, Volume 8, Issue 1, pp 232-248

Vertex packings: Structural properties and algorithms

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Abstract

We consider a binary integer programming formulation (VP) for the weighted vertex packing problem in a simple graph. A sufficient “local” optimality condition for (VP) is given and this result is used to derive relations between (VP) and the linear program (VLP) obtained by deleting the integrality restrictions in (VP). Our most striking result is that those variables which assume binary values in an optimum (VLP) solution retain the same values in an optimum (VP) solution. This result is of interest because variables are (0, 1/2, 1). valued in basic feasible solutions to (VLP) and (VLP) can be solved by a “good” algorithm. This relationship and other optimality conditions are incorporated into an implicit enumeration algorithm for solving (VP). Some computational experience is reported.

Supported by National Science Foundation Grant GK-32282X to Cornell University.
Work done at Cornell University, Yale University and the Mathematics Research Center of the University of Wisconsin under National Science Foundation Grants GK-32282X and GK-42095 and U.S. Army Contract No. DA-31-124-ARO-D-462, respectively.