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. For arbitrary k, the best-known approximation factor is 1.2965 due to Sharma and Vondrák while the best known inapproximability factor is 1.2 due to Angelidakis, Makarychev and Manurangsi. In this talk we show how to improve on the lower bound by constructing an integrality gap instance for the CKR relaxation. A technical challenge in improving the gap has been the lack of geometric tools to understand higher-dimensional simplices. We analyze the gap of the instance by viewing it as a convex combination of 2-dimensional instances and a uniform 3-dimensional instance. One of the byproducts from our proof technique is a generalization of a result on Sperner admissible labelings due to Mirzakhani and Vondrák.
Joint work with Karthakeyan Chanrdasekaran, Tamás Király and Vivek Madan.