Quasipolynomial Multicut-mimicking Networks and Kernels for Multiway Cut Problems

Abstract

We show the existence of an exact mimicking network of k O ( log k ) edges for minimum multicuts over a set of terminals in an undirected graph, where k is the total capacity of the terminals, i.e., the sum of the degrees of the terminal vertices. Furthermore, using the best available approximation algorithm for Small Set Expansion , we show that a mimicking network of k O ( log3 k ) edges can be computed in randomized polynomial time. As a consequence, we show quasipolynomial kernels for several problems, including Edge Multiway Cut , Group Feedback Edge Set for an arbitrary group, and Edge Multicut parameterized by the solution size and the number of cut requests. The result combines the matroid-based irrelevant edge approach used in the kernel for s -Multiway Cut with a recursive decomposition and sparsification of the graph along sparse cuts. This is the first progress on the kernelization of Multiway Cut problems since the kernel for s -Multiway Cut for constant value of s (Kratsch and Wahlström, FOCS 2012).