Towards the Characterization of Terminal Cut Functions: a Condition for Laminar Families

We study the following characterization problem. Given a set T of terminals and a (2^|T|-2)-dimensional vector π whose coordinates are indexed by proper subsets of T, is there a graph G that contains T, such that for all subsets ∅⊊ S⊊ T, π_S equals the value of the min-cut in G separating S from T∖ S? The only known necessary conditions are submodularity and a special class of linear inequalities given by Chaudhuri, Subrahmanyam, Wagner and Zaroliagis. Our main result is a new class of linear inequalities concerning laminar families, that generalize all previous ones. Using our new class of inequalities, we can generalize Karger's approximate min-cut counting result to graphs with terminals.