Facet defining inequalities among graph invariants: The system GraPHedron

Mélot, Hadrien

doi:10.1016/j.dam.2007.09.005

Cited by 32 publications

(21 citation statements)

References 11 publications

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“…However, the empirical evidence can certainly motivate the effort to reconsider the theoretical foundations of bounds and develop new conjectures. The use of inequalities in feature spaces to empirically support theoretical analysis of upper and lower bounds has already been proposed [39], and our proposed methodology of evolving new instances at target locations in the instance space can clearly lend weight to this effort, with augmented collections of instances strengthening empirical evidence.…”

Section: Empirical Analysis Of Upper and Lower Bounds Of Graph Featuresmentioning

confidence: 98%

Generating new test instances by evolving in instance space

Smith‐Miles

Bowly

2015

Computers & Operations Research

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Section: Empirical Analysis Of Upper and Lower Bounds Of Graph Featuresmentioning

confidence: 98%

Generating new test instances by evolving in instance space

Smith‐Miles

Bowly

2015

Computers & Operations Research

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“…Of course, it generates the graphs only for "small" values of n (the number of values of n can be increased if the class of graphs is more restricted; see [44] for details). Then, for each selected invariant, the system computes and stores its values for all the graphs which were generated.…”

Section: Data Generationmentioning

confidence: 99%

“…Then, for each selected invariant, the system computes and stores its values for all the graphs which were generated. Storing the values is useful especially when the computations of the invariant take a long time (actually, computing invariants is the most time-consuming part of the whole process; for details, see Mélot [44]). …”

Section: Data Generationmentioning

confidence: 99%

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Linear inequalities among graph invariants: Using GraPHedron to uncover optimal relationships

et al. 2008

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Optimality of a linear inequality in finitely many graph invariants is defined through a geometric approach. For a fixed number of graph vertices, consider all the tuples of values taken by the invariants on a selected class of graphs. Then form the polytope which is the convex hull of all these tuples. By definition, the optimal linear inequalities correspond to the facets of this polytope. They are finite in number, are logically independent, and generate precisely all the linear inequalities valid on the class of graphs. The computer system GraPHedron, developed by some of the authors, is able to produce experimental data about such inequalities for a "small" number of vertices. It greatly helps in conjecturing optimal linear inequalities, which are then hopefully proved for any number of vertices. Two examples are investigated here for the class of connected graphs. First, all the optimal linear inequalities for the stability number and the number of edges are obtained. To this aim, a problem of Ore (1962) related to the Turán Theorem (1941) is solved. Second, several optimal inequalities are established for three invariants: the maximum degree, the irregularity, and the diameter.

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“…The authors wish to thank Martine Labbé, with whom this research was initiated, and Hadrien Mélot, author of the GraPHedron software [16], which was used to formulate the initial conjectures.…”

Section: Acknowledgmentsmentioning

confidence: 99%

Improved approximation bounds for edge dominating set in dense graphs

Cardinal¹,

Langerman²,

Levy³

2009

Theoretical Computer Science

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a b s t r a c tWe analyze the simple greedy algorithm that iteratively removes the endpoints of a maximum-degree edge in a graph, where the degree of an edge is the sum of the degrees of its endpoints. This algorithm provides a 2-approximation to the minimum edge dominating set and minimum maximal matching problems. We refine its analysis and give an expression of the approximation ratio that is strictly less than 2 in the cases where the input graph has n vertices and at least n 2 edges, for > 1/2. This ratio is shown to be asymptotically tight for > 1/2.

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Facet defining inequalities among graph invariants: The system GraPHedron

Cited by 32 publications

References 11 publications

Generating new test instances by evolving in instance space

Generating new test instances by evolving in instance space

Linear inequalities among graph invariants: Using GraPHedron to uncover optimal relationships

Improved approximation bounds for edge dominating set in dense graphs

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