Subexponential Parameterized Algorithms for Cut and Cycle Hitting Problems on H<-Minor-Free Graphs

We design the first subexponential-time (parameterized) algorithms for several cut and cycle-hitting problems on H-minor free graphs. In particular, we obtain the following results (where k is the solution-size parameter). time algorithms for Edge Bipartization and Odd Cycle Transversal; a time algorithm for Edge Multiway Cut and a time algorithm for Vertex Multiway Cut (with undeletable terminals), where r is the number of terminals to be separated; a time algorithm for Edge Multicut and a time algorithm for Vertex Multicut (with undeletable terminals), where r is the number of terminal pairs to be separated; a time algorithm for Group Feedback Edge Set and a time algorithm for Group Feedback Vertex Set, where g is the size of the group. In addition, our approach also gives time algorithms for all above problems with the exception of time for Edge/Vertex Multicut and time for Group Feedback Edge/Vertex Set. All of our FPT algorithms (the first four items above) are randomized, as they use known randomized kernelization algorithms as sub-routines. We obtain our results by giving a new decomposition theorem on graphs of bounded genus, or more generally, an h-almost-embeddable graph for an arbitrary but fixed constant h. Our new decomposition theorem generalizes known Contraction Decomposition Theorem. Prior studies on this topic exhibited that the classes of planar graphs [Klein, SICOMP, 2008], graphs of bounded genus [Demaine, Hajiaghayi and Mohar, Combinatorica 2010] and H-minor free graphs [Demaine, Hajiaghayi and Kawarabayashi, STOC 2011] admit a Contraction Decomposition Theorem. In particular we show the following. Let G be a graph of bounded genus, or more generally, an h-almost-embeddable graph for an arbitrary but fixed constant h. Then for every p ∊ ℕ, there exist disjoint sets Z1, …, Zp ⊆ V(G) such that for every i ∊ 1, …, p and every Z′ ⊆ Zi, the treewidth of G/(Zi′) is upper bounded by O(p + |Z′|), where the constant hidden in O(·) depends on h. Here G/(Zi′) denotes the graph obtained from G by contracting every edge with both endpoints in Zi′. When Z′ = , this corresponds to classical Contraction Decomposition Theorem.