The Complexity of König Subgraph Problems and Above-Guarantee Vertex Cover

Mishra, Sambit Kumar; Raman, Venkatesh; Saurabh, Saket; Sikdar, Somnath; Subramanian, C. R.

doi:10.1007/s00453-010-9412-2

Cited by 35 publications

(27 citation statements)

References 34 publications

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“…Our work is closely related to that of Mishra et al [15] on vertex-removal and edge-removal problems to attain the König-Egerváry graph property. Similar to stability, KEG is not a monotone property.…”

Section: Related Workmentioning

confidence: 94%

“…Assuming the Unique Games Conjecture, no constant-factor approximation may exist for the problem. We note that the reductions used in [15] will likely not be helpful for proving hardness for the stabilizer problem as the graphs constructed in the reduction are stable. On the positive side, the authors show that, for a given graph G = (V, E) one can efficiently find a KEG (and hence stable) subgraph with at least 3|E|/5 edges.…”

Section: Related Workmentioning

confidence: 99%

“…In this paper, we study graphs G with the property ν(G) = τ f (G). The class of such graphs, known as stable graphs, subsumes the well-studied class of König-Egerváry (KEG) graphs (e.g., see [20,13,14,15]) for which ν(G) = τ (G). Stable graphs arise quite naturally in the study of cooperative matching games introduced by Shapley and Shubik in their seminal paper [19].…”

Section: Introductionmentioning

confidence: 99%

“…Readers familiar with the literature of bipartite graphs would immediately recognize that the stabilizer problem closely resembles the optimization problems of deleting the minimum number of edges to convert a given graph into a KEG or a bipartite graph, both of which have been well-studied in the literature (e.g., see [1,15]). The investigation of structural properties of unstable graphs has a long history (e.g., see [21,3,17]), but there are few algorithmic results on how to convert an unstable graph to a stable graph.…”

Section: Introductionmentioning

confidence: 99%

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Finding small stabilizers for unstable graphs

et al. 2014

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Abstract. An undirected graph G = (V, E) is stable if its inessential vertices (those that are exposed by at least one maximum matching) form a stable set. We call a set of edges F ⊆ E a stabilizer if its removal from G yields a stable graph. In this paper we study the following natural edgedeletion question: given a graph G = (V, E), can we find a minimumcardinality stabilizer? Stable graphs play an important role in cooperative game theory. In the classic matching game introduced by Shapley and Shubik [19] we are given an undirected graph G = (V, E) where vertices represent players, and we define the value of each subset S ⊆ V as the cardinality of a maximum matching in the subgraph induced by S. The core of such a game contains all fair allocations of the value of V among the players, and is well-known to be non-empty iff graph G is stable. The stabilizer problem addresses the question of how to modify the graph to ensure that the core is non-empty. We show that this problem is vertex-cover hard. We then prove that there is a minimum-cardinality stabilizer that avoids some maximum matching of G. We use this insight to give efficient approximation algorithms for sparse graphs and for regular graphs.

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Section: Related Workmentioning

confidence: 94%

Section: Related Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Finding small stabilizers for unstable graphs

et al. 2014

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“…Vertex Cover Above Maximum Matching (also known as Above Guarantee Vertex Cover) is one of the most popular such parameterizations of Vertex Cover that have been studied in the literature [8,19,24,26,27,29,31]. The problem is to decide if G has a vertex cover of size at most µ(G) + k, where k is the above guarantee parameter and was shown to be FPT by a parameter preserving reduction to Almost 2-SAT and an O * (15 k ) algorithm for the same [27,31]. The O * (2.3146 k ) algorithm in [24] is the current fastest algorithm for Vertex Cover Above Maximum Matching.…”

Section: Generalized Above Guarantee Vertex Covermentioning

confidence: 99%

Parameterized Algorithms for (r,l)-Partization

Krithika¹,

Narayanaswamy²

2013

JGAA

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We consider the (r, l)-Partization problem of finding a set of at most k vertices whose deletion results in a graph that can be partitioned into r independent sets and l cliques. Restricted to perfect graphs and split graphs, we describe sequacious fixed-parameter tractability results for (r, 0)-Partization, parameterized by k and r. For (r, l)-Partization where r + l = 2, we describe an O * (2 k ) algorithm for perfect graphs. We then study the parameterized complexity hardness of a generalization of the Above Guarantee Vertex Cover by a reduction from (r, l)-Partization.

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Finding Small Stabilizers for Unstable Graphs

Bock

Chandrasekaran

Könemann

et al. 2014

Integer Programming and Combinatorial Optimization

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An undirected graph G = (V, E) is stable if its inessential vertices (those that are exposed by at least one maximum matching) form a stable set. We call a set of edges F ⊆ E a stabilizer if its removal from G yields a stable graph. In this paper we study the following natural edge-deletion question: given a graph G = (V, E), can we find a minimum-cardinality stabilizer? Stable graphs play an important role in cooperative game theory. In the classic matching game introduced by Shapley and Shubik [19] we are given an undirected graph G = (V, E) where vertices represent players, and we define the value of each subset S ⊆ V as the cardinality of a maximum matching in the subgraph induced by S. The core of such a game contains all fair allocations of the value of V among the players, and is well-known to be non-empty iff graph G is stable. The stabilizer problem addresses the question of how to modify the graph to ensure that the core is non-empty. We show that this problem is vertex-cover hard. We then prove that there is a minimum-cardinality stabilizer that avoids some maximum matching of G. We use this insight to give efficient approximation algorithms for sparse graphs and for regular graphs.

show abstract

The Complexity of König Subgraph Problems and Above-Guarantee Vertex Cover

Cited by 35 publications

References 34 publications

Finding small stabilizers for unstable graphs

Finding small stabilizers for unstable graphs

Parameterized Algorithms for (r,l)-Partization

Finding Small Stabilizers for Unstable Graphs

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