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

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(α· ∈^-2) for graphs with arboricity1 bounded by α. 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(α· ∈^-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.