Steiner Cut Dominants

Abstract

For a subset T of nodes of an undirected graph G, a T-Steiner cut is a cut δ(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 τ, there is a finite set of inequalities such that for every pair (G,T) with |T|\ <= τthe 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.

Figures (10)

• Figure 1: A Steiner tree with its facet weights (the dark nodes are the terminals).
• Figure 2: A Steiner cactus with its facet weights (the dark nodes are the terminals).
• Figure 3: An example of a facet inducing Steiner graph that is not a Steiner cactus with facet weights in minimum integer form and right-hand-side two. A root basis is formed by the six cuts defined by the single terminals and the three cuts defined by the pairs of adjacent terminals.
• Figure 4: Notations used for $\mathop{\mathrm{Y}}\nolimits\!\nabla$-reduction s.
• Figure 5: The two possibilities for the subgraph of $G$ induced by the non-terminal node $v$ and its three neighbors when a $\mathop{\mathrm{Y}}\nolimits\!\nabla$-reduction is applied.

Theorems & Definitions (43)

• Remark 1
• Remark 2
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Theorem 1
• proof