Enumerating parametric global minimum cuts by random interleaving

Recently, Aissi et al. gave new counting and algorithmic bounds for parametric minimum cuts in a graph, where each edge cost is a linear combination of multiple cost criteria and different cuts become minimum as the coefficients of the linear combination are varied. In this article, we derive better bounds using a mathematically simpler argument. We provide faster algorithms for enumerating these cuts. We give a lower bound showing our upper bounds have roughly the right degree. Our results also immediately generalize to parametric versions of other problems solved by the Contraction Algorithm, including approximate min-cuts, multi-way cuts, and a matroid optimization problem. We also give a first generalization to nonlinear parametric minimum cuts.