Local Enumeration: The Not-All-Equal Case

2025-01-06

AI Summary

This work studies the local enumeration problem NAE-Enum(k,t): enumerating all Not-All-Equal (NAE) satisfying assignments of Hamming weight exactly t(n) for k-CNF formulas devoid of low-weight satisfying solutions. This problem offers a streamlined route to refuting the Super Strong Exponential Time Hypothesis (SSETH). We introduce, for the first time, a local enumeration model under NAE constraints and prove its sufficiency for challenging SSETH. Building on the classical Enum(3,n/2) problem, we formulate and solve NAE-Enum(3,n/2) via a novel algorithm combining randomization, Hamming-weight-constrained analysis, and NAE-semantics-based pruning—augmented by polynomial-time preprocessing and a divide-and-conquer strategy. Our algorithm solves NAE-Enum(3,n/2) in expected time poly(n)·6^{n/4} ≈ 1.565^n, matching the information-theoretic lower bound and improving upon the prior best bound of 1.598^n.

Technology Category

• Search and Optimization: Local Search
• Constraint Satisfaction and Optimization: Satisfiability
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

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 $poly(n) cdot 6^{frac{n}{4}}$.

Problem

• NAE-Enum(k, t)
• Optimal Solution Method
• Mathematical Hypothesis Challenge

Innovation

• NAE-Enum(k, t)
• Optimized Time Efficiency
• Improved Gurumukhani Method