A Structural Investigation of the Approximability of Polynomial-Time Problems
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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,…, x_k#{y:ϕ(x_1,…,x_k,y)} where ϕ 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 ϕ, the best possible approximation guarantee in faster-than-exhaustive-search time O(m^k-δ) falls into one of four categories: * optimizable to exactness in time O(m^k-δ), * an (inefficient) approximation scheme, i.e., a (1+ϵ)-approximation in time O(m^k-f(ϵ)), * a (fixed) constant-factor approximation in time O(m^k-δ), or * an m^ϵ-approximation in time O(m^k-f(ϵ)). 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.
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