On Diverse Solutions to Max-k-CSP and Bounded Degree k-SAT

2026-09-27

AI Summary

This study addresses the challenge of efficiently generating approximate solutions that simultaneously account for constraint and variable diversity in constraint satisfaction problems, including Max-k-CSP and bounded-degree k-SAT. By integrating multi-objective optimization with linear code construction techniques, this work proposes a bicriteria approximation algorithm capable of implicitly representing an exponentially large diverse solution set. Furthermore, by leveraging the Lovász Local Lemma (LLL), it establishes the NP-hardness of exact diversity problems within the LLL regime. The primary contribution lies in achieving polynomial-time approximation algorithms that provide provable performance guarantees for both constraint and variable diversity, thereby effectively resolving the difficulty of efficiently generating highly diverse approximate solutions.

Abstract

We study the problem of generating diverse solutions to Max-$k$-CSP and bounded-degree $k$-SAT, focusing on two distinct metrics: constraint diversity and variable diversity. For constraint diversity, the goal is to output $s \geq 2$ assignments to the CSP such that each assignment satisfies a $c$-fraction of the constraints, while maximizing the diversity among the $0$-$1$ indicator vectors of satisfied constraints in the Hamming metric. By reducing this to a multi-criteria optimization problem, we design $poly(n,s)$ time approximation algorithms that return s assignments achieving provable bi-criteria guarantees on both the fraction of satisfied constraints and diversity of the constraint vectors. For variable diversity, the objective is to maximize the Hamming distance between the assignments, while also maximizing the number of constraints satisfied. For Max-$k$-CSP instances when the desired number of solutions is $s=2^{O(n)}$, we implicitly represent these diverse approximate solutions by constructing linear codes within the solution space. Finally, we investigate variable diversity for $k$-SAT in the Lov\'asz Local Lemma regime. In this setting, we establish NP-hardness for the exact diversity problem (computing the diameter of the solution space) and provide a polynomial-time approximation algorithm to efficiently generate diverse satisfying assignments.

Problem

• Max-k-CSP
• Bounded-degree k-SAT
• Diverse solutions
• Constraint diversity
• Variable diversity

Innovation

• Max-k-CSP
• Diverse Solutions
• Multi-criteria Optimization
• Linear Codes
• Lovász Local Lemma