Optimal Sparsification for Some Binary CSPs Using Low-Degree Polynomials

Abstract

This article 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 ⊈ 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 −ε ) bits for any ε > 0. For the N ot -A ll -E qual sat problem, a compression to size  Õ ( n d −1 ) exists. We put these results in a common framework by analyzing the compressibility of CSPs with a binary domain. 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 E xact S atisfiability 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−ε ) bits for any  ε > 0, unless NP ⊆ coNP/poly.