arXiv:2611.16302 · v1

Smoothed Analysis for Distributed MIS

首次提交 2026-11-12 12:56

版本更新 2026-11-12 12:56

雷达观测 2026-11-13 14:16

摘要

We study smoothed analysis of the distributed Maximal Independent Set (MIS) problem. Luby's algorithm computes an MIS in $O(\log n)$ rounds, while the recent $Ω(\log n/\log\log n)$ lower bound of Khoury and Schild (2026) shows that it is nearly optimal even in the LOCAL model. We ask whether small random perturbations can overcome this worst-case barrier. Given an arbitrary base graph $G$ on $n$ nodes, we consider the smoothed graph $\mathcal{G}_s=G\triangle G(n,p)$, where $G(n,p)$ is an independent random graph and $\triangle$ denotes symmetric difference. Thus, each possible edge is independently toggled with probability $p$. We establish nearly tight bounds on the smoothing needed for faster distributed MIS. For every base graph $G$, when $p=β\ln n/n$ for a sufficiently large constant $β$, a randomized LOCAL algorithm, given only $\mathcal{G}_s$, computes an MIS of $\mathcal{G}_s$ in $O(\sqrt{\log n})$ rounds with high probability. This is almost quadratically faster than the worst-case lower bound. Conversely, if $p\le(1/2-δ)\ln n/n$ for a fixed constant $δ>0$, there exists a base graph on which every randomized LOCAL MIS algorithm requires $Ω(\log n/\log\log n)$ rounds on the smoothed graph. Hence, the smoothing parameter required for this speedup is optimal up to a constant factor. These results show that smoothed analysis can overcome worst-case lower-bound barriers for distributed local graph problems.