  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 $d$.
