  We present a framework for deterministically rounding a dynamic fractional
matching. Applying our framework in a black-box manner on top of existing
fractional matching algorithms, we derive the following new results: (1) The
first deterministic algorithm for maintaining a $(2-\delta)$-approximate
maximum matching in a fully dynamic bipartite graph, in arbitrarily small
polynomial update time. (2) The first deterministic algorithm for maintaining a
$(1+\delta)$-approximate maximum matching in a decremental bipartite graph, in
polylogarithmic update time. (3) The first deterministic algorithm for
maintaining a $(2+\delta)$-approximate maximum matching in a fully dynamic
general graph, in small polylogarithmic (specifically, $O(\log^4 n)$) update
time. These results are respectively obtained by applying our framework on top
of the fractional matching algorithms of Bhattacharya et al. [STOC'16],
Bernstein et al. [FOCS'20], and Bhattacharya and Kulkarni [SODA'19].
  Prior to our work, there were two known general-purpose rounding schemes for
dynamic fractional matchings. Both these schemes, by Arar et al. [ICALP'18] and
Wajc [STOC'20], were randomized.
  Our rounding scheme works by maintaining a good {\em matching-sparsifier}
with bounded arboricity, and then applying the algorithm of Peleg and Solomon
[SODA'16] to maintain a near-optimal matching in this low arboricity graph. To
the best of our knowledge, this is the first dynamic matching algorithm that
works on general graphs by using an algorithm for low-arboricity graphs as a
black-box subroutine. This feature of our rounding scheme might be of
independent interest.
