Online Weighted Bipartite Matching with a Sample

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 competitive-ratio. 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 $1/e$.