Steiner Cut Dominants

TL;DR

This paper characterizes the facet-defining inequalities of the T-Steiner cut dominant in graphs, providing finite descriptions for small T and extending known results for s-t cuts.

Contribution

It introduces a finite set of inequalities for the T-Steiner cut dominant for small T and describes how these can be generated through simple operations.

Findings

Finite inequality sets for |T| <= 5

Bounded coefficients in inequality descriptions

Extension of s-t-cut dominant descriptions

Abstract

For a subset T of nodes of an undirected graph G, a T-Steiner cut is a cut \delta(S) where S intersects both T and the complement of T. The T-Steiner cut dominant} of G is the dominant CUT_+(G,T) of the convex hull of the incidence vectors of the T-Steiner cuts of G. For T={s,t}, this is the well-understood s-t-cut dominant. Choosing T as the set of all nodes of G, we obtain the \emph{cut dominant}, for which an outer description in the space of the original variables is still not known. We prove that, for each integer \tau, there is a finite set of inequalities such that for every pair (G,T) with |T|\ <= \tau the non-trivial facet-defining inequalities of CUT_+(G,T) are the inequalities that can be obtained via iterated applications of two simple operations, starting from that set. In particular, the absolute values of the coefficients and of the right-hand-sides in a description of CUT_+(G,T) by integral inequalities can be bounded from above by a function of |T|. For all |T| <= 5 we provide descriptions of CUT_+(G,T) by facet defining inequalities, extending the known descriptions of s-t-cut dominants.