[Paper Review] The Locality of Distributed Symmetry Breaking

Leonid Barenboim, Michael Elkin+2 more|arXiv (Cornell University)|Feb 9, 2012

Complexity and Algorithms in GraphsComputer Science39 references86 citations

TL;DR

This paper introduces a novel technique to overcome the union bound barrier in distributed symmetry breaking by reducing large-scale randomized problems to small, polylogarithmic-sized components. It achieves the first significant improvements in randomized complexity for maximal independent sets (MIS), maximal matchings, and vertex coloring, with key results including an MIS algorithm running in $O(\log^2\Delta + 2^{O(\sqrt{\log\log n})})$ time and a provably optimal maximal matching algorithm in $O(\log\Delta + \log^4\log n)$ time.

ABSTRACT

Symmetry breaking problems are among the most well studied in the field of distributed computing and yet the most fundamental questions about their complexity remain open. In this paper we work in the LOCAL model (where the input graph and underlying distributed network are identical) and study the randomized complexity of four fundamental symmetry breaking problems on graphs: computing MISs (maximal independent sets), maximal matchings, vertex colorings, and ruling sets. A small sample of our results includes - An MIS algorithm running in $O(\log^2Δ+ 2^{O(\sqrt{\log\log n})})$ time, where $Δ$ is the maximum degree. This is the first MIS algorithm to improve on the 1986 algorithms of Luby and Alon, Babai, and Itai, when $\log n \ll Δ\ll 2^{\sqrt{\log n}}$, and comes close to the $Ω(\log Δ)$ lower bound of Kuhn, Moscibroda, and Wattenhofer. - A maximal matching algorithm running in $O(\logΔ+ \log^4\log n)$ time. This is the first significant improvement to the 1986 algorithm of Israeli and Itai. Moreover, its dependence on $Δ$ is provably optimal. - A method for reducing symmetry breaking problems in low arboricity/degeneracy graphs to low degree graphs. (Roughly speaking, the arboricity or degeneracy of a graph bounds the density of any subgraph.) Corollaries of this reduction include an $O(\sqrt{\log n})$-time maximal matching algorithm for graphs with arboricity up to $2^{\sqrt{\log n}}$ and an $O(\log^{2/3} n)$-time MIS algorithm for graphs with arboricity up to $2^{(\log n)^{1/3}}$. Each of our algorithms is based on a simple, but powerful technique for reducing a randomized symmetry breaking task to a corresponding deterministic one on a poly$(\log n)$-size graph.

Motivation & Objective

• To overcome the fundamental union bound barrier that limits the effectiveness of randomized algorithms in distributed symmetry breaking.
• To develop faster randomized algorithms for core distributed problems: MIS, maximal matching, ruling sets, and vertex coloring.
• To achieve provably optimal or near-optimal time complexities in the LOCAL model, especially for graphs with bounded degree or low arboricity.
• To establish a general reduction framework that transforms large-scale symmetry breaking problems into small, manageable subproblems.
• To clarify the relationship between randomized and deterministic complexity in distributed symmetry breaking by identifying inherent limitations.

Proposed method

• Reducing the original $n$-node symmetry breaking problem to a set of disjoint ${\mathrm{poly}}(\log n)$-size subproblems, where randomized strategies can succeed with high probability.
• Using a scale-based decomposition of the graph into low-degree components, followed by iterative application of a tree-based independent set algorithm.
• Applying a modified version of the $\mathsf{TreeIndependentSet}$ and $\mathsf{TreeMIS}$ algorithms with adjusted thresholds and iteration counts to handle high-girth graphs.
• Introducing a new invariant that bounds the number of high-degree neighbors in each component, ensuring bounded failure probability.
• Leveraging the fact that subgraphs of maximum degree $O(\log n)$ can be solved in $2^{O(\sqrt{\log\log n})}$ time using known techniques.
• Reducing symmetry breaking in low-arboricity graphs to low-degree graphs via structural decomposition, enabling faster algorithms on sparse graphs.

Experimental results

Research questions

• RQ1 Can we break the $\Omega(\log \Delta)$ lower bound for MIS in the LOCAL model using randomized algorithms?
• RQ2 Is it possible to achieve a provably optimal dependence on $\Delta$ for maximal matching in the randomized LOCAL model?
• RQ3 Can we reduce the complexity of $(\Delta+1)$-coloring below $O(\log \Delta + \sqrt{\log n})$?
• RQ4 Does the union bound barrier fundamentally limit randomized symmetry breaking, and if so, can it be overcome?
• RQ5 Can we achieve $o(\log n)$-time algorithms for MIS and maximal matching in general graphs?

Key findings

• An MIS algorithm runs in $O(\log^2\Delta + 2^{O(\sqrt{\log\log n})})$ time, improving on the 1986 Luby–Alon–Babai–Itai algorithms when $\log n \ll \Delta \ll 2^{\sqrt{\log n}}$.
• A maximal matching algorithm runs in $O(\log\Delta + \log^4\log n)$ time, representing the first significant improvement since the 1986 Israeli–Itai algorithm and achieving optimal dependence on $\Delta$.
• A $(\Delta+1)$-coloring algorithm runs in $O(\log\Delta + 2^{O(\sqrt{\log\log n})})$ time, improving on the prior $O(\log\Delta + \sqrt{\log n})$ bound.
• For graphs of arboricity up to $2^{\sqrt{\log n}}$, a maximal matching algorithm runs in $O(\sqrt{\log n})$ time via the reduction to low-degree graphs.
• For graphs of arboricity up to $2^{(\log n)^{1/3}}$, an MIS algorithm runs in $O(\log^{2/3}n)$ time using the same reduction technique.
• The paper identifies the union bound as a fundamental barrier and conjectures that randomized complexity is tied to deterministic complexity, implying that $2^{O(\sqrt{\log\log n})}$ terms may be unavoidable without deterministic improvements.