On Continuous Optimization for Constraint Satisfaction Problems

2025-10-06

AI Summary

This paper addresses general constraint satisfaction problems (CSPs) involving finite-domain variables and complex constraints. We propose FourierCSP, the first framework to extend continuous local search (CLS) from Boolean satisfiability (SAT) to CSPs. Our core method introduces a finite-domain generalization of the Walsh–Fourier transform, enabling compact multilinear polynomial representations that directly encode constraint semantics—eliminating the need for auxiliary variables or high-dimensional embeddings. By integrating projected gradient optimization over a continuous relaxation space, FourierCSP achieves encoding-free, differentiable constraint modeling with theoretical convergence guarantees. Experiments across multiple CSP benchmark suites demonstrate that FourierCSP is both scalable and competitive in solution quality, significantly broadening the class of discrete constraint problems amenable to efficient continuous optimization techniques.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Constraint satisfaction problems (CSPs) are fundamental in mathematics, physics, and theoretical computer science. While conflict-driven clause learning Boolean Satisfiability (SAT) solvers have achieved remarkable success and become the mainstream approach for Boolean satisfiability, recent advances show that modern continuous local search (CLS) solvers can achieve highly competitive results on certain classes of SAT problems. Motivated by these advances, we extend the CLS framework from Boolean SAT to general CSP with finite-domain variables and expressive constraints. We present FourierCSP, a continuous optimization framework that generalizes the Walsh-Fourier transform to CSP, allowing for transforming versatile constraints to compact multilinear polynomials, thereby avoiding the need for auxiliary variables and memory-intensive encodings. Our approach leverages efficient evaluation and differentiation of the objective via circuit-output probability and employs a projected gradient optimization method with theoretical guarantees. Empirical results on benchmark suites demonstrate that FourierCSP is scalable and competitive, significantly broadening the class of problems that can be efficiently solved by CLS techniques.

Problem

• Extending continuous local search from Boolean SAT to general constraint satisfaction problems
• Developing FourierCSP framework using Walsh-Fourier transform for constraint representation
• Enabling efficient continuous optimization for finite-domain CSPs without auxiliary variables

Innovation

• Generalizes Walsh-Fourier transform to CSP
• Transforms constraints to compact multilinear polynomials
• Uses projected gradient optimization with guarantees