steiner tree problem (and thus Ω(log 2 n) hard to approximate). On the positive side, we give algorithms which work well when the number of writers (AND nodes) or the number of readers (OR nodes) are small. We also consider a variant that we call permanent deadlock resolution where we cannot specify an execution order for the surviving processes; they should get completed even if they were scheduled adversarially. When all processes are writers (AND nodes), we give an O(log n log log n) approximation for this problem. Finally we give an LProunding approach and discuss some other natural variants. 1
The Checkpoint Problem
"... In this paper we consider the checkpoint problem. The input consists of an undirected graph G, a set of sourcedestination pairs {(s1, t1),..., (sk, tk)}, and a collection P of paths connecting the (si, ti) pairs. A feasible solution is a multicut E ′ ; namely, a set of edges whose removal disconnec ..."
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In this paper we consider the checkpoint problem. The input consists of an undirected graph G, a set of sourcedestination pairs {(s1, t1),..., (sk, tk)}, and a collection P of paths connecting the (si, ti) pairs. A feasible solution is a multicut E ′ ; namely, a set of edges whose removal disconnects every sourcedestination pair. For each p ∈ P we define cpE ′(p) = p ∩ E ′ . In the sum checkpoint (SCP) problem the goal is to minimize ∑ p∈P cpE ′(p), while in the maximum checkpoint (MCP) problem the goal is to minimize maxp∈P cpE ′(p). These problem have several natural applications, e.g., in urban transportation and network security. In a sense, they combine the multicut problem and the minimum membership set cover problem. For the sum objective we show that weighted SCP is equivalent, with respect to approximability, to undirected multicut. Thus there exists an O(log n) approximation for SCP in general graphs. Our current approximability results for the max objective have a wide gap: we provide an approximation factor of O ( √ n log n/opt) for MCP and a hardness of 2 under the assumption P ̸ = NP. The hardness holds for trees, in which case we can obtain an asymptotic approximation factor of 2. Finally we show strong hardness for the wellknown problem of finding a path with minimum forbidden pairs, which in a sense can be considered the dual to the checkpoint problem. Despite various works on this problem, hardness of approximation was not known prior to this work. We show that the problem cannot be approximated within c n for some constant c> 0, unless P = NP. This is the strongest type of hardness possible. It carries over to directed acyclic graphs and is a huge improvement over the plain NPhardness of Gabow (SIAM J. Comp 2007, pages 1648–1671).