Abstract (EN)

We investigate the bicriteria global minimum cut problem where each edge is evaluated by two nonnegative cost functions. The parametric complexity of such a problem is the number of linear segments in the parametric curve when we take all convex combinations of the criteria. We prove that the parametric complexity of the global minimum cut problem is O(|V|3). As a consequence, we show that the number of non-dominated points is O(|V|7) and give the first strongly polynomial time algorithm to compute these points. These results improve on significantly the super-polynomial bound on the parametric complexity given by Mulmuley [11], and the pseudo-polynomial time algorithm of Armon and Zwick [1] to solve this bicriteria problem. We extend some of these results to arbitrary cost functions and more than two criteria, and to global minimum cuts in hypergraphs.