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

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 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,√(loglog 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].

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