A Theory of Spectral CSP Sparsification
Sanjeev Khanna, Aaron Putterman, Madhu Sudan
TL;DR
The paper defines spectral energy for CSPs and introduces spectral sparsifiers that preserve this energy across all fractional assignments, uniting graph, hypergraph, and CSP sparsification. It develops a permutation-based quadratic form representation, a crossing matrix, and a code-theoretic framework to sparsify CSPs via code sparsification, achieving near-quadratic-size sparsifiers for all field-affine CSPs in randomized polynomial time. For XOR CSPs with even arity, it proves a Cheeger-type inequality tying the second eigenvalue of a CSP Laplacian to expansion, and extends the framework to general field-affine predicates with a log^2(p) factor. The results substantially generalize spectral sparsification from graphs and hypergraphs to broad CSP families, enabling efficient approximate representations that preserve energy and enabling analytic tools like Cheeger inequalities in this broader setting.
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 $P(x_1, \dots x_r) = \mathbf{1}[\sum a_i x_i \neq b \mod p]$. 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.