Multicriteria Global Minimum Cuts

We consider two multicriteria versions of the global minimum cut problem in undirected graphs In the k-criteria setting, each edge of the input graph has k non-negative costs associated with it These costs are measured in separate, non interchangeable, units In the AND-version of the problem, purchasing an edge requires the payment of all the k costs associated with it In the OR-version, an edge can be purchased by paying any one of the k-costs associated with it Given k bounds b1,b2,...,bk, the basic multicriteria decision problem is whether there exists a cut C of the graph that can be purchased using a budget of bi units of the i-th criterion, for 1≤ i≤ k. We show that the AND-version of the multicriteria global minimum cut problem is polynomial for any fixed number k of criteria The OR-version of the problem, on the other hand, is NP-hard even for k=2, but can be solved in pseudo-polynomial time for any fixed number k of criteria It also admits an FPTAS Further extensions, some applications, and multicriteria versions of two other optimization problems are also discussed.