Further facet generating procedures for vertex packing polytopes

Abstract

Padberg [4] and Nemhauser and Trot ter [3] have shown how certain facets of a graph can be obtained f rom facets of its node induced subgraphs. In a related vein Chvatal [2] has shown cases where knowledge of the integer hull of certain substructures of the graph allows one to obtain the integer hull (complete set of facets) for the given graph. Here we examine three further facet generation procedures. The procedures are valid for general independence systems. Throughout this note we only consider nontrivial f a c e t s t h o s e of the fo rm 27=1 ~rjxi ~< ~'* where ~rj/> 0, ~'* > 0. In each case we first describe graphically a method to generate a graph G' f rom a graph G. We then describe the generation of a polytope X ' f rom a polytope X, which can be the ver tex packing polytopes of G ' and G respectively. Finally, we show how a facet of X produces a f ace t of X ' , and demonstra te with a graphical example. 1. Given a graph G = (V, E ) where V = {vl, v2 . . . . . v,}, we construct a new graph G ' = (V ' , E ' ) where each node v~ is joined to a new node v.+, by an edge. Each node v,+i is in turn joined to a new node v0 by an edge. See Fig. 1. If Xs is the vertex-packing polytope of some node induced subgraph Gs of G, S C_ V, and X ' is the ver tex-packing polytope of G ' , we show below how facets of Xs conver t to facets of X ' . Consider the polytopes

DOI: 10.1007/BF01580383

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Cite this paper

@article{Wolsey1976FurtherFG, title={Further facet generating procedures for vertex packing polytopes}, author={Laurence A. Wolsey}, journal={Math. Program.}, year={1976}, volume={11}, pages={158-163} }