Quasipolynomial Multicut-mimicking Networks and Kernels for Multiway Cut Problems

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 Wahlstrom, FOCS 2012).