scholarly journals On local maximum stable set greedoids

2012 ◽  
Vol 312 (3) ◽  
pp. 588-596 ◽  
Author(s):  
Vadim E. Levit ◽  
Eugen Mandrescu
Keyword(s):  
Local Maximum ◽  
Stable Set ◽  
Maximum Stable Set

scholarly journals VERY WELL-COVERED GRAPHS OF GIRTH AT LEAST FOUR AND LOCAL MAXIMUM STABLE SET GREEDOIDS

2011 ◽  
Vol 03 (02) ◽  
pp. 245-252 ◽  
Author(s):  
VADIM E. LEVIT ◽  
EUGEN MANDRESCU
Keyword(s):  
Perfect Matching ◽  
Local Maximum ◽  
Stable Set ◽  
Stable Sets ◽  
Maximum Cardinality ◽  
Math Program ◽  
The Family ◽  
Maximum Stable Set ◽  
Vertex Set ◽  
Appl Math

A maximum stable set in a graph G is a stable set of maximum cardinality. S is a local maximum stable set of G, and we write S ∈ Ψ(G), if S is a maximum stable set of the subgraph induced by S ∪ N(S), where N(S) is the neighborhood of S. Nemhauser and Trotter Jr. [Vertex packings: structural properties and algorithms, Math. Program.8 (1975) 232–248], proved that any S ∈ Ψ(G) is a subset of a maximum stable set of G. In [Levit and Mandrescu, A new greedoid: the family of local maximum stable sets of a forest, Discrete Appl. Math.124 (2002) 91–101] we have shown that the family Ψ(T) of a forest T forms a greedoid on its vertex set. The cases where G is bipartite, triangle-free, well-covered, while Ψ(G) is a greedoid, were analyzed in [Levit and Mandrescu, Local maximum stable sets in bipartite graphs with uniquely restricted maximum matchings, Discrete Appl. Math.132 (2004) 163–174], [Levit and Mandrescu, Triangle-free graphs with uniquely restricted maximum matchings and their corresponding greedoids, Discrete Appl. Math.155 (2007) 2414–2425], [Levit and Mandrescu, Well-covered graphs and greedoids, Proc. 14th Computing: The Australasian Theory Symp. (CATS2008), Wollongong, NSW, Conferences in Research and Practice in Information Technology, Vol. 77 (2008) 89–94], respectively. In this paper we demonstrate that if G is a very well-covered graph of girth ≥4, then the family Ψ(G) is a greedoid if and only if G has a unique perfect matching.

scholarly journals Greedoids on vertex sets of B-joins of graphs

2014 ◽  
Vol 30 (3) ◽  
pp. 335-344
Author(s):  
VADIM E. LEVIT ◽  
◽  
EUGEN MANDRESCU ◽  
Keyword(s):  
Bipartite Graph ◽  
Sufficient Conditions ◽  
Local Maximum ◽  
Stable Set ◽  
Stable Sets ◽  
The Family ◽  
Maximum Stable Set ◽  
Vertex Set ◽  
Necessary And Sufficient ◽  
Vertex Sets

Let Ψ(G) be the family of all local maximum stable sets of graph G, i.e., S ∈ Ψ(G) if S is a maximum stable set of the subgraph induced by S ∪ N(S), where N(S) is the neighborhood of S. It was shown that Ψ(G) is a greedoid for every forest G [15]. The cases of bipartite graphs, triangle-free graphs, and well-covered graphs, were analyzed in [16, 17, 18, 19, 20, 24]. If G1, G2 are two disjoint graphs, and B is a bipartite graph having E(B) as an edge set and bipartition {V (G1), V (G2)}, then by B-join of G1, G2 we mean the graph B (G1, G2) whose vertex set is V (G1) ∪ V (G2) and edge set is E(G1) ∪ E(G2) ∪ E (B). In this paper we present several necessary and sufficient conditions for Ψ(B (G1, G2)) to form a greedoid, an antimatroid, and a matroid, in terms of Ψ(G1), Ψ(G2) and E (B).

scholarly journals Local maximum stable set greedoids stemming from very well-covered graphs

2012 ◽  
Vol 160 (12) ◽  
pp. 1864-1871 ◽  
Author(s):  
Vadim E. Levit ◽  
Eugen Mandrescu
Keyword(s):  
Local Maximum ◽  
Stable Set ◽  
Maximum Stable Set

scholarly journals Polyhedral approximations of the semidefinite cone and their application

2021 ◽  
Author(s):  
Yuzhu Wang ◽  
Akihiro Tanaka ◽  
Akiko Yoshise
Keyword(s):  
Initial Approximation ◽  
Semidefinite Relaxation ◽  
Stable Set ◽  
Cutting Plane Methods ◽  
Diagonally Dominant Matrices ◽  
Maximum Stable Set ◽  
Simple Expansion ◽  
Diagonally Dominant ◽  
Polyhedral Approximations ◽  
Semidefinite Cone

AbstractWe develop techniques to construct a series of sparse polyhedral approximations of the semidefinite cone. Motivated by the semidefinite (SD) bases proposed by Tanaka and Yoshise (Ann Oper Res 265:155–182, 2018), we propose a simple expansion of SD bases so as to keep the sparsity of the matrices composing it. We prove that the polyhedral approximation using our expanded SD bases contains the set of all diagonally dominant matrices and is contained in the set of all scaled diagonally dominant matrices. We also prove that the set of all scaled diagonally dominant matrices can be expressed using an infinite number of expanded SD bases. We use our approximations as the initial approximation in cutting plane methods for solving a semidefinite relaxation of the maximum stable set problem. It is found that the proposed methods with expanded SD bases are significantly more efficient than methods using other existing approximations or solving semidefinite relaxation problems directly.

scholarly journals A polyhedral study of the maximum stable set problem with weights on vertex-subsets

2016 ◽  
Vol 210 ◽  
pp. 223-234
Author(s):  
Manoel Campêlo ◽  
Victor A. Campos ◽  
Ricardo C. Corrêa ◽  
Diego Delle Donne ◽  
Javier Marenco ◽  
...  
Keyword(s):  
Stable Set ◽  
Maximum Stable Set ◽  
Polyhedral Study ◽  
Stable Set Problem

Construction of a Maximum Stable Set with $k$-Extensions

2005 ◽  
Vol 14 (03) ◽  
pp. 311 ◽  
Author(s):  
PETER L. HAMMER ◽  
IGOR E. ZVEROVICH
Keyword(s):  
Stable Set ◽  
Maximum Stable Set

scholarly journals Handelman’s hierarchy for the maximum stable set problem

2013 ◽  
Vol 60 (3) ◽  
pp. 393-423 ◽  
Author(s):  
Monique Laurent ◽  
Zhao Sun
Keyword(s):  
Stable Set ◽  
Maximum Stable Set ◽  
Stable Set Problem

scholarly journals Maximum stable set formulations and heuristics based on continuous optimization

2002 ◽  
Vol 94 (1) ◽  
pp. 137-166 ◽  
Author(s):  
Samuel Burer ◽  
Renato D.C. Monteiro ◽  
Yin Zhang
Keyword(s):  
Continuous Optimization ◽  
Stable Set ◽  
Maximum Stable Set

scholarly journals Average case polyhedral complexity of the maximum stable set problem

2016 ◽  
Vol 160 (1-2) ◽  
pp. 407-431 ◽  
Author(s):  
Gábor Braun ◽  
Samuel Fiorini ◽  
Sebastian Pokutta
Keyword(s):  
Stable Set ◽  
Average Case ◽  
Maximum Stable Set ◽  
Stable Set Problem
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