We present two deterministic dynamic algorithms for the maximum matching problem. (1) An algorithm that maintains a (2 + )-approximate maximum matching in general graphs with O(poly(log n, 1/ )) update time. (2) An algorithm that maintains an αK approximation of the value of the maximum matching with O(n 2/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(log n)-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 randomized polylogarithmic update time algorithm [Baswana et al. FOCS 2011]. Result (2) achieves a better-than-two approximation with arbitrarily small polynomial update time on bipartite graphs. Previously the best update time for this problem was O(m 1/4 ) [Bernstein et al. ICALP 2015], where m is the current number of edges in the graph.
Some years later a deterministic (2 + ǫ)-approximate matching algorithm with amortized poly(log n, 1/ǫ) update time algorithm was presented by Bhattacharya et al. REF .
New deterministic approximation algorithms for fully dynamic matching
STOC 2016
Mathematics Computer Science