Strongly polynomial bounds for multiobjective and parametric global minimum cuts in graphs and hypergraphs

We consider multiobjective and parametric versions of the global minimum cut problem in undirected graphs and bounded-rank hypergraphs with multiple edge cost functions. For a fixed number of edge cost functions, we show that the total number of supported non-dominated (SND) cuts is bounded by a polynomial in the numbers of nodes and edges, i.e., is strongly polynomial. This bound also applies to the combinatorial facet complexity of the problem, i.e., the maximum number of facets (linear pieces) of the parametric curve for the parametrized (linear combination) objective, over the set of all parameter vectors such that the parametrized edge costs are nonnegative and the parametrized cut costs are positive. We sharpen this bound in the case of two objectives (the bicriteria problem), for which we also derive a strongly polynomial upper bound on the total number of non-dominated (Pareto efficient) cuts. In particular, the bicriteria global minimum cut problem in an n-node graph admits O(n3 log n) SND cuts and O(n5 log n) non-dominated (Pareto efficient) cuts. These results significantly improve on earlier graph cut results by Mulmuley (1999) and Armon and Zwick (2006). They also imply that the parametric curve and all SND cuts, and, for the bicriteria problems, all Pareto efficient cuts, can be computed in strongly polynomial time when the number of objectives is fixed.