2015/02/03 by de Verdière, Éric Colin
#Computational Geometry (cs.CG) #Data Structures and Algorithms (cs.DS) #F.2.2 #FOS: Computer and information sciences #G.2.2 #I.3.5
paper · doi:10.48550/arxiv.1502.00911
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(sqrtg2+gt) 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, a.k.a. multiterminal cut problem; even for this special case, no dedicated algorithm was known for graphs embedded on surfaces.