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On Continuous Optimization for Constraint Satisfaction Problems

Yunuo Cen, Zixuan Wang, Jintao Zhang, Zhiwei Zhang, Xuanyao Fong

TL;DR

This work extends continuous local search from Boolean SAT to general finite-domain CSPs by introducing FourierCSP, which uses an orthonormal Walsh-Fourier basis to represent constraints and a randomized rounding scheme that yields a differentiable objective via circuit-output probability. Constraints are evaluated efficiently through multi-valued decision diagrams (MDDs), and a projected gradient method with convergence guarantees guides the search within the simplex product space, enabling GPU-accelerated optimization. Empirical results on task scheduling and graph coloring-with-hashing benchmarks show FourierCSP solving large, non-Boolean CSPs more effectively than several complete solvers in certain regimes, and significantly outperforming prior CLS approaches that required Booleanization. The work broadens the applicability of CLS to CSP and suggests a viable, scalable path for solving complex finite-domain problems with continuous optimization techniques and modern parallel hardware.

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.

On Continuous Optimization for Constraint Satisfaction Problems

TL;DR

This work extends continuous local search from Boolean SAT to general finite-domain CSPs by introducing FourierCSP, which uses an orthonormal Walsh-Fourier basis to represent constraints and a randomized rounding scheme that yields a differentiable objective via circuit-output probability. Constraints are evaluated efficiently through multi-valued decision diagrams (MDDs), and a projected gradient method with convergence guarantees guides the search within the simplex product space, enabling GPU-accelerated optimization. Empirical results on task scheduling and graph coloring-with-hashing benchmarks show FourierCSP solving large, non-Boolean CSPs more effectively than several complete solvers in certain regimes, and significantly outperforming prior CLS approaches that required Booleanization. The work broadens the applicability of CLS to CSP and suggests a viable, scalable path for solving complex finite-domain problems with continuous optimization techniques and modern parallel hardware.

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.

Paper Structure

This paper contains 42 sections, 14 theorems, 71 equations, 4 figures, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Contributions
  3. Preliminaries
  4. Constraint Satisfaction Problem
  5. Walsh-Fourier Transform
  6. Theoretical Framework
  7. Continuous Local Search
  8. Limitations of Previous CLS Solvers
  9. FourierCSP
  10. Reduction by Randomized Rounding
  11. Circuit-Output Probability
  12. Projected Gradient
  13. Implementation Techniques
  14. Experimental Results
  15. Benchmarks and Encodings
  16. ...and 27 more sections

Key Result

Theorem 1

Having fixed the Fourier bases, $f$ is uniquely expressible as where $\hat{f}(\alpha) \in \mathbb{R}$ is called Fourier coefficient, and it satisfies

Figures (4)

  • Figure 1: Results on task scheduling problems. The numbers in the legend are the average PAR-2 score of the solvers.
  • Figure 2: Results on graph coloring with hashing query problems. The numbers in the legend are (the average relative score, #Win of the solvers).
  • Figure 3: Decision diagrams of the test cases with 4 variables. Left: $c_1, c_2, c_3$; Mid: $c_{13}, c_{14}, c_{15}$; Right: $c_{25}, c_{26}, c_{27}$.
  • Figure 4: Decision diagrams of the test cases with 8 variables. Left: $c_5, c_6, c_7$; Mid: $c_{17}, c_{18}, c_{19}$; Right: $c_{29}, c_{30}, c_{31}$.

Theorems & Definitions (23)

  • Definition 1: Fourier Bases
  • Theorem 1: Walsh-Fourier Expansion o2014analysis
  • Definition 2: Objective
  • Definition 3: Randomized Rounding
  • Definition 4: Probability Space
  • Theorem 2: Expectation
  • Theorem 3: Reduction
  • Definition 5: Circuit-Output Probability
  • Corollary 1
  • Lemma 1: Top-Down Evaluation
  • ...and 13 more