Testing the Supermodular-Cut Condition - Tanigawa, Shin-Ichi; Yoshida, Yuichi (Algorithmica, 2015)
For a function 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.