Lower Bounds for Maximal Matchings and Maximal Independent Sets

Abstract

There are distributed graph algorithms for finding maximal matchings and maximal independent sets in O(∆ + log * n) communication rounds; here n is the number of nodes and ∆ is the maximum degree. The lower bound by Linial (1987Linial ( , 1992 shows that the dependency on n is optimal: these problems cannot be solved in o(log * n) rounds even if ∆ = 2.However, the dependency on ∆ is a long-standing open question, and there is currently an exponential gap between the upper and lower bounds.We prove that the upper bounds are tight. We show that maximal matchings and maximal independent sets cannot be found in o(∆ + log log n/ log log log n) rounds with any randomized algorithm in the LOCAL model of distributed computing.As a corollary, it follows that there is no deterministic algorithm for maximal matchings or maximal independent sets that runs in o(∆ + log n/ log log n) rounds; this is an improvement over prior lower bounds also as a function of n.

A Topological Perspective on Distributed Network Algorithms

“…However, all these achievements were based on an operational approach, using sophisticated algorithmic techniques and tools solely from graph theory. Similarly, the existing lower bounds on the round-complexity of tasks in the LOCAL model [32,35,8,23,3] were obtained using graph theoretical arguments only. The question of whether adopting a higher dimensional approach by using topology would help in the context of local computing, be it for a better conceptual understanding of the algorithms, or providing stronger technical tools for lower bounds, is, to our knowledge, entirely open.…”

Efficient Load-Balancing through Distributed Token Dropping

“…• Maximal matching on bipartite graphs can be solved in (Δ) rounds [17], but not in (Δ) rounds [4]. • Maximal fractional matching can be solved in (Δ) rounds [3], but not in (Δ) rounds [14].…”

“…By prior work, we do not have any nontrivial lower bounds for stable orientations, and the best upper bound is (Δ 5 ). The recent advances in the techniques for proving lower bounds [4][5][6]22] suggest that now would be a good time to revisit the stable orientation problem and see how far we can get in closing the gap between upper and lower bounds. In this work we take the first steps in this direction, by improving the upper bound to (Δ 4 ) and by proving a lower bound of Ω(Δ).…”

“…The proof of the theorem is a simple reduction from the bipartite maximal matching problem to the token dropping game. In a recent work, [4] showed that the bipartite maximal matching problem cannot be solved in (Δ + log /log log ) rounds in the LOCAL model of distributed computing. Consider a bipartite graph = ( ∪ , ) that is an input instance to the maximal matching problem.…”

Network Decomposition and Distributed Derandomization (Invited Paper)

“…Most notably, the gap for the coloring problem is much wider. While for MIS and maximal matching, deterministic algorithms cannot go much further below an O(log n) round complexity, thanks to a recent Ω(log n/ log log n) lower bound of Balliu et al [5], for the (Δ + 1) coloring, there is no such obstacle known and (significantly) sublogarithmic complexities are plausible. The best known lower bound for the round complexity of (Δ + 1) coloring remains at Ω(log * n) [36,37], even when restricted to deterministic algorithms.…”