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

Ainesh Bakshi

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arXiv · 2608.05331

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

Ainesh Bakshi

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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).

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Ainesh Bakshi. 2026-08-05. On the Approximability of Boolean Max-$k$-CSP. https://arxiv.org/abs/2608.05331

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