Dynamic Maximal Matching in Clique Networks
Minming Li, Peter Robinson, Xianbin Zhu
TL;DR
This work studies batch-dynamic maximal matching in a distributed setting where $n$ vertices are partitioned across $k$ players in the $k$-clique model and updates arrive in batches of at most $\ell$ edges. It establishes information-theoretic lower bounds on the update time for both oblivious and adaptive adversaries and presents a randomized algorithm with initialization time $O( ceil( n / (\beta k) ) \log n )$ rounds and update times that scale with the batch size as $O( ceil( \ell / (\beta k) ) \log(\beta k) )$ for the oblivious case and $O( ceil( \ell / \sqrt{\beta k} ) \log(\beta k) )$ for the adaptive case. The results quantify the trade-offs between batch size, bandwidth, and partitioning randomness, showing near-tight bounds up to polylog factors. The approach leverages efficient information dissemination, randomized initialization via a maximal matching algorithm adapted to the congested-clique setting, and a multi-phase update procedure that handles deletions and insertions with controlled communication. The methods extend to the congested clique, achieving communication-efficiency where the update message complexity scales with the number of edge changes.
Abstract
We consider the problem of computing a maximal matching with a distributed algorithm in the presence of batch-dynamic changes to the graph topology. We assume that a graph of $n$ nodes is vertex-partitioned among $k$ players that communicate via message passing. Our goal is to provide an efficient algorithm that quickly updates the matching even if an adversary determines batches of $\ell$ edge insertions or deletions. Assuming a link bandwidth of $O(β\log n)$ bits per round, for a parameter $β\ge 1$, we first show a lower bound of $Ω( \frac{\ell\,\log k}{β\,k^2\log n})$ rounds for recomputing a matching assuming an oblivious adversary who is unaware of the initial (random) vertex partition as well as the current state of the players, and a stronger lower bound of $Ω(\frac{\ell}{β\,k\log n})$ rounds against an adaptive adversary, who may choose any balanced (but not necessarily random) vertex partition initially and who knows the current state of the players. We also present a randomized algorithm that has an initialization time of $O( \lceil\frac{n}{β\,k}\rceil\log n )$ rounds, while achieving an update time that that is independent of $n$: In more detail, the update time is $O( \lceil \frac{\ell}{β\,k} \rceil \log(β\,k))$ against an oblivious adversary, who must fix all updates in advance. If we consider the stronger adaptive adversary, the update time becomes $O( \lceil \frac{\ell}{\sqrt{β\,k}}\rceil \log(β\,k))$ rounds.