Distributed Symmetry Breaking in Hypergraphs

Fundamental local symmetry breaking problems such as Maximal Independent Set (MIS) and coloring have been recognized as important by the community, and studied extensively in (standard) graphs. In particular, fast (i.e., polylogarithmic run time) algorithms are well-established for MIS and $\Delta +1$-coloring in both the LOCAL and CONGEST distributed computing models. On the other hand, comparatively much less is known on the complexity of distributed symmetry breaking in {\em hypergraphs}. In this paper, we study the distributed complexity of symmetry breaking in hypergraphs by presenting distributed algorithms for a variety of fundamental problems under a natural distributed computing model for hypergraphs. We then present a key result of this paper --- an $\tilde{O}(\Delta^{o(1)})$-round hypergraph MIS algorithm in the CONGEST model where $\Delta$ is the maximum degree of the hypergraph. To demonstrate the usefulness of hypergraph MIS, we present applications of our hypergraph algorithm to solving other problems. In particular, in standard graphs, the algorithm yields fast distributed algorithms for the {\em balanced minimal dominating set} problem (left open in Harris et al. [ICALP 2013]) and the {\em minimal connected dominating set problem}. We also present distributed algorithms for coloring, maximal matching, and maximal clique in hypergraphs. Our work shows that while some local symmetry breaking problems such as coloring can be solved in polylogarithmic rounds in both the LOCAL and CONGEST models, for many other hypergraph problems such as MIS, hitting set, and maximal clique, it remains challenging to obtain polylogarithmic time algorithms in the CONGEST model.