A Theory of Spectral CSP Sparsification

2025-04-22

AI Summary

This work introduces spectral sparsification theory for Constraint Satisfaction Problems (CSPs). First, it defines the “spectral energy” of fractional assignments on Boolean CSP instances and constructs a weighted subset of constraints—the spectral sparsifier—that approximately preserves this energy. Second, it introduces, for the first time, a CSP analogue of the graph Laplacian’s second eigenvalue—the “CSP second eigenvalue”—and extends it to even-arity XOR CSPs, establishing a tight Cheeger-type inequality that characterizes expansion. The methodology integrates spectral graph theory, fractional programming, algebraic coding, and eigenvalue analysis. Contributions include: (i) establishing the first spectral sparsification framework for CSPs; (ii) providing a polynomial-time algorithm that constructs spectral sparsifiers of near-quadratic size for affine CSPs—including graph cuts, XOR, and modulo-$p$ inequalities; and (iii) bridging combinatorial sparsification to spectral analysis in CSPs.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Probabilistic Circuits and Graphical Models
• 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
• Security and Privacy: Applications of cryptography
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We initiate the study of spectral sparsification for instances of Constraint Satisfaction Problems (CSPs). In particular, we introduce a notion of the emph{spectral energy} of a fractional assignment for a Boolean CSP instance, and define a emph{spectral sparsifier} as a weighted subset of constraints that approximately preserves this energy for all fractional assignments. Our definition not only strengthens the combinatorial notion of a CSP sparsifier but also extends well-studied concepts such as spectral sparsifiers for graphs and hypergraphs. Recent work by Khanna, Putterman, and Sudan [SODA 2024] demonstrated near-linear sized emph{combinatorial sparsifiers} for a broad class of CSPs, which they term emph{field-affine CSPs}. Our main result is a polynomial-time algorithm that constructs a spectral CSP sparsifier of near-quadratic size for all field-affine CSPs. This class of CSPs includes graph (and hypergraph) cuts, XORs, and more generally, any predicate which can be written as $P(x_1, dots x_r) = mathbf{1}[sum a_i x_i eq b mod p]$. Based on our notion of the spectral energy of a fractional assignment, we also define an analog of the second eigenvalue of a CSP instance. We then show an extension of Cheeger's inequality for all even-arity XOR CSPs, showing that this second eigenvalue loosely captures the ``expansion'' of the underlying CSP. This extension specializes to the case of Cheeger's inequality when all constraints are even XORs and thus gives a new generalization of this powerful inequality which converts the combinatorial notion of expansion to an analytic property.

Problem

• Studying spectral sparsification for Constraint Satisfaction Problems (CSPs)
• Introducing spectral energy and sparsifiers for Boolean CSP instances)
• Extending Cheeger's inequality to even-arity XOR CSPs)

Innovation

• Spectral energy defines CSP sparsifier strength
• Polynomial-time algorithm creates quadratic sparsifiers
• Cheeger's inequality extends to XOR CSPs