Improving the Integrality Gap for Multiway Cut

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 Vondr\'{a}k, 2014] while the best known inapproximability factor is $1.2$ due to [Angelidakis, Makarychev and Manurangsi, 2017]. In this work, we improve on the lower bound to $1.20016$ 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. Our instance is a non-trivial $3$-dimensional instance that overcomes this technical challenge. We analyze the gap of the instance by viewing it as a convex combination of $2$-dimensional instances and a uniform 3-dimensional instance. We believe that this technique could be exploited further to construct instances with larger integrality gap. One of the ingredients of our proof technique is a generalization of a result on \emph{Sperner admissible labelings} due to [Mirzakhani and Vondr\'{a}k, 2015] that might be of independent combinatorial interest.