On the Approximability of Boolean Max-$k$-CSP

2026-08-05

AI Summary

This work addresses the Boolean Max-k-CSP problem, which seeks to maximize the number of satisfied constraints in a Boolean constraint satisfaction instance where each constraint involves exactly k variables. We propose a polynomial-time algorithm that improves the best-known approximation ratio from 0.626612·k/2^k to k/2^k. This advancement is achieved by establishing a novel Gaussian comparison inequality and integrating techniques from coding theory originally developed to resolve the weak simplex conjecture. Under the Unique Games Conjecture, our result nearly matches the known NP-hardness threshold, thereby substantially advancing both the theoretical understanding and algorithmic performance for this fundamental optimization problem.

Abstract

Consider the problem of maximizing the number of satisfied constraints of an arbitrary boolean constraint satisfaction problem with arity $k$. We obtain a polynomial time algorithm that achieves a $(k/2^k)$-approximation, improving on the previous best guarantee of $0.626612\; k/2^k$, due to Makarychev and Makarychev (arXiv:1206.3603). Assuming the Unique Games Conjecture, De and Mossel (arXiv:1202.5258) showed that achieving an approximation ratio better than $(k+1)/2^k$ for odd $k$ and $(k+2)/2^k$ for even $k$, is NP-hard. The main technical ingredient is an extension of a recently established Gaussian comparison inequality, used to resolve the Weak Simplex Conjecture in coding theory (arXiv:2607.14087).

Problem

• Boolean Max-k-CSP
• approximability
• constraint satisfaction problem
• Unique Games Conjecture
• approximation ratio

Innovation

• Boolean Max-k-CSP
• approximation algorithm
• Gaussian comparison inequality
• Unique Games Conjecture
• constraint satisfaction problem