Online Algorithms for Maximum Cardinality Matching with Edge Arrivals

Online Matching with General Arrivals

“…(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.) On the positive side, as pointed out by Buchbinder et al [3], the edge arrival model has proven challenging, and results beating the 1 /2 competitive ratio were only achieved under various relaxations, including: random order edge arrival [14], bounded number of arrival batches [20], on trees, either with or without preemption [3,24], and for bounded-degree graphs [3]. The above papers all asked whether there exists a randomized ( 1 /2 + Ω(1))-competitive algorithm for adversarial edge arrivals (see also Open Question 17 in Mehta's survey [22]).…”

“…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].…”

“…As discussed above, for this one-sided arrival model, the problem is thoroughly understood (even down to lower-order error terms [11]). Wang and Wong [25] proved that general vertex arrivals are strictly harder than one-sided bipartite arrivals, providing an upper bound of 0.625 < 1 − 1 /e for the more general problem, later improved by Buchbinder et al [3] to 2 3+φ 2 ≈ 0.593. Clearly, the general vertex arrival model is no harder than the online edge arrival model but is it easier ?…”

“…Edge Arrivals. All prior upper bounds in the online literature [3,10,11,16,18] can be rephrased as upper bounds for fractional algorithms; i.e., algorithms which immediately and irrevocably assign each edge e a value x e on arrival, so that x is contained in the fractional matching polytope,…”

“…With the exception of [3], the core difficulty of these hard instances is uncertainty about "identity" of vertices (in particular, which vertices will neighbor which vertices in the following arrivals). Our hardness instances rely on uncertainty about the "time horizon".…”