Optimal Polynomial-time Compression for Boolean Max CSP

Abstract

In the Boolean maximum constraint satisfaction problem – Max CSP ( Γ ) – one is given a collection of weighted applications of constraints from a finite constraint language  Γ , over a common set of variables, and the goal is to assign Boolean values to the variables so that the total weight of satisfied constraints is maximized. There exists a concise dichotomy theorem providing a criterion on Γ for the problem to be polynomial-time solvable and stating that otherwise it becomes NP-hard. We study the NP-hard cases through the lens of kernelization and provide a complete characterization of  Max CSP ( Γ ) with respect to the optimal compression size. Namely, we prove that Max CSP ( Γ ) parameterized by the number of variables  n is either polynomial-time solvable, or there exists an integer d ≥ 2 depending on Γ , such that: (1) An instance of Max CSP ( Γ ) can be compressed into an equivalent instance with \(\mathcal {O}(n^d\log n) \) bits in polynomial time, (2) Max CSP( Γ ) does not admit such a compression to \(\mathcal {O}(n^{d-\varepsilon }) \) bits unless NP⊆co-NP/poly. Our reductions are based on interpreting constraints as multilinear polynomials combined with the framework of ‘constraint implementations’, formerly used in the context of APX-hardness. As another application of our reductions, we reveal tight connections between optimal running times for solving Max CSP( Γ ) . More precisely, we show that obtaining a running time of the form \(\mathcal {O}(2^{(1-\varepsilon)n}) \) for particular classes of Max CSP s is as hard as breaching this barrier for Max d - SAT for some d .