A Structural Investigation of the Approximability of Polynomial-Time Problems

We initiate the systematic study of a recently introduced polynomial-time analogue of MaxSNP, which includes a large number of well-studied problems (including Nearest and Furthest Neighbor in the Hamming metric, Maximum Inner Product, optimization variants of $k$-XOR and Maximum $k$-Cover). Specifically, MaxSP$_k$ denotes the class of $O(m^k)$-time problems of the form $\max_{x_1,\dots, x_k} \#\{y:\phi(x_1,\dots,x_k,y)\}$ where $\phi$ is a quantifier-free first-order property and $m$ denotes the size of the relational structure. Assuming central hypotheses about clique detection in hypergraphs and MAX3SAT, we show that for any MaxSP$_k$ problem definable by a quantifier-free $m$-edge graph formula $\phi$, the best possible approximation guarantee in faster-than-exhaustive-search time $O(m^{k-\delta})$ falls into one of four categories: * optimizable to exactness in time $O(m^{k-\delta})$, * an (inefficient) approximation scheme, i.e., a $(1+\epsilon)$-approximation in time $O(m^{k-f(\epsilon)})$, * a (fixed) constant-factor approximation in time $O(m^{k-\delta})$, or * an $m^\epsilon$-approximation in time $O(m^{k-f(\epsilon)})$. We obtain an almost complete characterization of these regimes, for MaxSP$_k$ as well as for an analogously defined minimization class MinSP$_k$. As our main technical contribution, we rule out approximation schemes for a large class of problems admitting constant-factor approximations, under the Sparse MAX3SAT hypothesis posed by (Alman, Vassilevska Williams'20). As general trends for the problems we consider, we find: (1) Exact optimizability has a simple algebraic characterization, (2) only few maximization problems do not admit a constant-factor approximation; these do not even have a subpolynomial-factor approximation, and (3) constant-factor approximation of minimization problems is equivalent to deciding whether the optimum is equal to 0.