Multi-Multiway Cut Problem on Graphs of Bounded Branch Width.

The Multi-Multiway Cut problem proposed by Avidor and Langberg [2] is a natural generalization of Multicut and Multiway Cut problems. That is, given a simple graph G and c sets of vertices S 1, ⋯ ,S c , the problem asks for a minimum set of edges whose removal disconnects every pair of vertices in S i for all 1 ≤ i ≤ c. In [13], the authors asked whether the problem is polynomial time solvable for fixed c on trees. In this paper, we give both a logical approach and a dynamic programming approach to the Multi-Multiway Cut problem on graphs of bounded branch width, which is exactly the class of graphs with bounded treewidth. In fact, for fixed c and branch width k, we show that the Multi-Multiway Cut problem can be solved in linear time, thus affirmatively answer the question in [13].