Faster Deterministic Distributed MIS and Approximate Matching

We present an (O) over tilde (log(2) n) round deterministic distributed algorithm for the maximal independent set problem. By known reductions, this round complexity extends also to maximal matching, Delta+ 1 vertex coloring, and 2 Delta- 1 edge coloring. These four problems are among the most central problems in distributed graph algorithms and have been studied extensively for the past four decades. This improved round complexity comes closer to the (Omega) over tilde (log n) lower bound of maximal independent set and maximal matching [Balliu et al. FOCS '19]. The previous best known deterministic complexity for all of these problems was Phi (log(3) n). Via the shattering technique, the improvement permeates also to the corresponding randomized complexities, e.g., the new randomized complexity of. + 1 vertex coloring is now (O) over tilde (log(2) log n) rounds. Our approach is a novel combination of the previously known (and seemingly orthogonal) two methods for developing fast deterministic algorithms for these problems, namely global derandomization via network decomposition (see e.g., [Rozhon, Ghaffari STOC'20; Ghaffari, Grunau, Rozhon SODA'21; Ghaffari et al. SODA'23]) and local rounding of fractional solutions (see e.g., [Fischer DISC'17; Harris FOCS'19; Fischer, Ghaffari, Kuhn FOCS'17; Ghaffari, Kuhn FOCS'21; Faour et al. SODA'23]). We consider a relaxation of the classic network decomposition concept, where instead of requiring the clusters in the same block to be non-adjacent, we allow each node to have a small number of neighboring clusters. We also show a deterministic algorithm that computes this relaxed decomposition faster than standard decompositions. We then use this relaxed decomposition to signiffcantly improve the integrality of certain fractional solutions, before handing them to the local rounding procedure that now has to do fewer rounding steps.