Local Enumeration: The Not-All-Equal Case

TL;DR

This paper introduces a simplified local enumeration problem focused on Not-All-Equal solutions, demonstrating that solving this problem efficiently is sufficient to challenge the Super Strong Exponential Time Hypothesis (SSETH).

Contribution

The authors refine existing algorithms to optimally solve the NAE-Enum(3, n/2) problem, establishing a tight expected runtime bound of approximately 6^{n/4}.

Findings

NAE-Enum(3, n/2) can be solved in expected polynomial times 6^{n/4}.

Solving NAE-Enum is sufficient to break SSETH.

Refined algorithm matches the lower bound, proving optimality.

Abstract

Gurumukhani et al. (CCC'24) proposed the local enumeration problem Enum(k, t) as an approach to break the Super Strong Exponential Time Hypothesis (SSETH): for a natural number $k$ and a parameter $t$, given an $n$-variate $k$-CNF with no satisfying assignment of Hamming weight less than $t(n)$, enumerate all satisfying assignments of Hamming weight exactly $t(n)$. Furthermore, they gave a randomized algorithm for Enum(k, t) and employed new ideas to analyze the first non-trivial case, namely $k = 3$. In particular, they solved Enum(3, n/2) in expected $1.598^n$ time. A simple construction shows a lower bound of $6^{\frac{n}{4}} \approx 1.565^n$. In this paper, we show that to break SSETH, it is sufficient to consider a simpler local enumeration problem NAE-Enum(k, t): for a natural number $k$ and a parameter $t$, given an $n$-variate $k$-CNF with no satisfying assignment of Hamming weight less than $t(n)$, enumerate all Not-All-Equal (NAE) solutions of Hamming weight exactly $t(n)$, i.e., those that satisfy and falsify some literal in every clause. We refine the algorithm of Gurumukhani et al. and show that it optimally solves NAE-Enum(3, n/2), namely, in expected time $poly(n) \cdot 6^{\frac{n}{4}}$.