Dynamic (1 + ε)-Approximate matchings: A density-sensitive approach

David Peleg, Shay Solomon

Research output: Contribution to conferencePaperpeer-review


Approximate matchings in fully dynamic graphs have been intensively studied in recent years. Gupta and Peng [FOCS'13J presented a deterministic algorithm for maintaining fully dynamic (1 +?)-approximate maximum cardinality matching (MCM) in general graphs with worst-case update time 0(?m ?-2), for any ? > 0, where m denotes the current number of edges in the graph. Despite significant research efforts, this ?m update time barrier remains the state-of-the-art even if amortized time bounds and randomization are allowed or the approximation factor is allowed to increase from 1 + ? to 2-?, and even in basic graph families such as planar graphs. This paper presents a simple deterministic algorithm whose performance depends on the density of the graph. Specifically, we maintain fully dynamic (1 + ?)-approximate MCM with worst-case update time O(? ?-2) for graphs with arboricity1 bounded by ?. The update time bound holds even if the arboricity bound a changes dynamically. Since the arboricity ranges between 1 and ?m, our density-sensitive bound O(? ?-2) naturally generalizes the O(?m ?-2) bound of Gupta and Peng. For the family of bounded arboricity graphs (which includes forests, planar graphs, and graphs excluding a fixed minor), in the regime ? = O(1) our update time reduces to a constant. This should be contrasted with the previous best 2-approximation results for bounded arboricity graphs, which achieve either an O(logn) worst-case bound (Kopelowitz et al., ICALP'14) or an O(?logn) amortized bound (He et al., ISAAC'14), where n stands for the number of vertices in the graph. En route to this result, we provide local algorithms of independent interest for maintaining fully dynamic approximate matching and vertex cover.
Original languageEnglish
Number of pages18
StatePublished - 2016
Event27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016 -
Duration: 1 Jan 20161 Jan 2016


Conference27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016


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