Distributed symmetry breaking on power graphs via sparsification

Abstract 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 $$G^k$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>G</mml:mi> <mml:mi>k</mml:mi> </mml:msup> </mml:math> , where each node of the communication network G is connected to all of its k -hop neighbors. A $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> -ruling set is a set of non-adjacent nodes such that each node in G has a ruling neighbor within $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> 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 $$G^k$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>G</mml:mi> <mml:mi>k</mml:mi> </mml:msup> </mml:math> in essentially the same time as they can be computed in G . Our main contribution is a deterministic $${{\,\textrm{poly}\,}}(k,\log n)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mspace/> <mml:mtext>poly</mml:mtext> <mml:mspace/> </mml:mrow> <mml:mo>(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>,</mml:mo> <mml:mo>log</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> time algorithm for computing k -ruling sets of $$G^k$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>G</mml:mi> <mml:mi>k</mml:mi> </mml:msup> </mml:math> , which (for k &gt; 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].