Multiway Cut, Pairwise Realizable Distributions, and Descending Thresholds

TL;DR

This paper introduces improved approximation algorithms for the Multiway Cut problem, achieving a new best-known factor of approximately 1.2965 through innovative rounding schemes and analysis.

Contribution

It develops new rounding schemes and combines existing methods to improve the approximation factor for Multiway Cut, including a scheme verified only by computer.

Findings

Improved approximation factor to 1.2965 verified by computer.

Tight example showing the limit of previous rounding schemes.

Introduction of descending thresholds in rounding schemes.

Abstract

We design new approximation algorithms for the Multiway Cut problem, improving the previously known factor of 1.32388 [Buchbinder et al., 2013]. We proceed in three steps. First, we analyze the rounding scheme of Buchbinder et al., 2013 and design a modification that improves the approximation to (3+sqrt(5))/4 (approximately 1.309017). We also present a tight example showing that this is the best approximation one can achieve with the type of cuts considered by Buchbinder et al., 2013: (1) partitioning by exponential clocks, and (2) single-coordinate cuts with equal thresholds. Then, we prove that this factor can be improved by introducing a new rounding scheme: (3) single-coordinate cuts with descending thresholds. By combining these three schemes, we design an algorithm that achieves a factor of (10 + 4 sqrt(3))/13 (approximately 1.30217). This is the best approximation factor that we are able to verify by hand. Finally, we show that by combining these three rounding schemes with the scheme of independent thresholds from Karger et al., 2004, the approximation factor can be further improved to 1.2965. This approximation factor has been verified only by computer.