  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.
