Multicuts in planar and bounded-genus graphs with bounded number of terminals

Abstract

Given an undirected, edge-weighted graph G together with pairs of vertices, called pairs of terminals, the minimum multicut problem asks for a minimum-weight set of edges such that, after deleting these edges, the two terminals of each pair belong to different connected components of the graph. Relying on topological techniques, we provide a polynomial-time algorithm for this problem in the case where G is embedded on a fixed surface of genus g (e.g., when G is planar) and has a fixed number t of terminals. The running time is a polynomial of degree O[formula presented] in the input size. In the planar case, our result corrects an error in an extended abstract by Bentz [Int.Workshop on Parameterized and Exact Computation, 109– 119, 2012]. The minimum multicut problem is also a generalization of the multiway cut problem, also known as the multiterminal cut problem; even for this special case, no dedicated algorithm was known for graphs embedded on surfaces.