Online Algorithms for Maximum Cardinality Matching with Edge Arrivals

In the adversarial edge arrival model for maximum cardinality matching, edges of an unknown graph are revealed one-by-one in an arbitrary order, and should be irrevocably accepted or rejected. Here, the goal of an online algorithm is to maximize the number of accepted edges while maintaining a feasible matching at any point in time. For this model, the standard greedy heuristic is $$\nicefrac {1}{2}$$12-competitive, and on the other hand, no algorithm that outperforms this ratio is currently known, even for very simple graphs. We present a clean Min-Index framework for devising a family of randomized algorithms, and provide a number of positive and negative results in this context. Among these results, we present a $$\nicefrac {5}{9}$$59-competitive algorithm when the underlying graph is a forest, and prove that this ratio is best possible within the Min-Index framework. In addition, we prove a new general upper bound of $$\frac{2}{3+1/\phi ^2}\approx 0.5914$$23+1/ϕ2≈0.5914 on the competitiveness of any algorithm in the edge arrival model. Interestingly, while this result slightly falls short of the currently best $$\frac{1}{1+\ln 2} \approx 0.5906$$11+ln2≈0.5906 bound by Epstein et al. (Inf Comput 259(1):31–40, 2018), it holds even for an easier model in which vertices along with their adjacent edges arrive online. As a result, we improve on the currently best upper bound of 0.6252 for the latter model, due to Wang and Wong (in: Proceedings of the 42nd ICALP, 2015).