Multicuts in Planar and Bounded-Genus Graphs with Bounded Number of Terminals

TL;DR

This paper presents a polynomial-time algorithm for the minimum multicut problem in graphs embedded on fixed surfaces with bounded genus and terminals, improving understanding of cut problems in topologically constrained graphs.

Contribution

It introduces a topological approach to solve the multicut problem efficiently in bounded-genus graphs with a fixed number of terminals, correcting previous errors in the planar case.

Findings

Polynomial-time algorithm for fixed-genus graphs with terminals

Corrects previous errors in planar multicut algorithms

Extends to multiway cut problem in surface-embedded graphs

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(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.