On the Locality of Hall's Theorem
Sebastian Brandt, Yannic Maus, Ananth Narayanan, Florian Schager, Jara Uitto
TL;DR
This work introduces a novel distributed design technique that yields near-tight, logarithmic-time algorithms for fundamental LOCAL problems by computing and composing local solutions derived from a distributed Hall’s theorem. Central to the approach is the concept of Hall graphs and a distributed Hal l-theorem that guarantees every node lies in a small-diameter Hall subgraph, enabling $O(\log n)$-round solutions for tasks such as $3\Delta/2$-edge coloring and hypergraph sinkless orientation. The authors obtain a deterministic $O(\Delta^2 \log n)$-round algorithm for $3\Delta/2$-edge coloring and show corollaries for constant-degree graphs, while also delivering tight or near-tight bounds for HSO and related problems, including bipartite maximum matching and weak splitting. They further provide randomized algorithms with polylogarithmic improvements under favorable parameter regimes and establish a linear lower bound for HSO in the $\delta = r$ regime, underscoring the limitations of sublinear deterministic approaches. Overall, the paper advances deterministic locality by bridging lower and upper bounds through a modular framework built on distributed Hall subgraphs and their interplay with hypergraph problems.
Abstract
The last five years of research on distributed graph algorithms have seen huge leaps of progress, both regarding algorithmic improvements and impossibility results: new strong lower bounds have emerged for many central problems and exponential improvements over the state of the art have been achieved for the runtimes of many algorithms. Nevertheless, there are still large gaps between the best known upper and lower bounds for many important problems. The current lower bound techniques for deterministic algorithms are often tailored to obtaining a logarithmic bound and essentially cannot be used to prove lower bounds beyond $Ω(\log n)$. In contrast, the best deterministic upper bounds are often polylogarithmic, raising the fundamental question of how to resolve the gap between logarithmic lower and polylogarithmic upper bounds and finally obtain tight bounds. We develop a novel algorithm design technique aimed at closing this gap. In essence, each node finds a carefully chosen local solution in $O(\log n)$ rounds and we guarantee that this solution is consistent with the other nodes' solutions without coordination. The local solutions are based on a distributed version of Hall's theorem that may be of independent interest and motivates the title of this work. We showcase our framework by improving on the state of the art for the following fundamental problems: edge coloring, bipartite saturating matchings and hypergraph sinkless orientation. In particular, we obtain an asymptotically optimal $O(\log n)$-round algorithm for $3Δ/2$-edge coloring in bounded degree graphs. The previously best bound for the problem was $O(\log^4 n)$ rounds, obtained by plugging in the state-of-the-art maximal independent set algorithm from arXiv:2303.16043 into the $3Δ/2$-edge coloring algorithm from arXiv:1711.05469 .