Tight Competitive Ratios of Classic Matching Algorithms in the Fully Online Model

Abstract

Huang et al. (STOC 2018) introduced the fully online matching problem, a generalization of the classic online bipartite matching problem in that it allows all vertices to arrive online and considers general graphs. They showed that the ranking algorithm by Karp et al. (STOC 1990) is strictly better than 0.5-competitive and the problem is strictly harder than the online bipartite matching problem in that no algorithms can be (1 − 1/e)-competitive.This paper pins down two tight competitive ratios of classic algorithms for the fully online matching problem. For the fractional version of the problem, we show that a natural instantiation of the water-filling algorithm is 2 − √ 2 ≈ 0.585-competitive, together with a matching hardness result. Interestingly, our hardness result applies to arbitrary algorithms in the edgearrival models of the online matching problem, improving the state-of-art 1 1+ln 2 ≈ 0.5906 upper bound. For integral algorithms, we show a tight competitive ratio of ≈ 0.567 for the ranking algorithm on bipartite graphs, matching a hardness result by Huang et al. (STOC 2018).

Online Matching with General Arrivals

“…We note that any α-competitive online matching under general vertex arrivals is α-competitive in the less restrictive model of Huang et al As observed by Huang et al, for their model an optimal approach might as well be greedy; i.e., an unmatched vertex v should always be matched at its departure time if possible. In particular, Huang et al [15,16], showed that the ranking algorithm of Karp et al is optimal in this model, giving a competitive ratio of ≈ 0.567. For general vertex arrivals, however, ranking (and indeed any maximal matching algorithm) is no better than 1 /2 competitive, as is readily shown by a path on three edges with the internal vertices arriving first.…”

“…On the hardness front, the problem is known to be strictly harder than the one-sided vertex arrival model of Karp et al [18], which admits a competitive ratio of 1 − 1 /e ≈ 0.632. In particular, Epstein et al [10] gave an upper bound of 1 1+ln 2 ≈ 0.591 for this problem, recently improved by Huang et al [16] to 2− √ 2 ≈ 0.585. (Both bounds apply even to online algorithms with preemption; i.e., allowing edges to be removed from the matching in favor of a newly-arrived edge.)…”

“…The answer is "yes" for fractional algorithms, as shown by combining our Theorem 1.1 with the 0.526-competitive fractional online matching algorithm under general vertex arrivals of Wang and Wong [25]. For integral online matching, however, the problem has proven challenging, and the only positive results for this problem, too, are for various relaxations, such as restriction to trees, either with or without preemption [3,4,24], for bounded-degree graphs [3], or (recently) allowing vertices to be matched during some known time interval [15,16].…”

“…We elaborate on the last relaxation above. In the model recently studied by Huang et al [15,16] vertices have both arrival and departure times, and edges can be matched whenever both their endpoints are present. (One-sided vertex arrivals is a special case of this model with all online vertices departing immediately after arrival and offline vertices departing at ∞.)…”

“…Online matching and many extensions of this problem under one-sided bipartite vertex arrivals were widely studied over the years, both under adversarial and stochastic arrival models. See recent work [5,15,16,17] and the excellent survey of Mehta [22] for further references on this rich literature.…”