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 highlights, we, in particular, obtain the following results. — We obtain a deterministic \(O(\log^{2}\Delta\cdot\log n)\) -round algorithm for computing an MIS in the \(\mathsf{LOCAL}\) model and an almost as efficient \(O(\log^{2}\Delta\cdot\log\log\Delta\cdot\log n)\) -round deterministic MIS algorithm in the \(\mathsf{CONGEST}\) 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. — We obtain efficient deterministic distributed algorithms for rounding fractional solutions for maximum (weighted) independent set and minimum (weighted) set cover. In particular, for any constant \(\varepsilon > 0\) , we give a deterministic \(O(\log^{2}\Delta+\log^{*}n)\) -round algorithm for computing an independent set of size \((1/2-\varepsilon)\,{\cdot}\,n/\deg_{\mathrm{avg}}\) , and we give deterministic \(O(\log^{2}(\Delta W)+\log^{*}n)\) -round algorithms for computing a \((1-\varepsilon)/\Delta\) -approximation of maximum weight independent set, and for computing a \((1-\varepsilon)/r\) -approximation of maximum weight matching in hypergraphs of rank \( r \) . For minimum set cover instances with sets of size at most \( s \) and where each element is contained in at most \( t \) sets, we show that an \(O(\log s)\) -approximation can be computed in time \(O(\log s\cdot\log^{2}t+\log^{*}n)\) .