A Theory of Spectral CSP Sparsification

Abstract

We initiate the study of spectral sparsification for instances of Constraint Satisfaction Problems (CSPs). In particular, we introduce a notion of the spectral energy of a fractional assignment for a Boolean CSP instance, and define a 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 combinatorial sparsifiers for a broad class of CSPs, which they term 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(x₁, … x_r) = 𝟏[∑ a_i x_i ≠ 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.
Perhaps the most important effect of spectral sparsification is that it has led to certifiable sparsifiers for graphs and hypergraphs. This aspect remains open in our case even for XOR CSPs since the eigenvalues we describe in our Cheeger inequality are not known to be efficiently computable. Computing this efficiently, and/or finding other ways to certifiably sparsify CSPs are open questions emerging from our work. Another important open question is determining which classes of CSPs have near-linear size spectral sparsifiers.