Minimum Edge Dominating Sets

Horton, J. D.; Kilakos, K.

doi:10.1137/0406030

Cited by 104 publications

(64 citation statements)

References 14 publications

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“…By Proposition 3, MIN-POM belongs to NP. To show NP-hardness, we give a reduction from the NP-complete restriction of MMM to subdivision graphs [6] (given a graph G, the subdivision graph of G is obtained by subdividing each edge e = {u, w} into two edges {u, v e }, {v e , w}, where v e is a new vertex corresponding to e).…”

Section: Theorem 2 Min-pom Is Np-completementioning

confidence: 99%

Pareto Optimality in House Allocation Problems

Abraham

Cechlárová

Manlove

et al. 2004

Lecture Notes in Computer Science

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Abstract. We study Pareto optimal matchings in the context of house allocation problems. We present an O( √ nm) algorithm, based on Gale's Top Trading Cycles Method, for finding a maximum cardinality Pareto optimal matching, where n is the number of agents and m is the total length of the preference lists. By contrast, we show that the problem of finding a minimum cardinality Pareto optimal matching is NP-hard, though approximable within a factor of 2. We then show that there exist Pareto optimal matchings of all sizes between a minimum and maximum cardinality Pareto optimal matching. Finally, we introduce the concept of a signature, which allows us to give a characterization, checkable in linear time, of instances that admit a unique Pareto optimal matching.

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Section: Theorem 2 Min-pom Is Np-completementioning

confidence: 99%

Pareto Optimality in House Allocation Problems

Abraham

Cechlárová

Manlove

et al. 2004

Lecture Notes in Computer Science

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“…The problem of focus in our paper is the maximal independent edge set problem (Cormen et al, 2009) wherein we want to find the largest set of independent edges such that no two edges have a common end vertex. Note that heuristics (Horton & Kilakos, 1993) for the minimum edge dominating set problem cannot be applied to determine the maximal a(di)ssortative matching. Because, heuristics for the minimum edge set problem are more likely to determine the set of edges such that each edge in the set covers a larger number of adjacent edges.…”

Section: Related Workmentioning

confidence: 99%

“…It is equivalent to the problem of finding a minimum edge dominating set (Horton & Kilakos, 1993) -to find the smallest set of edges of the graph such that each edge in the set covers itself and covers one or more adjacent edges as well as satisfies the matching constraint (no two edges in the set have a common end vertex). The problem of focus in our paper is the maximal independent edge set problem (Cormen et al, 2009) wherein we want to find the largest set of independent edges such that no two edges have a common end vertex.…”

Section: Related Workmentioning

confidence: 99%

On the Conduciveness of Random Network Graphs for Maximal Assortative or Maximal Dissortative Matching

Meghanathan

2015

CIS

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A maximal matching of a graph is the set of edges such that the addition of an edge to this set violates the property of matching (i.e., no two edges of the matching share a vertex). We use the notion of assortative index (ranges from -1 to 1) to evaluate the extent of similarity of the end vertices constituting the edges of a matching. A maximal matching of the edges whose assortative index is as close as possible to 1 is referred to as maximal assortative matching (MAM) and a maximal matching of the edges whose assortative index is as close as possible to -1 is referred to as maximal dissortative matching (MDM). We present algorithms to determine the MAM and MDM of the edges in a network graph. Through extensive simulations, we conclude that random network graphs are more conducive for maximal dissortative matching rather than maximal assortative matching. We observe the assortative index of an MDM on random network graphs to be relatively more closer to the targeted optimal value of -1 compared to the assortative index of an MAM to the targeted optimal value of 1.

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“…Another consequence of our results is. In [11], it was shown that STABLE SET is NP-complete on squares of the subdivision of some graph (i.e. the squares of the total graph of some graph).…”

Section: Further Considerationsmentioning

confidence: 99%

Complexity of Finding Graph Roots with Girth Conditions

et al. 2010

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Graph G is the square of graph H if two vertices x, y have an edge in G if and only if x, y are of distance at most two in H . Given H it is easy to compute its square H 2 , however Motwani and Sudan proved that it is NP-complete to determine if a given graph G is the square of some graph H (of girth 3). In this paper we consider the characterization and recognition problems of graphs that are squares of graphs of small girth, i.e. to determine if G = H 2 for some graph H of small girth. The main results are the following.• There is a graph theoretical characterization for graphs that are squares of some graph of girth at least 7. A corollary is that if a graph G has a square root H of girth at least 7 then H is unique up to isomorphism.• There is a polynomial time algorithm to recognize if G = H 2 for some graph H of girth at least 6.• It is NP-complete to recognize if G = H 2 for some graph H of girth 4. These results almost provide a dichotomy theorem for the complexity of the recognition problem in terms of girth of the square roots. The algorithmic and graph theoretical results generalize previous results on tree square roots, and provide polynomial time algorithms to compute a graph square root of small girth if it exists. Some open questions and conjectures will also be discussed.

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Minimum Edge Dominating Sets

Cited by 104 publications

References 14 publications

Pareto Optimality in House Allocation Problems

Pareto Optimality in House Allocation Problems

On the Conduciveness of Random Network Graphs for Maximal Assortative or Maximal Dissortative Matching

Complexity of Finding Graph Roots with Girth Conditions

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