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Local Enumeration: The Not-All-Equal Case

Mohit Gurumukhani, Ramamohan Paturi, Michael Saks, Navid Talebanfard

TL;DR

This work reframes SSETH-breaking potential by focusing on the Not-All-Equal local enumeration problem $NAE$-Enum$(k,t)$. It refines the transversal-tree based TreeSearch framework to show that solving $NAE$-Enum$(3,n/2)$ optimally suffices to break SSETH, achieving an expected time of $poly(n)\cdot 6^{n/4}$. The results tie algorithmic bounds to hypergraph Turán problems and depth-3 majority complexity, providing tight combinatorial and probabilistic analyses and interpreting the bound as a potential route to stronger circuit lower bounds. The work also outlines a path for extending the approach to larger $k$ and discusses the broader implications for exact exponential-time algorithms for SAT-like problems.

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

Local Enumeration: The Not-All-Equal Case

TL;DR

This work reframes SSETH-breaking potential by focusing on the Not-All-Equal local enumeration problem -Enum. It refines the transversal-tree based TreeSearch framework to show that solving -Enum optimally suffices to break SSETH, achieving an expected time of . The results tie algorithmic bounds to hypergraph Turán problems and depth-3 majority complexity, providing tight combinatorial and probabilistic analyses and interpreting the bound as a potential route to stronger circuit lower bounds. The work also outlines a path for extending the approach to larger and discusses the broader implications for exact exponential-time algorithms for SAT-like problems.

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 and a parameter , given an -variate -CNF with no satisfying assignment of Hamming weight less than , enumerate all satisfying assignments of Hamming weight exactly . Furthermore, they gave a randomized algorithm for Enum(k, t) and employed new ideas to analyze the first non-trivial case, namely . In particular, they solved Enum(3, n/2) in expected time. A simple construction shows a lower bound of . 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 and a parameter , given an -variate -CNF with no satisfying assignment of Hamming weight less than , enumerate all Not-All-Equal (NAE) solutions of Hamming weight exactly , 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 .
Paper Structure (31 sections, 15 theorems, 30 equations)

This paper contains 31 sections, 15 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Lower bounds for local enumeration
  3. Our Contributions
  4. Depth-3 complexity of Majority function.
  5. Hypergraph Turán problems.
  6. Combinatorial interpretation.
  7. Proof Strategy
  8. Transversal trees and the TreeSearch algorithm
  9. Transversals and Transversal trees
  10. Important definitions
  11. Constructing the tree
  12. The TreeSearch algorithm for enumerating minimum-size transversals
  13. Pruning under random edge ordering
  14. Clause Selection Criterion
  15. The disjoint stage
  16. ...and 16 more sections

Key Result

Theorem 1.1

For $n\ge 0, t\le n/2$, let $F$ be an arbitrary $n$-variate $3$-CNF where every satisfying assignment has Hamming weight at least $t$. Then, the number of NAE satisfying assignments of $F$ of Hamming weight exactly $t$ is at most $6^{\frac{n}{4}}$. Furthermore, we can enumerate these solutions in ex

Theorems & Definitions (84)

  • Theorem 1.1
  • Definition 2.1: Transversals
  • Definition 2.2: Transversal number
  • Definition 2.3: Simplification of clause / formula
  • Definition 2.4
  • Proposition 2.5
  • Remark 2.6
  • Definition 2.7: Superfluous edge
  • Proposition 2.8
  • Definition 2.9: Survival and survival probability
  • ...and 74 more