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21
Approximating Minimum Feedback Sets and Multicuts in Directed Graphs
 ALGORITHMICA
, 1998
"... This paper deals with approximating feedback sets in directed graphs. We consider two related problems: the weighted feedback vertex set (fvs) problem, and the weighted feedback edge set problem (fes). In the fvs (resp. fes) problem, one is given a directed graph with weights (each of which is at le ..."
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Cited by 97 (3 self)
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This paper deals with approximating feedback sets in directed graphs. We consider two related problems: the weighted feedback vertex set (fvs) problem, and the weighted feedback edge set problem (fes). In the fvs (resp. fes) problem, one is given a directed graph with weights (each of which is at least 1) on the vertices (resp. edges), and is asked to find a subset of vertices (resp. edges) with minimum total weight that intersects every directed cycle in the graph. These problems are among the classical NPHard problems and have many applications. We also consider a generalization of these problems: subsetfvs and subsetfes, in which the feedback set has to intersect only a subset of the directed cycles in the graph. This subset consists of all the cycles that go through a distinguished input subset of vertices and edges, denoted by X . This generalization is also NPHard even when X = 2. We present approximation algorithms for the subsetfvs and subsetfes problems. The first algorithm we present achieves an approximation factor of O(log2 X). The second algorithm achieves an approximation factor of O(min(log tau log log tau; log n log log n)), where tau is the value of the optimum fractional solution of the problem at hand, and n is the number of vertices in the graph. We also define a multicut problem in a special type of directed networks which we call circular networks, and show that the subsetfes and subsetfvs problems are equivalent to this multicut problem. Another contribution of our paper is a combinatorial algorithm that computes a (1 + epsilon) approximation to the fractional optimal feedback vertex set. Computing the approximate solution is much simpler and more efficient than general linear programming methods. All of our algorithms use this approximate solution.
The Importance of Being Biased
, 2002
"... The Minimum Vertex Cover problem is the problem of, given a graph, finding a smallest set of vertices that touches all edges. We show that it is NPhard to approximate this problem 1.36067, improving on the previously known hardness result for a 6 factor. 1 ..."
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Cited by 85 (8 self)
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The Minimum Vertex Cover problem is the problem of, given a graph, finding a smallest set of vertices that touches all edges. We show that it is NPhard to approximate this problem 1.36067, improving on the previously known hardness result for a 6 factor. 1
The Approximability of Constraint Satisfaction Problems
 SIAM J. Comput
, 2001
"... We study optimization problems that may be expressed as "Boolean constraint satisfaction problems." An instance of a Boolean constraint satisfaction problem is given by m constraints applied to n Boolean variables. Di#erent computational problems arise from constraint satisfaction problems depending ..."
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Cited by 69 (2 self)
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We study optimization problems that may be expressed as "Boolean constraint satisfaction problems." An instance of a Boolean constraint satisfaction problem is given by m constraints applied to n Boolean variables. Di#erent computational problems arise from constraint satisfaction problems depending on the nature of the "underlying" constraints as well as on the goal of the optimization task. Here we consider four possible goals: Max CSP (Min CSP) is the class of problems where the goal is to find an assignment maximizing the number of satisfied constraints (minimizing the number of unsatisfied constraints). Max Ones (Min Ones) is the class of optimization problems where the goal is to find an assignment satisfying all constraints with maximum (minimum) number of variables set to 1. Each class consists of infinitely many problems and a problem within a class is specified by a finite collection of finite Boolean functions that describe the possible constraints that may be used.
Rounding via Trees: Deterministic Approximation Algorithms for Group Steiner Trees and kmedian
"... ..."
Fast Approximate Graph Partitioning Algorithms
 SIAM Journal on Computing
, 1999
"... . We study graph partitioning problems on graphs with edge capacities and vertex weights. The problems of bbalanced cuts and kbalanced partitions are unified into a new problem called minimum capacity #separators. A #separator is a subset of edges whose removal partitions the vertex set into con ..."
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Cited by 53 (6 self)
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. We study graph partitioning problems on graphs with edge capacities and vertex weights. The problems of bbalanced cuts and kbalanced partitions are unified into a new problem called minimum capacity #separators. A #separator is a subset of edges whose removal partitions the vertex set into connected components such that the sum of the vertex weights in each component is at most # times the weight of the graph. We present a new and simple O(log n) approximation algorithm for minimum capacity #separators which is based on spreading metrics yielding an O(log n)approximation algorithm both for bbalanced cuts and kbalanced partitions. In particular, this result improves the previous best known approximation factor for kbalanced partitions in undirected graphs by a factor of O(log k). We enhance these results by presenting a version of the algorithm that obtains an O(log opt)approximation factor. The algorithm is based on a technique called spreading metrics that enables us to...
Constraint satisfaction: The approximability of minimization problems
 Proc. 12th Annual Conference on Structure in Complexity Theory, IEEE
, 1997
"... This paper continues the work initiated by Creignou [5] and Khanna, Sudan and Williamson [15] who classify maximization problems derived from Boolean constraint satisfaction. Here we study the approximability of minimization problems derived thence. A problem in this framework is characterized by a ..."
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Cited by 45 (5 self)
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This paper continues the work initiated by Creignou [5] and Khanna, Sudan and Williamson [15] who classify maximization problems derived from Boolean constraint satisfaction. Here we study the approximability of minimization problems derived thence. A problem in this framework is characterized by a collection�of “constraints” (i.e., functions����������) and an instance of a problem is constraints drawn from�applied to specified subsets of Boolean variables. We study the two minimization analogs of classes studied in [15]: in one variant, namely MIN CSP�, the objective is to find an assignment to minimize the number of unsatisfied constraints, while in the other, namely MIN ONES�, the goal is to find a satisfying assignment with minimum number of ones. These two classes together capture an entire spectrum of important minimization problems including Min Cut, vertex cover, hitting set with bounded size sets, integer programs with two variables per inequality, graph bipartization, clause deletion in CNF formulae, and nearest codeword. Our main result is that there exists a finite partition of the space of all constraint sets such that for any given�, the approximability of MIN CSP�and MIN ONES� is completely determined by the partition containing it. Moreover, we present a compact set of rules that determines which partition contains a given family�. Our classification identifies the central elements governing the approximability of problems in these classes, by unifying a large collection algorithmic and hardness of approximation results. When contrasted with the work of [15], our results also serve to formally highlight inherent differences between maximization and minimization problems.
Approximating Minimum Subset Feedback Sets in Undirected Graphs with Applications to Multicuts
, 1996
"... Let G = (V; E) be a weighted undirected graph where all weights are at least one. We consider the following generalization of feedback set problems. Let S ae V be a subset of the vertices. A cycle is called interesting if it intersects the set S. A subset feedback edge (vertex) set is a subset of th ..."
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Cited by 17 (0 self)
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Let G = (V; E) be a weighted undirected graph where all weights are at least one. We consider the following generalization of feedback set problems. Let S ae V be a subset of the vertices. A cycle is called interesting if it intersects the set S. A subset feedback edge (vertex) set is a subset of the edges (vertices) that intersects all interesting cycles. In minimum subset feedback problems the goal is to find such sets of minimumweight. The case in which S consists of a single vertex is equivalent to the multiway cut problem, in which the goal is to separate a given set of terminals. Hence, the subset feedback problem is NPcomplete, and also generalizes the multiway cut problem. We provide a polynomialtime algorithm for approximating the subset feedback edge set problem that achieves an approximation factor of two. For the subset feedback vertex set problem we achieve an approximation factor of minf2\Delta; O(log jSj); O(log ø )g, where \Delta is the maximum degree in G and ø ...
Oblivious routing on nodecapacitated and directed graphs
 IN PROCEEDINGS OF THE 16TH ACMSIAM SYMPOSIUM ON DISCRETE ALGORITHMS (SODA), 2005
, 2005
"... Oblivious routing algorithms for general undirected networks were introduced by Räcke [17], and this work has led to many subsequent improvements and applications. Comparatively little is known about oblivious routing in general directed networks, or even in undirected networks with node capacities. ..."
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Cited by 16 (8 self)
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Oblivious routing algorithms for general undirected networks were introduced by Räcke [17], and this work has led to many subsequent improvements and applications. Comparatively little is known about oblivious routing in general directed networks, or even in undirected networks with node capacities. We present the first nontrivial upper bounds for both these cases, providing algorithms for kcommodity oblivious routing problems with competitive ratio O (√ k log(n)) for undirected nodecapacitated graphs and O (√ k n 1/4 log(n)) for directed graphs. In the special case that all commodities have a common source or sink, our upper bound becomes O ( √ n log(n)) in both cases, matching the lower bound up to a factor of log(n). The lower bound (which first appeared in [6]) is obtained on a graph with very high degree. We show that in fact the degree of a graph is a crucial parameter for nodecapacitated oblivious routing in undirected graphs, by providing an O(∆ polylog(n))competitive oblivious routing scheme for graphs of degree ∆. For the directed case, however, we show that the lower bound of Ω (√ n) still holds in lowdegree graphs. Finally, we settle an open question about routing problems in which all commodities share a common source or sink. We show that even in this simplified scenario there are networks in which no oblivious routing algorithm can achieve a competitive ratio better than Ω(log n).