A constraint satisfaction problem (CSP), $\textsf{Max-CSP}(\mathcal{F})$, is specified by a finite set of constraints $\mathcal{F} \subseteq \{[q]^k \to \{0,1\}\}$ for positive integers $q$ and $k$. An instance of the problem on $n$ variables is given by $m$ applications of constraints from $\mathcal{F}$ to subsequences of the $n$ variables, and the goal is to find an assignment to the variables that satisfies the maximum number of constraints. In the $(\gamma,\beta)$-approximation version of the problem for parameters $0 \leq \beta < \gamma \leq 1$, the goal is to distinguish instances where at least $\gamma$ fraction of the constraints can be satisfied from instances where at most $\beta$ fraction of the constraints can be satisfied. In this work we consider the approximability of this problem in the context of sketching algorithms and give a dichotomy result. Specifically, for every family $\mathcal{F}$ and every $\beta < \gamma$, we show that either a linear sketching algorithm solves the problem in polylogarithmic space, or the problem is not solvable by any sketching algorithm in $o(\sqrt{n})$ space.
翻译:限制满意度问题( CSP) $\ textsf{ max- CSP} (\ mathcal{F}) $, 限制满意度问题( CSP) $\ textsf{ max- CSP} (\ mathcal{F}) 美元, 由一组有限的限制来指定 $\ mathcal{F} 美元[q] k k\ k 至 0. 0, 1 美元。 美元变量问题的例子由 $\ macal{ F} 至 $n 变量的后继( mathcathcal{ {F} ) 提供, 目标是找到一个符合最大限制数量的变量的指定 。 在 $( gamma,\ beta) $- a adcol- access regrequemember exmination exminational $ glasmaislation.