Better Balance by Being Biased: A 0.8776-Approximation for Max Bisection

TL;DR

This paper presents a nearly optimal approximation algorithm for Max Bisection, improving previous results and showing that the bisection constraint does not significantly increase problem difficulty, with implications for related problems like Max 2-Sat.

Contribution

It introduces a 0.8776-approximation algorithm for Max Bisection, nearly matching the UGC-based hardness bound, and provides an optimal algorithm for a Max 2-Sat variant under UGC.

Findings

Achieved a 0.8776-approximation for Max Bisection.

Max Bisection's approximation ratio is close to the UGC hardness bound.

Provided an optimal algorithm for a Max 2-Sat variant under UGC.

Abstract

Recently Raghavendra and Tan (SODA 2012) gave a 0.85-approximation algorithm for the Max Bisection problem. We improve their algorithm to a 0.8776-approximation. As Max Bisection is hard to approximate within $\alpha_{GW} + \epsilon \approx 0.8786$ under the Unique Games Conjecture (UGC), our algorithm is nearly optimal. We conjecture that Max Bisection is approximable within $\alpha_{GW}-\epsilon$, i.e., the bisection constraint (essentially) does not make Max Cut harder. We also obtain an optimal algorithm (assuming the UGC) for the analogous variant of Max 2-Sat. Our approximation ratio for this problem exactly matches the optimal approximation ratio for Max 2-Sat, i.e., $\alpha_{LLZ} + \epsilon \approx 0.9401$, showing that the bisection constraint does not make Max 2-Sat harder. This improves on a 0.93-approximation for this problem due to Raghavendra and Tan.