On the minimum s-t cut problem with budget constraints

We consider in this paper the budgeted minimum s - t cut problem. Suppose that we are given a directed graph G = ( V , A ) with two distinguished nodes s and t , k non-negative cost functions c 1 , … , c k : A → Z + , and k - 1 budget bounds b 1 , … , b k - 1 . The goal is to find a s - t cut C satisfying the budget constraints c h ( C ) ⩽ b h , for h = 1 , … , k - 1 , and whose cost c k ( C ) is minimum. In this paper we discuss the linear relaxation of the problem and introduce a strict partial ordering on its solutions. We give a necessary and sufficient condition for which it has an integral optimal minimal (with respect to this ordering) basic solution. We also show that recognizing whether this is the case is NP-hard.