We present two deterministic dynamic algorithms for the maximum matching
problem. (1) An algorithm that maintains a (2+ϵ)-approximate maximum
matching in general graphs with O(poly(logn,1/ϵ)) update
time. (2) An algorithm that maintains an αK approximation of the {\em
value} of the maximum matching with O(n2/K) update time in bipartite
graphs, for every sufficiently large constant positive integer K. Here,
1≤αK<2 is a constant determined by the value of K. Result (1)
is the first deterministic algorithm that can maintain an o(logn)-approximate maximum matching with polylogarithmic update time, improving
the seminal result of Onak et al. [STOC 2010]. Its approximation guarantee
almost matches the guarantee of the best {\em randomized} polylogarithmic
update time algorithm [Baswana et al. FOCS 2011]. Result (2) achieves a
better-than-two approximation with {\em arbitrarily small polynomial} update
time on bipartite graphs. Previously the best update time for this problem was
O(m1/4) [Bernstein et al. ICALP 2015], where m is the current number of
edges in the graph.Comment: To appear in STOC 201