Min-CSPs on Complete Instances III: Nearly Optimal Characterizations of Approximation for Symmetric CSPs

This paper studies complete Min-$k$-CSPs: an instance on $n$ variables has one nontrivial constraint for every $k$-set of variables (so $\binom{n}{k}$ constraints in total), and the goal is to find an assignment that minimizes the number of unsatisfied constraints. They are motivated by the connections to machine learning and data science, as well as the possibility of being testbeds for new algorithmic techniques. Previous recent results presented polynomial-time algorithms for the decision version of every complete Boolean $k$-CSP, which is a necessary condition for any multiplicative approximations for the minimization version, but nontrivial approximation results have been limited to fragile CSPs, 2-SAT, and NAE-3-SAT, leaving the approximability of most complete Boolean Min-CSPs open, including complete Min-$k$-SAT. In this paper, we resolve this question, up to nearly matching factors, for the natural class of symmetric Boolean predicates. Given $A \subseteq \{0, \dots, k\}$, Min-CSP($A$) is a Min-$k$-CSP where each constraint is specified by $k$ literals and is satisfied if the number of true literals belongs to $A$. Let $w(A)$ be the maximum number of consecutive integers in $A$. Our main results for complete Min-CSP($A$) are (1) a QPTAS for $w(A) \leq 2$ and (2) an $\widetilde{O}(n^{w(A)-3})$-approximation for $w(A) \geq 3$, as well as almost matching hardness results for the latter. Our QPTAS for $w(A)=2$ significantly improves the previous $O(1)$-approximation for Min-2-SAT and the $\operatorname{polylog}(n)$-approximation for Min-NAE-3-SAT, as well as extends the PTAS for fragile CSPs (i.e., $w(A) = 1$).