走进广场-论二次可解问题的复杂性
Into the Square - On the Complexity of Quadratic-Time Solvable Problems
摘要：
本文将分析几个二次时间可解的问题，并将它们分为两类：在真正的次二次时间（即，在时间$O(n^{2-ε})$为某些$ε>0$）可解的问题和不是的问题，除非众所周知的强指数时间假设（SETH）是假的。特别是，我们将证明一些二次时间可解问题确实比预期的更容易。我们将提供一个计算有向图在时间上的传递闭包的算法$O(mn^{\frac{ω+1}{4}})$，其中$m$表示传递闭包中的边数，$ω$是矩阵乘法的指数。作为一个副作用，我们将证明我们的算法运行在时间$O(n^{\frac{5}{3}})$，如果传递闭包是稀疏的。如果我们想检查一个图是否是传递的，通过将m替换为图本身的边数，同样的时间界限也成立。据我们所知，这是稀疏传递有向图识别的最快算法。最后，我们将把我们的算法应用于相似性图识别问题（可以追溯到1941年），获得第一个真正的次二次算法。本文的第二部分涉及硬度结果。从一个人工的二次时间可解的变化的k-SAT问题，我们将构建一个图的卡普约简，证明一个真正的次二次时间算法中的任何问题的图伪造SETH。所分析的问题有：计算子集图，求支配集，计算顶点的介数中心度，计算最小贴近中心度，计算顶点对的双曲性。我们也将能够包括在我们的框架中已经出现在文献中的三个证明，关于图直径计算，字符串的局部对齐和向量的正交性。
英文摘要：
This paper will analyze several quadratic-time solvable problems, and will classify them into two classes: problems that are solvable in truly subquadratic time (that is, in time $O(n^{2-ε})$ for some $ε>0$) and problems that are not, unless the well known Strong Exponential Time Hypothesis (SETH) is false. In particular, we will prove that some quadratic-time solvable problems are indeed easier than expected. We will provide an algorithm that computes the transitive closure of a directed graph in time $O(mn^{\frac{ω+1}{4}})$, where $m$ denotes the number of edges in the transitive closure and $ω$ is the exponent for matrix multiplication. As a side effect, we will prove that our algorithm runs in time $O(n^{\frac{5}{3}})$ if the transitive closure is sparse. The same time bounds hold if we want to check whether a graph is transitive, by replacing m with the number of edges in the graph itself. As far as we know, this is the fastest algorithm for sparse transitive digraph recognition. Finally, we will apply our algorithm to the comparability graph recognition problem (dating back to 1941), obtaining the first truly subquadratic algorithm. The second part of the paper deals with hardness results. Starting from an artificial quadratic-time solvable variation of the k-SAT problem, we will construct a graph of Karp reductions, proving that a truly subquadratic-time algorithm for any of the problems in the graph falsifies SETH. The analyzed problems are the following: computing the subset graph, finding dominating sets, computing the betweenness centrality of a vertex, computing the minimum closeness centrality, and computing the hyperbolicity of a pair of vertices. We will also be able to include in our framework three proofs already appeared in the literature, concerning the graph diameter computation, local alignment of strings and orthogonality of vectors.
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。
