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17
Edge dominating set: efficient enumerationbased exact algorithms
 Proceedings 2nd International Workshop on Parameterized and Exact Computation, IWPEC 2006
, 2006
"... Abstract. We analyze edge dominating set from a parameterized perspective. More specifically, we prove that this problem is in FPT for general (weighted) graphs. The corresponding algorithms rely on enumeration techniques. In particular, we show how the use of compact representations may speed up th ..."
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Abstract. We analyze edge dominating set from a parameterized perspective. More specifically, we prove that this problem is in FPT for general (weighted) graphs. The corresponding algorithms rely on enumeration techniques. In particular, we show how the use of compact representations may speed up the decision algorithm. 1
Approximability of sparse integer programs
 In Proc. 17th ESA
, 2009
"... The main focus of this paper is a pair of new approximation algorithms for sparse integer programs. First, for covering integer programs {min cx: Ax ≥ b,0 ≤ x ≤ d} where A has at most k nonzeroes per row, we give a kapproximation algorithm. (We assume A, b, c, d are nonnegative.) For any k ≥ 2 and ..."
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The main focus of this paper is a pair of new approximation algorithms for sparse integer programs. First, for covering integer programs {min cx: Ax ≥ b,0 ≤ x ≤ d} where A has at most k nonzeroes per row, we give a kapproximation algorithm. (We assume A, b, c, d are nonnegative.) For any k ≥ 2 and ǫ> 0, if P = NP this ratio cannot be improved to k − 1 − ǫ, and under the unique games conjecture this ratio cannot be improved to k − ǫ. One key idea is to replace individual constraints by others that have better rounding properties but the same nonnegative integral solutions; another critical ingredient is knapsackcover inequalities. Second, for packing integer programs {max cx: Ax ≤ b,0 ≤ x ≤ d} where A has at most k nonzeroes per column, we give a 2 k k 2approximation algorithm. This is the first polynomialtime approximation algorithm for this problem with approximation ratio depending only on k, for any k> 1. Our approach starts from iterated LP relaxation, and then uses probabilistic and greedy methods to recover a feasible solution. Note added after publication: This version includes subsequent developments: a O(k 2) approximation for the latter problem using the iterated rounding framework, and several literature reference updates including a O(k)approximation for the same problem by Bansal et al.
Path hitting in acyclic graphs
 In Proceedings of the 14th Annual European Symposium on Algorithms
, 2006
"... Abstract. An instance of the path hitting problem consists of two families of paths, D and H, in a common undirected graph, where each path in H is associated with a nonnegative cost. We refer to D and H as the sets of demand and hitting paths, respectively. When p ∈ H and q ∈ D share at least one ..."
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Abstract. An instance of the path hitting problem consists of two families of paths, D and H, in a common undirected graph, where each path in H is associated with a nonnegative cost. We refer to D and H as the sets of demand and hitting paths, respectively. When p ∈ H and q ∈ D share at least one mutual edge, we say that p hits q. The objective is to find a minimum cost subset of H whose members collectively hit those of D. In this paper we provide constant factor approximation algorithms for path hitting, confined to instances in which the underlying graph is a tree, a spider, or a star. Although such restricted settings may appear to be very simple, we demonstrate that they still capture some of the most basic covering problems in graphs. 1
Distributed algorithms for edge dominating sets
 In: Proc. 29th Annual ACM Symposium on Principles of Distributed Computing (PODC 2010
, 2010
"... An edge dominating set for a graph G is a set D of edges such that each edge of G is in D or adjacent to at least one edge in D. This work studies deterministic distributed approximation algorithms for finding minimumsize edge dominating sets. The focus is on anonymous portnumbered networks: th ..."
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An edge dominating set for a graph G is a set D of edges such that each edge of G is in D or adjacent to at least one edge in D. This work studies deterministic distributed approximation algorithms for finding minimumsize edge dominating sets. The focus is on anonymous portnumbered networks: there are no unique identifiers, but a node of degree d can refer to its neighbours by integers 1, 2,..., d. The present work shows that in the portnumbering model, edge dominating sets can be approximated as follows: in dregular graphs, to within 4 − 6/(d+ 1) for an odd d and to within 4−2/d for an even d; and in graphs with maximum degree ∆, to within 4 − 2/(∆ − 1) for an odd ∆ and to within 4 − 2/∆ for an even ∆. These approximation ratios are tight for all values of d and ∆: there are matching lower bounds.
POLYHEDRAL TECHNIQUES FOR GRAPHIC COVERING PROBLEMS
"... The motivation of this thesis is twofold: (i) designing approximation algorithms for NPhard covering problems in graphs by unearthing polyhedral roots to better understood problems, and (ii) a dual approach in which we hope to expand the foundation of wellunderstood polyhedra. Our design ..."
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Cited by 5 (3 self)
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The motivation of this thesis is twofold: (i) designing approximation algorithms for NPhard covering problems in graphs by unearthing polyhedral roots to better understood problems, and (ii) a dual approach in which we hope to expand the foundation of wellunderstood polyhedra. Our design
Improved Approximation Bounds for Edge Dominating Set in Dense Graphs
"... Abstract. We analyze the simple greedy algorithm that iteratively removes the endpoints of a maximumdegree edge in a graph, where the degree of an edge is the sum of the degrees of its endpoints. This algorithm provides a 2approximation to the minimum edge dominating set and minimum maximal matchi ..."
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Abstract. We analyze the simple greedy algorithm that iteratively removes the endpoints of a maximumdegree edge in a graph, where the degree of an edge is the sum of the degrees of its endpoints. This algorithm provides a 2approximation to the minimum edge dominating set and minimum maximal matching problems. We refine its analysis and give an expression of the approximation ratio that is strictly less than 2 in the cases where the input graph has n vertices and at edges, for ɛ> 1/2. This ratio is shown to be asymptotically tight for least ɛ ` n 2 ɛ> 1/2.
Approximation Hardness of Minimum Edge Dominating Set and Minimum Maximal Matching
"... We provide the rst interesting explicit lower bounds on ecient approximability for two closely related optimization problems in graphs, Minimum Edge Dominating Set and Minimum Maximal Matching. We show that it is NPhard to approximate the solution of both problems to within any constant factor ..."
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Cited by 4 (0 self)
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We provide the rst interesting explicit lower bounds on ecient approximability for two closely related optimization problems in graphs, Minimum Edge Dominating Set and Minimum Maximal Matching. We show that it is NPhard to approximate the solution of both problems to within any constant factor smaller than 6 . The result extends with negligible loss to bounded degree graphs and to everywhere dense graphs.
Integer Programming Formulations for the Minimum Weighted Maximal Matching Problem
 OPTIMIZATION LETTERS
"... Given an undirected graph, the problem of finding a maximal matching that has minimum total weight is NPhard. This problem has been studied extensively from a graph theoretical point of view. Most of the existing literature considers the problem in some restricted classes of graphs and give polyn ..."
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Cited by 2 (2 self)
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Given an undirected graph, the problem of finding a maximal matching that has minimum total weight is NPhard. This problem has been studied extensively from a graph theoretical point of view. Most of the existing literature considers the problem in some restricted classes of graphs and give polynomial time exact or approximation algorithms. On the contrary, we consider the problem on general graphs and approach it from an optimization point of view. In this paper, we develop integer programming formulations for the minimum weighted maximal matching problem and analyze their efficacy on randomly generated graphs. We also compare solutions found by a greedy approximation algorithm, which is based on the literature, against optimal solutions. Our results show that our integer programming formulations are able to solve medium size instances to optimality and suggest further research for improvement.
Decomposition algorithms for solving the minimum weight maximal matching problem. Networks
, 2013
"... We investigate the problem of finding a maximal matching that has minimum total weight on a given edgeweighted graph. Although the minimum weight maximal matching problem is NPhard in general, polynomial time exact or approximation algorithms on several restricted graph classes are given in the li ..."
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We investigate the problem of finding a maximal matching that has minimum total weight on a given edgeweighted graph. Although the minimum weight maximal matching problem is NPhard in general, polynomial time exact or approximation algorithms on several restricted graph classes are given in the literature. In this paper, we propose an exact algorithm for solving several variants of the problem on general graphs. In particular, we develop integer programming formulations for the problem and devise a decomposition algorithm, which is based on a combination of integer programming techniques and combinatorial matching algorithms. Our computational tests on a large suite of randomly generated graphs show that our decomposition approach significantly improves the solvability of the problem compared to the underlying integer programming formulation.
New Results on Polynomial Inapproximability and Fixed Parameter Approximability of edge dominating set
, 2012
"... An edge dominating set in a graph G =(V,E) is a subset S of edges such that each edge in E − S is adjacent to at least one edge in S. The edge dominating set problem, to find an edge dominating set of minimum size, is a basic and important NPhard problem that has been extensively studied in appro ..."
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An edge dominating set in a graph G =(V,E) is a subset S of edges such that each edge in E − S is adjacent to at least one edge in S. The edge dominating set problem, to find an edge dominating set of minimum size, is a basic and important NPhard problem that has been extensively studied in approximation algorithms and parameterized complexity. In this paper, we present improved hardness results and parameterized approximation algorithms for edge dominating set. More precisely, we first show that it is NPhard to approximate edge dominating set in polynomial time within a factor better than 1.18. Next, we give a parameterized approximation schema (with respect to the standard parameter) for the problem and, finally, we develop an O ∗ (1.821 τ)time exact algorithm where τ is the size of a minimum vertex cover of G.