通过自动自归约的分布式下界

中文摘要 AI 辅助

将轮消除发展成为一种通用技术[PODC 2019]标志着我们对分布式环境中许多图问题难解性理解的转折点，并带来了若干突破性结果。然而，轮消除技术似乎无法产生作为节点数$n$的函数的$\omega(\log\log n)$轮随机下界。最近，Khoury和Schild [FOCS 2025]引入了一种称为通过自归约的轮消除的新技术，它绕过了经典轮消除的局限性。利用这种方法，作者证明了在LOCAL模型中，任何用于最大匹配的随机算法都需要$\Omega(\sqrt{\log n})$轮。他们的优雅技术在某些方面类似于经典轮消除，但在其他方面根本不同。然而，它专门针对最大匹配，而非适用于广泛的问题类别。在本文中，我们证明自归约实际上是经典轮消除的一个特例，从而将其转变为一种通用方法。特别地，我们引入了一种新的衡量算法误差的方法，并表明在这种新度量下，经典轮消除确实可以产生$\omega(\log\log n)$随机下界。更具体地说，我们确定了一大类问题，对于这些问题，这种改进完全是黑盒的：一旦一个问题被证明属于该类，更强的随机下界就会自动从经典轮消除框架中得出。作为应用，我们为一系列图问题证明了$\Omega(\sqrt{\log n})$随机下界，即正则$2$-着色图上的最大匹配、$\frac{1}{k}$-积分匹配和最大$H$-打包。

英文摘要

The development of round elimination into a general-purpose technique [PODC 2019] marked a turning point in our understanding of the hardness of many graph problems in the distributed setting and led to several breakthrough results. However, the round elimination technique seems unable to yield randomized lower bounds of $ω(\log \log n)$ rounds as a function of the number $n$ of nodes. Very recently, Khoury and Schild [FOCS 2025] introduced a new technique called round elimination via self-reduction, which bypasses the limitations of classical round elimination. Using this approach, the authors show that any randomized algorithm for maximal matching requires $Ω(\sqrt{\log n})$ rounds in the LOCAL model. Their elegant technique is, in some respects, similar to classical round elimination while being fundamentally different in others. However, it is tailored specifically to maximal matching rather than being applicable to a broad class of problems. In this paper, we show that self-reduction is, in fact, a special case of classical round elimination, thereby turning it into a general-purpose approach. In particular, we introduce a new way to measure the error of an algorithm and show that, under this new measure, classical round elimination can indeed yield $ω(\log \log n)$ randomized lower bounds. More specifically, we identify a large class of problems for which this improvement is entirely black-box: once a problem is shown to belong to the class, stronger randomized lower bounds follow automatically from the classical round-elimination framework. As an application, we prove $Ω(\sqrt{\log n})$ randomized lower bounds for a range of graph problems, namely, maximal matching on regular $2$-colored graphs, $\frac{1}{k}$-integral matching, and maximal $H$-packing.

发表机构

• Gran Sasso Science Institute（格兰萨索科学研究所）

机构由 AI 辅助整理，请以论文原文为准。