Some Complexity Results for k-Cardinality Minimum Cut Problems

Many polynomially solvable combinatorial optimization problems (COP) become NP hard when we require solutions to satisfy an additional cardinality constraint. This family of problems has been considered only recently. We study a new problem of this family: the k-cardinality minimum cut problem. Given an undirected edge-weighted graph the k-cardinality minimum cut problem is to nd a partition of the vertex set V in two sets V1, V2 such that the number of the edges between V1 and V2 is exactly k and the sum of the weights of these edges is minimal. A variant of this problem is the k-cardinality minimum s-t cut problem where s and t are xed vertices and we have the additional request that s belongs to V1 and t belongs to V2. We also consider other variants where the number of edges of the cut is constrained to be either less or greater than k. For all these problems we show complexity results in the most signi cant graph classes.