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Given a graph, a maximal independent set (MIS) is a maximal subset ofpairwise non-adjacent vertices. Finding an MIS is a fundamental problem indistributed computing. Although the problem is extensively studied and wellunderstood in simple graphs, our knowledge is still quite limited when solvingit in hypergraphs, especially in the distributed CONGEST model. In this paper,we focus on linear hypergraphs---a class of hypergraphs in which any twohyperedges overlap on at most one node.  We first present a randomized algorithm for computing an MIS in linearhypergraphs. 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 aparallel 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 aGMIS, for each hyperedge $e$ in a hypergraph $\mathcal{H}$, we associate aninteger threshold $t_e$ in the range $[1, |e|-1]$, and the goal is to find amaximal subset $\mathcal{I}$ of vertices that do not violate any thresholdconstraints: $\forall e \in E(\mathcal{H}), |e \cap \mathcal{I}| \leq t_e$. Wehope that GMIS might capture a broader class of real-world problems than MIS;we also believe that GMIS is an interesting and challenging symmetry breakingproblem on its own.  Our second upper bound result is a distributed algorithm for computing a GMISin linear hypergraphs, subject to the constraint that the maximum hyperedgesize is bounded by some constant. Again, the algorithm has poly-logarithmicruntime and it works in the CONGEST model. It is obtained by generalizing ourprevious (linear) hypergraph MIS algorithm.

Efficient Distributed Computation of MIS and Generalized MIS in Linear Hypergraphs