Node-Packing Problems with Integer Rounding Properties

This paper considers an integer programming formulation of the node-packing problem $\max \{ 1 \cdot x:Ax\leqq w,x\geqq 0,x\,{\text{integral}}\} $, and its linear programming relaxation, $\max \{ 1 \cdot x:Ax \leqq w,x\geqq 0\} $, where A is the edge-node incidence matrix of a graph G and w is a nonnegative integral vector. An excluded subgraph characterization quantifying the difference between the values of these two programs is given. One consequence of this characterization is an explicit description for the “integer rounding” case. Specifically, a characterization is given for graphs G with the property that for every subgraph of G and for any choice of w the optimum objective function values of these two problems differ by less than unity.