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This paper analyzes to what extent it is possible to efficiently reduce thenumber of clauses in NP-hard satisfiability problems, without changing theanswer. Upper and lower bounds are established using the concept ofkernelization. Existing results show that if NP is not contained in coNP/poly,no efficient preprocessing algorithm can reduce n-variable instances of CNF-SATwith d literals per clause, to equivalent instances with $O(n^{d-e})$ bits forany 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 analyzingthe compressibility of binary CSPs. We characterize constraint types based onthe minimum degree of multivariate polynomials whose roots correspond to thesatisfying assignments, obtaining (nearly) matching upper and lower bounds inseveral settings. Our lower bounds show that not just the number ofconstraints, but also the encoding size of individual constraints plays animportant role. For example, for Exact Satisfiability with unbounded clauselength 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.

Optimal Sparsification for Some Binary CSPs Using Low-degree Polynomials