On the Dominant of the Multicut Polytope

Abstract

AbstractGiven a graph $$G=(V,E)$$ G = ( V , E ) and a set $$S \subseteq \left( {\begin{array}{c}V\\ 2\end{array}}\right) $$ S ⊆ V 2 of terminal pairs, the minimum multicut problem asks for a minimum edge set $$\delta \subseteq E$$ δ ⊆ E such that there is no s-t-path in $$G -\delta $$ G - δ for any $$\{s,t\}\in S$$ { s , t } ∈ S . For $$|S|=1$$ | S | = 1 this is the well known s-t-cut problem, but in general the minimum multicut problem is NP-complete, even if the input graph is a tree. The multicut polytope $$\textsc {MultC}^\square (G,S)$$ M U L T C □ ( G , S ) is the convex hull of all multicuts in G; the multicut dominant is given by $$\textsc {MultC}(G,S)=\textsc {MultC}^\square (G,S)+\mathbb {R}^E_{{\ge 0}}$$ M U L T C ( G , S ) = M U L T C □ ( G , S ) + R ≥ 0 E . The latter is the relevant object for the minimization problem. While polyhedra associated to several cut problems have been studied intensively there is only little knowledge for multicut. We investigate properties of the multicut dominant and in particular derive results on liftings of facet-defining inequalities. This yields a classification of all facet-defining path- and edge inequalities. Moreover, we investigate the effect of graph operations such as node splitting, edge subdivisions, and edge contractions on the multicut-dominant and its facet-defining inequalities. In addition, we introduce facet-defining inequalities supported on stars, trees, and cycles and show that the former two can be separated in polynomial time when the input graph is a tree.