Testing the Supermodular-Cut Condition

Shin-ichi Tanigawa, Yuichi Yoshida

Abstract: For a function \(f:2^{V} \to \mathbb {Z}_{+}\) on a finite set V with f(∅)=f(V)=0, a digraph D=(V,A) is called f-connected if it satisfies the f-cut condition, that is, δ D (X)≥f(X) for any X⊆V, where δ D (X) is the number of arcs from X to V∖X. We show that, for any crossing supermodular function f, the f-connectivity can be tested with a constant number of queries in the general digraph model with average degree bound. As immediate corollaries, we obtain constant-time testers for k-edge-connectivity, rooted-(k,l)-edge-connectivity, and the property of having k arc-disjoint arborescences. We also give a corresponding result for the undirected case.