Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Thursday, April 29, 2021, 12:15 pm

Duration: 30 minutes

Location: Zoom: conference room

Speaker: Sebastian Brandt

Improved Distributed Lower Bounds for MIS and Bounded Degree Dominating Sets on Trees

Given a graph G = (V,E) and a parameter k, a k-degree dominating set is a subset S of V such that S is a dominating set of G and in the induced subgraph G[S] every node has degree at most k. For k = 0, this definition coincides with the definition of a maximal independent set (MIS), while for k > 0, the problem of finding a k-degree dominating set is a natural relaxation of the MIS problem. Due to work by Barenboim and Elkin [PODC'08] and Ghaffari [SODA'16], it is known that, on trees, an MIS, and therefore also a k-degree dominating set for any k, can be computed in O(log n / log log n) rounds deterministically, and in O(sqrt(log n)) rounds randomized, in the distributed LOCAL model.

In this talk, we present Omega(sqrt(log n))-round lower bounds for the deterministic, and Omega(sqrt(log log n))-round lower bounds for the randomized computation of k-degree dominating sets on trees, both in the LOCAL model, improving, for k > 0, on the best previously known lower bounds of Omega(log*n)) due to Linial [FOCS'87] and Naor [J.Disc.Math.'91]. Our results also slightly improve on the best previously known lower bounds for MIS on trees due to Balliu, Brandt, and Olivetti [FOCS'20], while providing a simpler proof.

This is joint work with Alkida Balliu, Fabian Kuhn, and Dennis Olivetti.

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