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CONNECTED VERTEX COVERS IN DENSE GRAPHS
"... Abstract. We consider the variant of the minimum vertex cover problem in which we require that the cover induces a connected subgraph. We give new approximation results for this problem in dense graphs, in which either the minimum or the average degree is linear. In particular, we prove tight parame ..."
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Abstract. We consider the variant of the minimum vertex cover problem in which we require that the cover induces a connected subgraph. We give new approximation results for this problem in dense graphs, in which either the minimum or the average degree is linear. In particular, we prove tight parameterized upper bounds on the approximation returned by Savage’s algorithm, and extend a vertex cover algorithm from Karpinski and Zelikovsky to the connected case. The new algorithm approximates the minimum connected vertex cover problem within a factor strictly less than 2 on all dense graphs. All these results are shown to be tight. Finally, we introduce the price of connectivity for the vertex cover problem, defined as the worstcase ratio between the sizes of a minimum connected vertex cover and a minimum vertex cover. We prove that the price of connectivity is bounded by 2/(1 + ε) in graphs with average degree εn, and give a family of neartight examples. Key words: approximation algorithm, vertex cover, connected vertex cover, dense graph. 1.
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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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.
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.
A Tight Approximation Bound for the Stable Marriage Problem with Restricted Ties
"... The problem of finding a maximum cardinality stable matching in the presence of ties and unacceptable partners, called MAX SMTI, is a wellstudied NPhard problem. The MAX SMTI is NPhard even for highly restricted instances where (i) ties appear only in women’s preference lists and (ii) each tie ap ..."
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The problem of finding a maximum cardinality stable matching in the presence of ties and unacceptable partners, called MAX SMTI, is a wellstudied NPhard problem. The MAX SMTI is NPhard even for highly restricted instances where (i) ties appear only in women’s preference lists and (ii) each tie appears at the end of each woman’s preference list. The current best lower bounds on the approximation ratio for this variant are 1.1052 unless P=NP and 1.25 under the unique games conjecture, while the current best upper bound is 1.4616. In this paper, we improve the upper bound to 1.25, which matches the lower bound under the unique games conjecture. Note that this is the first special case of the MAX SMTI where the tight approximation bound is obtained. The improved ratio is achieved via a new analysis technique, which avoids the complicated casebycase analysis used in earlier studies. As a byproduct of our analysis, we show that the integrality gap of natural IP and LP formulations for this variant is 1.25. We also show that the unrestricted MAX SMTI cannot be approximated with less than 1.5 unless the approximation ratio of a certain special case of the minimum maximal matching problem can be improved.