• DOI:10.1145/3465084.3467901
• Corpus ID: 235352573

Improved Distributed Lower Bounds for MIS and Bounded (Out-)Degree Dominating Sets in Trees

@article{Balliu2021ImprovedDL,
  title={Improved Distributed Lower Bounds for MIS and Bounded (Out-)Degree Dominating Sets in Trees},
  author={Alkida Balliu and Sebastian Brandt and Fabian Kuhn and Dennis Olivetti},
  journal={Proceedings of the 2021 ACM Symposium on Principles of Distributed Computing},
  year={2021}
}

• A. Balliu, S. Brandt, +1 author D. Olivetti
• Published 4 June 2021
• Computer Science, Mathematics
• Proceedings of the 2021 ACM Symposium on Principles of Distributed Computing

Recently, Balliu, Brandt, and Olivetti [FOCS '20] showed the first ω(log n) lower bound for the maximal independent set (MIS) problem in trees. In this work we prove lower bounds for a much more relaxed family of distributed symmetry breaking problems. As a by-product, we obtain improved lower bounds for the distributed MIS problem in trees. For a parameter k and an orientation of the edges of a graph G, we say that a subset S of the nodes of G is a k-outdegree dominating set if S is a dominating set of G and if in the induced subgraph G[S], every node in S has outdegree at most k. Note that for k=0, this definition coincides with the definition of an MIS. For a given k, we consider the problem of computing a k-outdegree dominating set. We show that, even in regular trees of degree at most Δ, in the standard ŁOCAL model, there exists a constant ε>0 such that for k ≤Δ^ε, for the problem of computing a k-outdegree dominating set, any randomized algorithm requires at least Ω(min logΔ,√log log n ) rounds and any deterministic algorithm requires at least Ω(min logΔ,√log n ) rounds. The proof of our lower bounds is based on the recently highly successful round elimination technique. We provide a novel way to do simplifications for round elimination, which we expect to be of independent interest. Our new proof is considerably simpler than the lower bound proof in [FOCS '20]. In particular, our round elimination proof uses a family of problems that can be described by only a constant number of labels. The existence of such a proof for the MIS problem was believed impossible by the authors of [FOCS '20].