Optimal Polynomial-Time Compression for Boolean Max CSP

In the Boolean maximum constraint satisfaction problem—Max CSPΓ—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 CSPΓ with respect to the optimal compression size. Namely, we prove that Max CSPΓ 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 CSPΓ can be compressed into an equivalent instance with 𝒪(nd log n) bits in polynomial time,
(2)
Max CSPΓ does not admit such a compression to 𝒪(nd-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 CSPΓ. 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.