Mathematical Programming

, Volume 8, Issue 1, pp 232–248

# Vertex packings: Structural properties and algorithms

• G. L. Nemhauser
• L. E. TrotterJr.
Article

## 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.

## Keywords

Optimality Condition Feasible Solution Integer Programming Computational Experience Programming Formulation
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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