Facets of the Total Matching Polytope for bipartite graphs

TL;DR

This paper provides new formulations and inequalities for the Total Matching Polytope in bipartite graphs, including perfect formulations for trees, new facet-defining inequalities, and a complete description for complete bipartite graphs.

Contribution

It introduces balanced biclique inequalities and non-balanced lifted biclique inequalities, expanding the understanding of the polytope's structure in bipartite graphs.

Findings

Balanced biclique inequalities are always facet-defining.

Non-balanced lifted biclique inequalities are facet-defining for bipartite graphs.

Complete descriptions are provided for complete bipartite graphs.

Abstract

The Total Matching Polytope generalizes the Stable Set Polytope and the Matching Polytope. In this paper, we give the perfect formulation for Trees and we derive two new families of valid inequalities, the balanced biclique inequalities which are always facet-defining and the non-balanced lifted biclique inequalities obtained by a lifting procedure, which are facet-defining for bipartite graphs. Finally, we give a complete description for Complete Bipartite Graphs.