Improving the Integrality Gap for Multiway Cut

• Published in: arXiv.org
• Main Authors: Bérczi, Kristóf, Chandrasekaran, Karthekeyan, Király, Tamás, Madan, Vivek
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 21.11.2018
• Subjects: Combinatorial analysis; Computer Science - Data Structures and Algorithms; Lower bounds; Mathematics - Optimization and Control; Partitions
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1807.09735

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.