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14
A Linear Time Approximation Algorithm for Weighted Matchings in Graphs
, 2003
"... Approximation algorithms have so far mainly been studied for problems that are not known to have polynomial time algorithms for solving them exactly. Here we propose an approximation algorithm for the weighted matching problem in graphs which can be solved in polynomial time. The weighted matching p ..."
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Cited by 25 (3 self)
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Approximation algorithms have so far mainly been studied for problems that are not known to have polynomial time algorithms for solving them exactly. Here we propose an approximation algorithm for the weighted matching problem in graphs which can be solved in polynomial time. The weighted matching problem is to find a matching in an edge weighted graph that has maximum weight. The first polynomial time algorithm for this problem was given by Edmonds in 1965. The fastest known algorithm for the weighted matching problem has a running time of O(nm+n 2 log n). Many real world problems require graphs of such large size that this running time is too costly. Therefore there is considerable need for faster approximation algorithms for the weighted matching problem. We present a linear time approximation algorithm for the weighted matching problem with a performance ratio arbitrarily close to 2/3
A GraphTheoretic Network Security Game
 INT. J. AUTONOMOUS AND ADAPTIVE COMMUNICATIONS SYSTEMS
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Linear Time Local Improvements for Weighted Matchings in Graphs
 IN INTERNATIONAL WORKSHOP ON EXPERIMENTAL AND ECIENT ALGORITHMS (WEA), LNCS 2647
, 2003
"... Recently two different linear time approximation algorithms for the weighted matching problem in graphs have been suggested [5][17]. Both these ..."
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Cited by 11 (2 self)
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Recently two different linear time approximation algorithms for the weighted matching problem in graphs have been suggested [5][17]. Both these
Approximating weighted matchings in parallel
"... revised Version Abstract. algorithm for the weighted matching problem in graphs with an approximation ratio of (1 − ɛ). This improves the previously best approximation ratio of − ɛ) of an NC algorithm for this problem. ..."
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Cited by 8 (1 self)
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revised Version Abstract. algorithm for the weighted matching problem in graphs with an approximation ratio of (1 − ɛ). This improves the previously best approximation ratio of − ɛ) of an NC algorithm for this problem.
The Complexity of Finding Subgraphs Whose Matching Number Equals the Vertex Cover Number
 In Proceedings of the 18th International Symposium on Algorithms and Computation (ISAAC 2007), Springer LNCS
, 2007
"... Abstract. The class of graphs where the size of a minimum vertex cover equals that of a maximum matching is known as KönigEgerváry graphs. KönigEgerváry graphs have been studied extensively from a graph theoretic point of view. In this paper, we introduce and study the algorithmic complexity o ..."
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Cited by 7 (3 self)
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Abstract. The class of graphs where the size of a minimum vertex cover equals that of a maximum matching is known as KönigEgerváry graphs. KönigEgerváry graphs have been studied extensively from a graph theoretic point of view. In this paper, we introduce and study the algorithmic complexity of finding maximumKönigEgerváry subgraphs of a given graph. More specifically, we look at the problem of finding a minimum number of vertices or edges to delete to make the resulting graph KönigEgerváry. We show that both these versions are NPcomplete and study their complexity from the points of view of approximation and parameterized complexity. En route, we point out an interesting connection between the vertex deletion version and the A G V C problem where one is interested in the parameterized complexity of the V C problem when parameterized by the ‘additional number of vertices ’ needed beyond the matching size. This connection is of independent interest and could be useful in establishing the parameterized complexity of A G V C problem. 1
On Parallel Complexity of Maximum FMatching and the Degree Sequence Problem
, 1994
"... We present a randomized NC solution to the problem of constructing a maximum (cardinality) fmatching. As a corollary, we obtain a randomized NC algorithm for the problem of constructing a graph satisfying a sequence d 1 ; d 2 ;...; d n of equality degree constraints. We provide an optimal NC alg ..."
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Cited by 3 (0 self)
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We present a randomized NC solution to the problem of constructing a maximum (cardinality) fmatching. As a corollary, we obtain a randomized NC algorithm for the problem of constructing a graph satisfying a sequence d 1 ; d 2 ;...; d n of equality degree constraints. We provide an optimal NC algorithm for the decision version of the degree sequence problem and an approximation NC algorithm for the construction version of this problem. Our main result is an NC algorithm for constructing if possible a graph satisfying the degree constraints d 1 ; d 2 ;...; d n in case d i q P n j=1 d j =5 for i = 1; :::; n: 1 Introduction Finding a maximum (cardinality) matching in a graph is a fundamental problem in combinatorial optimization. It is a major open problem whether a maximum matching can be constructed by an NC algorithm. Achieving simultaneously a polylogtime and a polynomial number of processors is possible for this problem if random bits are used. Randomized NC algorithms ...
Parameterized Algorithm for Eternal Vertex Cover
, 2009
"... In this paper we initiate the study of a “dynamic ” variant of the classical Vertex Cover problem, the Eternal Vertex Cover problem, from parameterized algorithmic perspective. Klostermeyer and Mynhardt introduced the Eternal Vertex Cover problem, which consists in placing a minimum number of guards ..."
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Cited by 2 (2 self)
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In this paper we initiate the study of a “dynamic ” variant of the classical Vertex Cover problem, the Eternal Vertex Cover problem, from parameterized algorithmic perspective. Klostermeyer and Mynhardt introduced the Eternal Vertex Cover problem, which consists in placing a minimum number of guards on the vertices of a graph such that these guards can protect the graph from any sequence of attacks on its edges. In response to an attack, each guard is allowed either to stay in his vertex, or to move to a neighboring vertex. However, at least one guard has to fix the attacked edge by moving along it. The other guards may move to reconfigure and prepare for the next attack. Thus at every step the vertices occupied by guards form a vertex cover. We show that the problem admits a kernel of size 4 k (k +1)+2k, which shows that the problem is fixed parameter tractable when parameterized by the number of available guards k. Finally, we also provide an algorithm with running time O(2 O(k2) + (1.2738) k m + n) for Eternal Vertex Cover, where n is the number of vertices and m the number of edges of the input graph. In passing we also observe that Eternal Vertex Cover is NPhard but that there is a polynomial time 2approximation algorithm. 1
Expander Graphs and Gaps between Primes
"... The explicit construction of infinite families of dregular graphs which are Ramanujan is known only in the case d−1 is a prime power. In this paper, we consider the case when d − 1 is not a prime power. The main result is that by perturbing known Ramanujan graphs and using results about gaps betwee ..."
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The explicit construction of infinite families of dregular graphs which are Ramanujan is known only in the case d−1 is a prime power. In this paper, we consider the case when d − 1 is not a prime power. The main result is that by perturbing known Ramanujan graphs and using results about gaps between consecutive primes, we are able to construct infinite families of “almost ” Ramanujan graphs for almost every value of d. More precisely, for any fixed ǫ> 0 and for almost every value of d (in the sense of natural density), there are infinitely many dregular graphs such that all the nontrivial eigenvalues of the adjacency matrices of these graphs have absolute value less than (2 + ǫ) √ d − 1. 1
A Combinatoric Interpretation of Dual Variables for Weighted Matching Problems
, 2012
"... We consider four weighted matchingtype problems: the bipartite graph and general graph versions of matching and ffactors. The linear program duals for these problems are shown to be weights of certain subgraphs. Specifically the socalled y duals are the weights of certain maximum matchings or ff ..."
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We consider four weighted matchingtype problems: the bipartite graph and general graph versions of matching and ffactors. The linear program duals for these problems are shown to be weights of certain subgraphs. Specifically the socalled y duals are the weights of certain maximum matchings or ffactors; z duals (used for general graphs) are the weights of certain 2factors or 2ffactors. The y duals are canonical in a welldefined sense; z duals are canonical for matching and more generally for bmatchings (a special case of ffactors) but for ffactors their support can vary. As weights of combinatorial objects the duals are integral for given integral edge weights, and so they give new proofs that the linear programs for these problems are TDI. 1