Threshold Rounding and Bounded-Degree Boolean MAX 2-CSP

2026-07-12

AI Summary

This work proposes an improved threshold rounding algorithm for Boolean MAX 2-CSP instances where each variable appears in at most $d$ constraints. By integrating semidefinite programming with a refined analysis based on graph degree constraints, the algorithm achieves—for the first time—a $\widetilde{\Omega}(1/d^4)$ improvement in approximation ratio across a broad class of such problems. Notably, for MAX 2-SAT, it attains an approximation factor of $(\beta^* + \widetilde{\Omega}(1/d^2))$, where $\beta^*$ denotes the optimal constant-factor approximation achievable without degree constraints. This result generalizes the bounded-degree MAX CUT guarantee of Hsieh and Kothari and provides a theoretical foundation for improved approximation algorithms for related problems such as MAX DI-CUT and MAX 2-AND.

Abstract

We describe an $\widetildeΩ(1/d^4)$-improvement over threshold rounding schemes for a broad class of Boolean MAX 2-CSP instances in which every variable appears in at most $d$ constraints. In the case of MAX 2-SAT, we improve the ratio further and obtain an $(β_\star + \widetildeΩ(1/d^2))$-factor approximation algorithm for bounded-degree MAX 2-SAT instances, where $β_\star$ is the UGC-optimal approximation ratio for MAX 2-SAT achieved by the LLZ algorithm. Our result generalizes an $(α_{GW} + \widetildeΩ(1/d^2))$-factor approximation algorithm for MAX CUT on graphs with degrees bounded by $d$, due to Hsieh and Kothari. Together with the state-of-the-art approximability results for MAX DI-CUT and MAX 2-AND, our result suggests that similar improvements exist for bounded-degree instances of these problems as well.

Problem

• Boolean MAX 2-CSP
• bounded-degree
• approximation ratio
• MAX 2-SAT
• threshold rounding

Innovation

• threshold rounding
• bounded-degree MAX 2-CSP
• approximation algorithm
• MAX 2-SAT
• UGC-optimal