Improved compact formulations for metric and cut polyhedra

Abstract

Given a graph G=(V,E) with |V|=n and |E|=m, we consider the metric cone MET(G) and the metric polytope METP(G) defined on RE. These polyhedra are relaxations of several important problems in combinatorial optimization such as the max-cut problem and the multicommodity flow problem. They are known to have non-compact formulations via the cycle inequalities in the original space RE and compact (i.e. polynomial size) extended formulations via the triangle inequalities defined on the complete graph Kn. In this paper, we show that one can reduce the number of triangle inequalities to O(nm) and still have extended formulations for MET(G) and METP(G). This is particularly interesting for sparse graphs when m=O(n).