Improving the integrality gap for multiway cut

Abstract

In the multiway cut problem, we are given an undirected graph with non-negative edge weights and a collection of k terminal nodes, and the goal is to partition the node set of the graph into k non-empty parts each containing exactly one terminal so that the total weight of the edges crossing the partition is minimized. The multiway cut problem for \(k\ge 3\) is APX-hard. For arbitrary k, the best-known approximation factor is 1.2965 due to Sharma and Vondrak [12] while the best known inapproximability result due to Angelidakis, Makarychev and Manurangsi [1] rules out efficient algorithms to achieve an approximation factor that is less than 1.2. In this work, we improve on the lower bound to \(1.20016\) by constructing an integrality gap instance for the CKR relaxation.