Abstract
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 $$\nicefrac {1}{2}$$ -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 $$\nicefrac {5}{9}$$ -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 $$\frac{2}{3+1/\phi ^2}\approx 0.5914$$ on the competitiveness of any algorithm in the edge arrival model. Interestingly, while this result slightly falls short of the currently best $$\frac{1}{1+\ln 2} \approx 0.5906$$ 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).
Highlights
Graph matchings are cornerstone problems in combinatorial optimization, that have extensively been studied by the discrete mathematics, computer science, and operations research communities
In the most fundamental setting, given an undirected graph G = (V, E), our objective is to identify a maximum cardinality matching, namely, a subset of edges M ⊆ E without any vertices in common
Due to the breadth and depth of subsequent research on this one-sided arrival model, it is beyond the scope of this paper to provide a comprehensive literature review
Summary
Graph matchings are cornerstone problems in combinatorial optimization, that have extensively been studied by the discrete mathematics, computer science, and operations research communities. Wang and Wong demonstrated that this model is strictly harder than the one-sided vertex arrival model of Karp et al [14] by proving an upper bound of 0.6252 < 1 − 1/e They presented a fractional matching algorithm with a competitive ratio of 0.526. An even harder setting is the edge arrival model, where edges are revealed one-by-one in arbitrary order, and should be irrevocably accepted or rejected For this model, the simple greedy heuristic, that deterministically adds an arriving edge to the current matching whenever possible, is 1/2-competitive. Guruganesh and Singla [11] studied a different type of relaxation, in which edges arrive according to a uniformly-picked random permutation, rather than in an arbitrary adversarial order Under this assumption, they were able to design a (1/2 + δ)-competitive algorithm, for some absolute constant δ > 0. We obtain a clean framework that directly leads to a randomized online matching algorithm, whose formal statement is given in Algorithm 1
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