Local Distributed Rounding: Generalized to MIS, Matching, Set Cover, and Beyond.

We develop a general deterministic distributed method for locally rounding fractional solutions of graph problems for which the analysis can be broken down into analyzing pairs of vertices. Roughly speaking, the method can transform fractional/probabilistic label assignments of the vertices into integral/deterministic label assignments for the vertices, while approximately preserving a potential function that is a linear combination of functions, each of which depends on at most two vertices (subject to some conditions usually satisfied in pairwise analyses). The method unifies and significantly generalizes prior work on deterministic local rounding techniques [Ghaffari, Kuhn FOCS'21; Harris FOCS'19; Fischer, Ghaffari, Kuhn FOCS'17; Fischer DISC'17] to obtain polylogarithmic-time deterministic distributed solutions for combinatorial graph problems. Our general rounding result enables us to locally and efficiently derandomize a range of distributed algorithms for local graph problems, including maximal independent set (MIS), maximum-weight independent set approximation, and minimum-cost set cover approximation. As a highlight, we in particular obtain a deterministic $O(\log^2\Delta\cdot\log n)$-round algorithm for computing an MIS in the LOCAL model and an almost as efficient $O(\log^2\Delta\cdot\log\log\Delta\cdot\log n)$-round deterministic MIS algorithm in the CONGEST model. As a result, the best known deterministic distributed time complexity of the four most widely studied distributed symmetry breaking problems (MIS, maximal matching, $(\Delta+1)$-vertex coloring, and $(2\Delta-1)$-edge coloring) is now $O(\log^2\Delta\cdot\log n)$. Our new MIS algorithm is also the first direct polylogarithmic-time deterministic distributed MIS algorithm, which is not based on network decomposition.