Efficient Distributed Computation of MIS and Generalized MIS in Linear Hypergraphs

Given a graph, a maximal independent set (MIS) is a maximal subset of pairwise non-adjacent vertices. Finding an MIS is a fundamental problem in distributed computing. Although the problem is extensively studied and well understood in simple graphs, our knowledge is still quite limited when solving it in hypergraphs, especially in the distributed CONGEST model. In this paper, we focus on linear hypergraphs---a class of hypergraphs in which any two hyperedges overlap on at most one node. We first present a randomized algorithm for computing an MIS in linear hypergraphs. It has poly-logarithmic runtime and it works in the CONGEST model. The algorithm uses a network decomposition to achieve fast parallel processing. Within each cluster of the decomposition, we run a distributed variant of a parallel hypergraph MIS algorithm by Luczak and Szymanska. We then propose the concept of a generalized maximal independent set (GMIS) as an extension to the classical MIS in hypergraphs. More specifically, in a GMIS, for each hyperedge $e$ in a hypergraph $\mathcal{H}$, we associate an integer threshold $t_e$ in the range $[1, |e|-1]$, and the goal is to find a maximal subset $\mathcal{I}$ of vertices that do not violate any threshold constraints: $\forall e \in E(\mathcal{H}), |e \cap \mathcal{I}| \leq t_e$. We hope that GMIS might capture a broader class of real-world problems than MIS; we also believe that GMIS is an interesting and challenging symmetry breaking problem on its own. Our second upper bound result is a distributed algorithm for computing a GMIS in linear hypergraphs, subject to the constraint that the maximum hyperedge size is bounded by some constant. Again, the algorithm has poly-logarithmic runtime and it works in the CONGEST model. It is obtained by generalizing our previous (linear) hypergraph MIS algorithm.