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

Éric Colin de Verdière (1, 2)

1 CNRS - Centre National de la Recherche Scientifique
2 LIGM - Laboratoire d'Informatique Gaspard-Monge

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(sqrt{g^2+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.

Domaines: Algorithme et structure de données [cs.DS]; Géométrie algorithmique [cs.CG]; Mathématique discrète [cs.DM]

Éric Colin de Verdière. Multicuts in planar and bounded-genus graphs with bounded number of terminals. Algorithmica, 2017, 78 (4), pp.1206-1224. ⟨10.1007/s00453-016-0258-0⟩. ⟨hal-01582195⟩