基于在线矩阵-向量乘法猜想统一和增强动态问题的硬度

M. Henzinger, Sebastian Krinninger, Danupon Nanongkai, Thatchaphol Saranurak
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引用次数: 257

摘要

考虑下面的在线布尔矩阵向量乘法问题:我们给定一个n × n矩阵M,并将得到n个大小为n的列向量,记为v1,…, vn,一个接一个。在看到每个向量vi之后,我们必须在看到下一个向量之前输出乘积Mvi。一种朴素算法可以用总共O(n3)时间来解决这个问题,其运行时间可以略微提高到O(n3/log2 n) [Williams SODA'07]。我们证明了这个问题不存在真正的次三次(O(n3-ε))时间算法的猜想可以用来展示许多动态问题所共有的多项式时间硬度。对于许多问题,如子图连通性、Pagh问题、d-failure连通性、递减单源最短路径和递减传递闭包,这个猜想意味着紧硬度结果。因此,证明或反驳这个猜想将是非常有趣的,因为它要么暗示了几个严格的无条件下界,要么突破了阻碍这些问题进展的常见障碍。这一猜想也可能被认为是反对进一步改进这些问题的有力证据,因为反驳它将意味着组合布尔矩阵乘法和其他长期存在的问题的重大突破,如果术语“组合算法”被解释为“Strassen-like算法”[Ballard et al]。SPAA 11]。该猜想还导致以前基于不同问题和猜想的问题的硬度结果-例如3SUM,组合布尔矩阵乘法,三角形检测和多相-从而为证明动态算法的多项式硬度结果提供了统一的方法;一些新的证明也更简单,甚至变得微不足道。该猜想还导致更强的和新的,非平凡的,硬度的结果,例如,对于全动态密度子图和直径问题。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Unifying and Strengthening Hardness for Dynamic Problems via the Online Matrix-Vector Multiplication Conjecture
Consider the following Online Boolean Matrix-Vector Multiplication problem: We are given an n x n matrix M and will receive n column-vectors of size n, denoted by v1, ..., vn, one by one. After seeing each vector vi, we have to output the product Mvi before we can see the next vector. A naive algorithm can solve this problem using O(n3) time in total, and its running time can be slightly improved to O(n3/log2 n) [Williams SODA'07]. We show that a conjecture that there is no truly subcubic (O(n3-ε)) time algorithm for this problem can be used to exhibit the underlying polynomial time hardness shared by many dynamic problems. For a number of problems, such as subgraph connectivity, Pagh's problem, d-failure connectivity, decremental single-source shortest paths, and decremental transitive closure, this conjecture implies tight hardness results. Thus, proving or disproving this conjecture will be very interesting as it will either imply several tight unconditional lower bounds or break through a common barrier that blocks progress with these problems. This conjecture might also be considered as strong evidence against any further improvement for these problems since refuting it will imply a major breakthrough for combinatorial Boolean matrix multiplication and other long-standing problems if the term "combinatorial algorithms" is interpreted as "Strassen-like algorithms" [Ballard et al. SPAA'11]. The conjecture also leads to hardness results for problems that were previously based on diverse problems and conjectures -- such as 3SUM, combinatorial Boolean matrix multiplication, triangle detection, and multiphase -- thus providing a uniform way to prove polynomial hardness results for dynamic algorithms; some of the new proofs are also simpler or even become trivial. The conjecture also leads to stronger and new, non-trivial, hardness results, e.g., for the fully-dynamic densest subgraph and diameter problems.
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