Distributed symmetry breaking on power graphs via sparsification

• Published in: Distributed computing Vol. 38; no. 3; pp. 261 - 296
• Main Authors: Maus, Yannic, Peltonen, Saku, Uitto, Jara
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.09.2025; Springer Nature B.V
• Subjects: Algorithms; Broken symmetry; Communication; Communications networks; Computation; Computer Communication Networks; Computer Hardware; Computer Science; Computer Systems Organization and Communication Networks; Distributed processing; Graphs; Message passing; Run time (computers); Software Engineering/Programming and Operating Systems; Symmetry; Theory of Computation; Shattering; Maximal independent set; Power graphs; CONGEST model; Ruling sets; Distributed algorithm
• ISSN: 0178-2770; 1432-0452
• DOI: 10.1007/s00446-025-00485-9

In this paper we present efficient distributed algorithms for classical symmetry breaking problems, maximal independent sets (MIS) and ruling sets, in power graphs. We work in the standard CONGEST model of distributed message passing, where the communication network is abstracted as a graph G . Typically, the problem instance in CONGEST is identical to the communication network G , that is, we perform the symmetry breaking in G . In this work, we consider a setting where the problem instance corresponds to a power graph G k , where each node of the communication network G is connected to all of its k -hop neighbors. A β -ruling set is a set of non-adjacent nodes such that each node in G has a ruling neighbor within β hops; a natural generalization of an MIS. On top of being a natural family of problems, ruling sets (in power graphs) are well-motivated through their applications in the powerful shattering framework [BEPS JACM’16, Ghaffari SODA’19] (and others). We present randomized algorithms for computing maximal independent sets and ruling sets of G k in essentially the same time as they can be computed in G . Our main contribution is a deterministic poly ( k , log n ) time algorithm for computing k -ruling sets of G k , which (for k > 1) improves exponentially on the current state-of-the-art runtimes. Our main technical ingredient for this result is a deterministic sparsification procedure which may be of independent interest. We also revisit the shattering algorithm for MIS [BEPS JACM’16] and present different approaches for the post-shattering phase. Our solutions are algorithmically and analytically simpler (also in the LOCAL model) than existing solutions and obtain the same runtime as [Ghaffari SODA’16].