Optimal Sparsification for Some Binary CSPs Using Low-degree Polynomials

TL;DR

This paper investigates the limits of efficiently reducing the size of binary CSPs and satisfiability problems using kernelization, establishing bounds based on polynomial degree and encoding size, with implications for NP-hard problems.

Contribution

It provides a unified framework for analyzing the compressibility of binary CSPs through polynomial degree characterization, deriving nearly tight bounds for various problem types.

Findings

Lower bounds depend on the polynomial degree of constraints.

Encoding size of constraints affects compressibility.

Certain problems cannot be compressed below specific bounds unless NP ⊆ coNP/poly.

Abstract

This paper analyzes to what extent it is possible to efficiently reduce the number of clauses in NP-hard satisfiability problems, without changing the answer. Upper and lower bounds are established using the concept of kernelization. Existing results show that if NP is not contained in coNP/poly, no efficient preprocessing algorithm can reduce n-variable instances of CNF-SAT with d literals per clause, to equivalent instances with $O(n^{d-e})$ bits for any e > 0. For the Not-All-Equal SAT problem, a compression to size $\~O(n^{d-1})$ exists. We put these results in a common framework by analyzing the compressibility of binary CSPs. We characterize constraint types based on the minimum degree of multivariate polynomials whose roots correspond to the satisfying assignments, obtaining (nearly) matching upper and lower bounds in several settings. Our lower bounds show that not just the number of constraints, but also the encoding size of individual constraints plays an important role. For example, for Exact Satisfiability with unbounded clause length it is possible to efficiently reduce the number of constraints to n+1, yet no polynomial-time algorithm can reduce to an equivalent instance with $O(n^{2-e})$ bits for any e > 0, unless NP is a subset of coNP/poly.