We extend the standard online worst-case model to accommodate past experience which is available to the online player in many practical scenarios. We do this by revealing a random sample of the adversarial input to the online player ahead of time. The online player competes with the expected optimal value on the part of the input that arrives online. Our model bridges between existing online stochastic models (e.g., items are drawn i.i.d. from a distribution) and the online worst-case model. We also extend in a similar manner (by revealing a sample) the online random-order model.We study the classical secretary problem in our new models. In the worst-case model we present a simple online algorithm with optimal competitive-ratio for any sample size. In the random-order model, we also give a simple online algorithm with an almost tight competitiveratio for small sample sizes. Interestingly, we prove that for a large enough sample, no algorithm can be simultaneously optimal both in the worst-cast and random-order models.
We study the classical online bipartite matching problem: One side of the graph is known and vertices of the other side arrive online. It is well known that when the graph is edge-weighted, and vertices arrive in an adversarial order, no online algorithm has a nontrivial competitiveratio. To bypass this hurdle we modify the rules such that the adversary still picks the graph but has to reveal a random part (say half) of it to the player. The remaining part is given to the player in an adversarial order. This models practical scenarios in which the online algorithm has some history to learn from.This way of modeling a history was formalized recently by the authors (SODA 20) and was called the AOS model (for Adversarial Online with a Sample). It allows developing online algorithms for the secretary problem that compete even when the secretaries arrive in an adversarial order. Here we use the same model to attack the much more challenging matching problem.We analyze a natural algorithmic framework that decides how to match an arriving vertex v by applying an offline matching algorithm to v and the sample. We get roughly 1/4 of the maximum weight by applying the offline greedy matching algorithm to the sample and v. Our analysis ties the performance of this algorithm to the performance of the offline greedy matching on the online part and we also prove that it is tight. Surprisingly, when replacing greedy with an optimal algorithm for maximum matching, no constant competitive-ratio can be guaranteed when the size of the sample is comparable to the size of the online part. However, when the sample is quadratic in the size of the online part, we do get a competitive-ratio of 1/e.
We study the online facility location problem with uniform facility costs in the randomorder model. Meyerson's algorithm [FOCS'01] is arguably the most natural and simple online algorithm for the problem with several advantages and appealing properties. Its analysis in the random-order model is one of the cornerstones of random-order analysis beyond the secretary problem. Meyerson's algorithm was shown to be (asymptotically) optimal in the standard worstcase adversarial-order model and 8-competitive in the random order model. While this bound in the random-order model is the long-standing state-of-the-art, it is not known to be tight, and the true competitive-ratio of Meyerson's algorithm remained an open question for more than two decades.We resolve this question and prove tight bounds on the competitive-ratio of Meyerson's algorithm in the random-order model, showing that it is exactly 4-competitive. Following our tight analysis, we introduce a generic parameterized version of Meyerson's algorithm that retains all the advantages of the original version. We show that the best algorithm in this family is exactly 3-competitive. On the other hand, we show that no online algorithm for this problem can achieve a competitive-ratio better than 2. Finally, we prove that the algorithms in this family are robust to partial adversarial arrival orders.
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