如果说当前的团算法是最优的，那么Valiant的解析器也是最优的
If the Current Clique Algorithms are Optimal, so is Valiant's Parser
摘要：
CFG识别问题是：给定上下文无关文法$\mathcal{G}$和长度为$n$的字符串$w$，判断是否可以从$\mathcal{G}$获得$w$。这是最基本的解析问题，也是计算机科学的核心问题。Valiant在1975年的解析器在$O(n^ω)$时间内解决了这个问题，其中$ω<2.373$是矩阵乘法指数。多年来，已经提出了数十种解析算法，但Valiant的上限仍然是不败的。最好的组合算法具有适度的次立方$O(n^3/\log^3{n})$复杂度。Lee（JACM'01）提供的证据表明，CFG解析需要快速矩阵乘法，并且很难甚至不可能获得非常有效和实用的算法。Lee指出，任何一种在运行时间上解决更一般的解析问题的算法$O(|\mathcal{G}|\cdot n^{3-\varepsilon})$都可以转换为布尔矩阵乘法的一种令人惊讶的次立方算法。不幸的是，Lee的硬性结果要求语法大小为$|\mathcal{G}|=Ω(n^6)$。对于更相关的常数大小语法，我们一无所知。在这项工作中，我们证明了任何改进的Valiant的算法，即使是恒定大小的语法，无论是在运行时间或避免效率低下的快速矩阵乘法，将意味着一个突破性的算法的$k$ -集团问题：给定一个图上的$n$节点，决定是否有$k$，形成一个集团。除了对基本问题的复杂性进行分类外，我们的简化还使我们对更现代和研究更充分的立方时间问题获得了类似的下限，对于这些问题，更快的算法在实践中是非常可取的：RNA折叠，计算生物学中的一个中心问题，以及Dyck语言编辑距离，回答了Saha的一个开放问题（FOCS'14）。
英文摘要：
The CFG recognition problem is: given a context-free grammar $\mathcal{G}$ and a string $w$ of length $n$, decide if $w$ can be obtained from $\mathcal{G}$. This is the most basic parsing question and is a core computer science problem. Valiant's parser from 1975 solves the problem in $O(n^ω)$ time, where $ω<2.373$ is the matrix multiplication exponent. Dozens of parsing algorithms have been proposed over the years, yet Valiant's upper bound remains unbeaten. The best combinatorial algorithms have mildly subcubic $O(n^3/\log^3{n})$ complexity. Lee (JACM'01) provided evidence that fast matrix multiplication is needed for CFG parsing, and that very efficient and practical algorithms might be hard or even impossible to obtain. Lee showed that any algorithm for a more general parsing problem with running time $O(|\mathcal{G}|\cdot n^{3-\varepsilon})$ can be converted into a surprising subcubic algorithm for Boolean Matrix Multiplication. Unfortunately, Lee's hardness result required that the grammar size be $|\mathcal{G}|=Ω(n^6)$. Nothing was known for the more relevant case of constant size grammars. In this work, we prove that any improvement on Valiant's algorithm, even for constant size grammars, either in terms of runtime or by avoiding the inefficiencies of fast matrix multiplication, would imply a breakthrough algorithm for the $k$-Clique problem: given a graph on $n$ nodes, decide if there are $k$ that form a clique. Besides classifying the complexity of a fundamental problem, our reduction has led us to similar lower bounds for more modern and well-studied cubic time problems for which faster algorithms are highly desirable in practice: RNA Folding, a central problem in computational biology, and Dyck Language Edit Distance, answering an open question of Saha (FOCS'14).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
