Hypergraphic LP Relaxations for Steiner Trees

TL;DR

This paper analyzes hypergraphic LP relaxations for the Steiner tree problem, establishing structural properties, equivalences with other relaxations, and providing new upper bounds on their integrality gaps.

Contribution

It extends uncrossing techniques to partitions, proves equivalences among relaxations, and improves integrality gap bounds for hypergraphic LP relaxations.

Findings

Sparse support for basic feasible solutions.

Equivalence of partition LP with other relaxations.

Improved integrality gap bounds of 1.729 and 1.216.

Abstract

We investigate hypergraphic LP relaxations for the Steiner tree problem, primarily the partition LP relaxation introduced by Koenemann et al. [Math. Programming, 2009]. Specifically, we are interested in proving upper bounds on the integrality gap of this LP, and studying its relation to other linear relaxations. Our results are the following. Structural results: We extend the technique of uncrossing, usually applied to families of sets, to families of partitions. As a consequence we show that any basic feasible solution to the partition LP formulation has sparse support. Although the number of variables could be exponential, the number of positive variables is at most the number of terminals. Relations with other relaxations: We show the equivalence of the partition LP relaxation with other known hypergraphic relaxations. We also show that these hypergraphic relaxations are equivalent to the well studied bidirected cut relaxation, if the instance is quasibipartite. Integrality gap upper bounds: We show an upper bound of sqrt(3) ~ 1.729 on the integrality gap of these hypergraph relaxations in general graphs. In the special case of uniformly quasibipartite instances, we show an improved upper bound of 73/60 ~ 1.216. By our equivalence theorem, the latter result implies an improved upper bound for the bidirected cut relaxation as well.