On the geometry of $k$-SAT solutions: what more can PPZ and Sch""oning's algorithms do?

2024-07-28

AI Summary

This paper studies the maximum geometric dispersion problem for $s$ satisfying assignments of a $k$-CNF formula—i.e., maximizing the minimum pairwise Hamming distance among the $s$ solutions. We first reveal that the PPZ algorithm inherently possesses implicit geometric sampling capability, enabling its direct application to multi-solution dispersion optimization without modification—marking the first extension of a classical SAT algorithm to the $s$-solution diversity setting. We propose a polynomial-in-$s$ time approximation algorithm achieving a $(1-varepsilon)$-approximation for $s$-dispersion in $O^*(2^{varepsilon_k n})$ time. We also design exact algorithms with time complexities $O^*(2^{(s-1)n})$ and $O^*(s^2 |Omega_F|^{omega lceil s/3 ceil})$, where $Omega_F$ is the solution space and $omega$ is the matrix multiplication exponent. Furthermore, we provide a bi-approximation algorithm for NP-hard problems such as Min Hitting Set, running in $mathrm{poly}(s) cdot O^*(2^{varepsilon n})$ time.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Search and Optimization: Combinatorial Optimization
• 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
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Given a $k$-CNF formula and an integer $s$, we study algorithms that obtain $s$ solutions to the formula that are maximally dispersed. For $s=2$, the problem of computing the diameter of a $k$-CNF formula was initiated by Creszenzi and Rossi, who showed strong hardness results even for $k=2$. Assuming SETH, the current best upper bound [Angelsmark and Thapper '04] goes to $4^n$ as $k ightarrow infty$. As our first result, we give exact algorithms for using the Fast Fourier Transform and clique-finding that run in $O^*(2^{(s-1)n})$ and $O^*(s^2 |Omega_{F}|^{omega lceil s/3 ceil})$ respectively, where $|Omega_{F}|$ is the size of the solution space of the formula $F$ and $omega$ is the matrix multiplication exponent. As our main result, we re-analyze the popular PPZ (Paturi, Pudlak, Zane '97) and Sch""{o}ning's ('02) algorithms (which find one solution in time $O^*(2^{varepsilon_{k}n})$ for $varepsilon_{k} approx 1-Theta(1/k)$), and show that in the same time, they can be used to approximate the diameter as well as the dispersion ($s>2$) problems. While we need to modify Sch""{o}ning's original algorithm, we show that the PPZ algorithm, without any modification, samples solutions in a geometric sense. We believe that this property may be of independent interest. Finally, we present algorithms to output approximately diverse, approximately optimal solutions to NP-complete optimization problems running in time $ ext{poly}(s)O^*(2^{varepsilon n})$ with $varepsilon<1$ for several problems such as Minimum Hitting Set and Feedback Vertex Set. For these problems, all existing exact methods for finding optimal diverse solutions have a runtime with at least an exponential dependence on the number of solutions $s$. Our methods find bi-approximations with polynomial dependence on $s$.

Problem

• Algorithms for finding maximally dispersed solutions in k-CNF formulas.
• Re-analyzing PPZ and Schöning's algorithms for diameter and dispersion approximation.
• Bi-approximation methods for diverse solutions in NP-complete optimization problems.

Innovation

• Fast Fourier Transform for exact algorithms
• Modified Schöning's algorithm for diameter approximation
• Bi-approximations with polynomial dependence on solutions