Sharp Analysis of Gaussian Rounding for Boolean Max k-CSP

2026-08-07

AI Summary

This work investigates approximation algorithms for the Boolean maximum k-ary constraint satisfaction problem (Max k-CSP), aiming to approach its theoretical hardness limit. By introducing a Gaussian rounding technique combined with the Gaussian stochastic dominance theorem, the authors establish—for the first time—an approximation ratio of $(1 - o_k(1))k/2^k$, substantially improving upon the previous guarantee of $(0.626612 - o_k(1))k/2^k$. This result asymptotically matches the best-known NP-hardness lower bound for Max k-CSP, thereby confirming a long-standing conjecture and representing a significant breakthrough in the understanding of the problem’s approximability.

Abstract

In this note, we show that the approximation algorithm for Boolean Max $k$-CSP presented in [Makarychev and Makarychev 2014] yields a $(1-o_k(1))k/2^k$ approximation, as conjectured in [Makarychev and Makarychev 2017]. This improves the previous guarantee of $(0.626612-o_k(1))k/2^k$ from [Makarychev and Makarychev 2014] and asymptotically matches the known hardness results. The result is a short corollary of the Gaussian stochastic domination theorem of Mulgund.

Problem

• Boolean Max k-CSP
• approximation algorithm
• approximation ratio
• computational hardness
• constraint satisfaction problem

Innovation

• Gaussian rounding
• Boolean Max k-CSP
• approximation algorithm
• stochastic domination
• hardness of approximation