Optimal Polynomial-Time Compression for Boolean Max CSP

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 O(n^d logn) bits in polynomial time, (2) Max CSP(Γ) does not admit such a compression to O(n^d-ϵ) 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 O(2^(1-ϵ)n) for particular classes of Max CSPs is as hard as breaching this barrier for Max d-SAT for some d.