Node and Edge Averaged Complexities of Local Graph Problems

TL;DR

This paper investigates node and edge-averaged complexities of local graph problems, providing new lower bounds, algorithms, and resolving open questions in distributed symmetry breaking, especially for MIS, maximal matching, and sinkless orientation.

Contribution

It introduces novel lower bounds for node-averaged complexity, adapts known bounds to new contexts, and offers improved algorithms and complexity results for several fundamental graph problems.

Findings

Lower bounds for MIS node-averaged complexity in graphs and trees.

Edge-averaged complexity of maximal matching is constant, despite high node complexity.

Deterministic node-averaged complexity for sinkless orientation is logarithmic, with worst-case remaining logarithmic.

Abstract

The node-averaged complexity of a distributed algorithm running on a graph $G=(V,E)$ is the average over the times at which the nodes $V$ of $G$ finish their computation and commit to their outputs. We study the node-averaged complexity for some distributed symmetry breaking problems and provide the following results (among others): - The randomized node-averaged complexity of computing a maximal independent set (MIS) in $n$-node graphs of maximum degree $\Delta$ is at least $\Omega\big(\min\big\{\frac{\log\Delta}{\log\log\Delta},\sqrt{\frac{\log n}{\log\log n}}\big\}\big)$. This bound is obtained by a novel adaptation of the well-known KMW lower bound [JACM'16]. As a side result, we obtain the same lower bound for the worst-case randomized round complexity for computing an MIS in trees -- this essentially answers open problem 11.15 in the book of Barenboim and Elkin and resolves the complexity of MIS on trees up to an $O(\sqrt{\log\log n})$ factor. We also show that, $(2,2)$-ruling sets, which are a minimal relaxation of MIS, have $O(1)$ randomized node-averaged complexity. - For maximal matching, we show that while the randomized node-averaged complexity is $\Omega\big(\min\big\{\frac{\log\Delta}{\log\log\Delta},\sqrt{\frac{\log n}{\log\log n}}\big\}\big)$, the randomized edge-averaged complexity is $O(1)$. Further, we show that the deterministic edge-averaged complexity of maximal matching is $O(\log^2\Delta + \log^* n)$ and the deterministic node-averaged complexity of maximal matching is $O(\log^3\Delta + \log^* n)$. - Finally, we consider the problem of computing a sinkless orientation of a graph. The deterministic worst-case complexity of the problem is known to be $\Theta(\log n)$, even on bounded-degree graphs. We show that the problem can be solved deterministically with node-averaged complexity $O(\log^* n)$, while keeping the worst-case complexity in $O(\log n)$.