Optimal polynomial-time compression for Boolean Max CSP

In the Boolean maximum constraint satisfaction problem - Max CSP$(\Gamma)$ - one is given a collection of weighted applications of constraints from a finite constraint language $\Gamma$, 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 an elegant dichotomy theorem providing a criterion on $\Gamma$ 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$(\Gamma)$ with respect to the optimal compression size. Namely, we prove that Max CSP$(\Gamma)$ parameterized by the number of variables $n$ is either polynomial-time solvable, or there exists an integer $d \ge 2$ depending on $\Gamma$, such that 1. An instance of \textsc{Max CSP$(\Gamma)$} can be compressed into an equivalent instance with $O(n^d\log n)$ bits in polynomial time, 2. Max CSP$(\Gamma)$ does not admit such a compression to $O(n^{d-\epsilon})$ bits unless $\text{NP} \subseteq \text{co-NP} / \text{poly}$. Our reductions are based on interpreting constraints as multilinear polynomials combined with the framework of constraint implementations. As another application of our reductions, we reveal tight connections between optimal running times for solving Max CSP$(\Gamma)$. More precisely, we show that obtaining a running time of the form $O(2^{(1-\epsilon)n})$ for particular classes of Max CSPs is as hard as breaching this barrier for Max $d$-SAT for some $d$.