布尔Max-$k$-CSP的可近似性研究
On the Approximability of Boolean Max-$k$-CSP
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中文总结 AI 辅助
针对布尔Max-$k$-CSP问题，本文提出多项式时间算法将近似比提升至$k/2^k$，并结合前人结果明确了其近似比的NP困难阈值，核心技术为推广的高斯比较不等式。
中文摘要 AI 辅助
研究目标是最大化任意元数为$k$的布尔约束满足问题中被满足的约束数量。本文提出一种多项式时间算法，可达到$(k/2^k)$的近似比，优于Makarychev与Makarychev（arXiv:1206.3603）提出的此前最优保证值$0.626612\; k/2^k$。在唯一游戏猜想（Unique Games Conjecture）成立的前提下，De与Mossel（arXiv:1202.5258）证明：对于奇数$k$，达到优于$(k+1)/2^k$的近似比是NP困难的；对于偶数$k$，达到优于$(k+2)/2^k$的近似比是NP困难的。本文的核心技术是对近期用于解决编码理论中弱单纯形猜想（arXiv:2607.14087）的高斯比较不等式的推广。
英文摘要
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).
