We initiate the study of fine-grained completeness theorems for exact and approximate optimization in the polynomial-time regime. Inspired by the first completeness results for decision problems in P (Gao, Impagliazzo, Kolokolova, Williams, TALG 2019) as well as the classic class MaxSNP and MaxSNP-completeness for NP optimization problems (Papadimitriou, Yannakakis, JCSS 1991), we define polynomial-time analogues MaxSP and MinSP, which contain a number of natural optimization problems in P, including Maximum Inner Product, general forms of nearest neighbor search and optimization variants of the $k$-XOR problem. Specifically, we define MaxSP as the class of problems definable as $\max_{x_1,\dots,x_k} \#\{ (y_1,\dots,y_\ell) : \phi(x_1,\dots,x_k, y_1,\dots,y_\ell) \}$, where $\phi$ is a quantifier-free first-order property over a given relational structure (with MinSP defined analogously). On $m$-sized structures, we can solve each such problem in time $O(m^{k+\ell-1})$. Our results are: - We determine (a sparse variant of) the Maximum/Minimum Inner Product problem as complete under *deterministic* fine-grained reductions: A strongly subquadratic algorithm for Maximum/Minimum Inner Product would beat the baseline running time of $O(m^{k+\ell-1})$ for *all* problems in MaxSP/MinSP by a polynomial factor. - This completeness transfers to approximation: Maximum/Minimum Inner Product is also complete in the sense that a strongly subquadratic $c$-approximation would give a $(c+\varepsilon)$-approximation for all MaxSP/MinSP problems in time $O(m^{k+\ell-1-\delta})$, where $\varepsilon > 0$ can be chosen arbitrarily small. Combining our completeness with~(Chen, Williams, SODA 2019), we obtain the perhaps surprising consequence that refuting the OV Hypothesis is *equivalent* to giving a $O(1)$-approximation for all MinSP problems in faster-than-$O(m^{k+\ell-1})$ time.
翻译:我们开始研究精细的完整度理论, 以精确和最接近的方式优化 IP- SP- SP- 制度。 受以下决定问题的第一个完整度结果的启发: P (Gao, Impagliazzo, Kolokolova, Williams, TALG 2019) 以及典型的 MaxSNP 和 MaxSNP 20 问题( Papadimitriou, Yannakakis, JCSS 1991), 我们定义了 IP (x_ 1, dirioSP 和 MinSP, 包含一些自然优化问题的 P, 包括最大内部产品, 最接近的邻居搜索和优化变异 美元- Xokololor 问题。 具体来说, 我们定义的 MaxSP 类别问题类别为 $xxxx_ 1, dockrealtial- listal prouplemental dismoal a max. a max- primoal- matial matial max m max modeal- rial- tial mextial- extial tialtial m) a tium a tium a, tial tium tial tial tial- timoal- time tial tialtialtime- tial- tialtialtialtialtialtialtialtialtial exxxx_ 1, mocal- mocal- mocal- mocal- moxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx motialtialtialtial mocal- mocalti, mo- mocal- moal moal motial moal moal mo moxl moxl moxl moxxxxxxxxxxxxxxxxal motaltal motal