The Lifted-Cut Relaxation of the Steiner Forest Problem

In this essay we analyze properties of the lifted-cut linear programming relaxation of the Steiner Forest problem introduced by Könemann, Leonardi and Schäfer. We introduce this new formulation and prove some basic properties establishing that it is in fact a valid relaxation and analyzing the integrality gap. We are interested in analyzing the possible half-integrality of this relaxation and construct some families of graphs for which the unweighted minimum spanning tree instance defined by this relaxation has half-integral optimal solutions. Finally we described some computational and implementation issues, namely describing a polynomial sized flow formulation and rounding procedures that we used to find half-integral optimal solutions. We list the results of a computational study we conducted on Steiner Tree instances, stating half-integrality properties, as well as comparing the integrality gap with that of the standard undirected-cut relaxation.