scholarly journals A Set and Collection Lemma

10.37236/2514 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Vadim E. Levit ◽  
Eugen Mandrescu
Keyword(s):  
Independent Set ◽  
Independent Sets ◽  
Stable Set ◽  
Maximum Stable Set

A set $S\subseteq V(G)$ is independent if no two vertices from $S$ are adjacent. Let $\alpha\left( G\right) $ stand for the cardinality of a largest independent set.In this paper we prove that if $\Lambda$ is a nonempty collection of maximum independent sets of a graph $G$, and $S$ is an independent set, thenthere is a matching from $S-\bigcap\Lambda$ into $\bigcup\Lambda-S$, and$\left\vert S\right\vert +\alpha(G)\leq\left\vert \bigcap\Lambda\cap S\right\vert +\left\vert \bigcup\Lambda\cup S\right\vert $.Based on these findings we provide alternative proofs for a number of well-known lemmata, such as the "Maximum Stable Set Lemma" due to Claude Berge and the "Clique Collection Lemma" due to András Hajnal. 

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

Greedy Algorithms

1995 ◽  
Author(s):  
Raymond Greenlaw ◽  
H. James Hoover ◽  
Walter L. Ruzzo
Keyword(s):  
Greedy Algorithms ◽  
Independent Set ◽  
Independent Sets ◽  
Case Finding ◽  
Sequential Algorithm ◽  
Maximal Independent Set ◽  
Maximum Independent Set ◽  
Greedy Method ◽  
Numerical Order ◽  
Selection Of

We consider the selection of two basketball teams at a neighborhood playground to illustrate the greedy method. Usually the top two players are designated captains. All other players line up while the captains alternate choosing one player at a time. Usually, the players are picked using a greedy strategy. That is, the captains choose the best unclaimed player. The system of selection of choosing the best, most obvious, or most convenient remaining candidate is called the greedy method. Greedy algorithms often lead to easily implemented efficient sequential solutions to problems. Unfortunately, it also seems to be that sequential greedy algorithms frequently lead to solutions that are inherently sequential — the solutions produced by these algorithms cannot be duplicated rapidly in parallel, unless NC equals P. In the following subsections we will examine this phenomenon. We illustrate some of the important aspects of greedy algorithms using one that constructs a maximal independent set in a graph. An independent set is a set of vertices of a graph that are pairwise nonadjacent. A maximum independent set is such a set of largest cardinality. It is well known that finding maximum independent sets is NP-hard. An independent set is maximal if no other vertex can be added while maintaining the independent set property. In contrast to the maximum case, finding maxima? independent sets is very easy. Figure 7.1.1 depicts a simple polynomial time sequential algorithm computing a maximal independent set. The algorithm is a greedy algorithm: it processes the vertices in numerical order, always attempting to add the lowest numbered vertex that has not yet been tried. The sequential algorithm in Figure 7.1.1, having processed vertices 1,... , j -1, can easily decide whether to include vertex j. However, notice that its decision about j potentially depends on its decisions about all earlier vertices — j will be included in the maximal independent set if and only if all j' less than j and adjacent to it were excluded.

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 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 Hypergraph Independent Sets

2012 ◽  
Vol 22 (1) ◽  
pp. 9-20 ◽  
Author(s):  
JONATHAN CUTLER ◽  
A. J. RADCLIFFE
Keyword(s):  
Upper Bound ◽  
Independent Set ◽  
Independent Sets ◽  
Extremal Problems ◽  
Regularity Lemma ◽  
Uniform Hypergraph ◽  
Fixed Size

The study of extremal problems related to independent sets in hypergraphs is a problem that has generated much interest. There are a variety of types of independent sets in hypergraphs depending on the number of vertices from an independent set allowed in an edge. We say that a subset of vertices isj-independentif its intersection with any edge has size strictly less thanj. The Kruskal–Katona theorem implies that in anr-uniform hypergraph with a fixed size and order, the hypergraph with the mostr-independent sets is the lexicographic hypergraph. In this paper, we use a hypergraph regularity lemma, along with a technique developed by Loh, Pikhurko and Sudakov, to give an asymptotically best possible upper bound on the number ofj-independent sets in anr-uniform hypergraph.

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

Algebraic independence

1981 ◽  
Vol 46 (2) ◽  
pp. 377-384 ◽  
Author(s):  
Julia F. Knight
Keyword(s):  
Algebraic Closure ◽  
Algebraic Independence ◽  
Independent Set ◽  
Independent Sets ◽  
Large Cardinal ◽  
Similarity Type ◽  
Limit Cardinal ◽  
Free Set ◽  
Infinite Set ◽  
The Universe

This paper is concerned with algebraic independence in structures that are relatively simple for their size. It is shown that for κ a limit cardinal, if a structure of power at least κ is ∞ω-equivalent to a structure of power less than κ, then must contain an infinite set of algebraically independent elements. The same method of proof yields the fact that if σ is an Lω1ω-sentence (not necessarily complete) and σ has a model of power ℵω then some model of σ contains an infinite algebraically independent set.All structures are assumed to be of countable similarity type. Letters , etc. will be used to denote either a structure or the universe of the structure. If X ⊆ , the algebraic closure of X (in ), denoted by Cl(X), is the union of all finite sets that are weakly definable (in ) by Lωω-formulas with parameters from X. A set S is algebraically independent if for each a in S, a ∉ Cl(S – {a}). An algebraically independent set is sometimes called a “free” set (in [3] and [4], for example).It is known (see [5]) that any structure of power ℵn must have a set of n algebraically independent elements, and there are structures of power ℵn with no independent set of size n + 1. In power ℵω every structure will have arbitrarily large finite algebraically independent sets. However, it is consistent with ZFC that some models of power ℵω do not have any infinite algebraically independent set. Devlin [4] showed that if V = L, then for any cardinal κ, if every structure of power κ has an infinite algebraically independent set, then κ has a certain large cardinal property that ℵω can never possess.

A matrix approach to graph maximum stable set and coloring problems with application to multi-agent systems

Automatica ◽  
2012 ◽  
Vol 48 (7) ◽  
pp. 1227-1236 ◽  
Author(s):  
Yuzhen Wang ◽  
Chenghui Zhang ◽  
Zhenbin Liu
Keyword(s):  
Stable Set ◽  
Matrix Approach ◽  
Multi Agent Systems ◽  
Agent Systems ◽  
Maximum Stable Set ◽  
Multi Agent
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