An Improved Integrality Gap for the Calinescu-Karloff-Rabani Relaxation for Multiway Cut

TL;DR

This paper presents a new integrality gap instance for the Multiway Cut LP relaxation, achieving a better ratio than previous bounds for multiple terminals, and links this to hardness of approximation results.

Contribution

It constructs an improved integrality gap for the Multiway Cut LP relaxation, surpassing previous bounds and establishing new hardness of approximation results.

Findings

New integrality ratio of 6/(5+1/(k-1)) - ε for k ≥ 3

Improves upon the previous lower bound of 8/(7+1/(k-1)) for k ≥ 4

Establishes Unique Games hardness for approximating Multiway Cut at this ratio

Abstract

We construct an improved integrality gap instance for the Calinescu-Karloff-Rabani LP relaxation of the Multiway Cut problem. In particular, for $k \geqslant 3$ terminals, our instance has an integrality ratio of $6 / (5 + \frac{1}{k - 1}) - \varepsilon$, for every constant $\varepsilon > 0$. For every $k \geqslant 4$, our result improves upon a long-standing lower bound of $8 / (7 + \frac{1}{k - 1})$ by Freund and Karloff (2000). Due to Manokaran et al.'s result (2008), our integrality gap also implies Unique Games hardness of approximating Multiway Cut of the same ratio.