We introduce a weighted version of the ranking algorithm by (STOC 1990), and prove a competitive ratio of 0.6534 for the vertex-weighted online bipartite matching problem when online vertices arrive in random order. Our result shows that random arrivals help beating the 1-1 e barrier even in the vertex-weighted case. We build on the randomized primal-dual framework by (SODA 2013) and design a two dimensional gain sharing function, which depends not only on the rank of the offline vertex, but also on the arrival time of the online vertex. To our knowledge, this is the first competitive ratio strictly larger than 1-1 e for an online bipartite matching problem achieved under the randomized primal-dual framework. Our algorithm has a natural interpretation that offline vertices offer a larger portion of their weights to the online vertices as time goes by, and each online vertex matches the neighbor with the highest offer at its arrival.
@cite_9: We study online variants of weighted bipartite matching on graphs and hypergraphs. In our model for online matching, the vertices on the right-hand side of a bipartite graph are given in advance and the vertices on the left-hand side arrive online in random order. Whenever a vertex arrives, its adjacent edges with the corresponding weights are revealed and the online algorithm has to decide which of these edges should be included in the matching. The studied matching problems have applications, e.g., in online ad auctions and combinatorial auctions where the right-hand side vertices correspond to items and the left-hand side vertices to bidders.
@cite_1: In the classical secretary problem an employer would like to choose the best candidate among n competing candidates that arrive in a random order. In each iteration, one candidate's rank vis-a-vis previously arrived candidates is revealed and the employer makes an irrevocable decision about her selection. This basic concept of n elements arriving in a random order and irrevocable decisions made by an algorithm have been explored extensively over the years, and used for modeling the behavior of many processes. Our main contribution is a new linear programming technique that we introduce as a tool for obtaining and analyzing algorithms for the secretary problem and its variants. The linear program is formulated using judiciously chosen variables and constraints and we show a one-to-one correspondence between algorithms for the secretary problem and feasible solutions to the linear program. Capturing the set of algorithms as a linear polytope holds the following immediate advantages: Computing the optimal algorithm reduces to solving a linear program. Proving an upper bound on the performance of any algorithm reduces to finding a feasible solution to the dual program. Exploring variants of the problem is as simple as adding new constraints, or manipulating the objective function of the linear program. We demonstrate these ideas by exploring some natural variants of the secretary problem. In particular, using our approach, we design optimal secretary algorithms in which the probability of selecting a candidate at any position is equal. We refer to such algorithms as position independent and these algorithms are motivated by the recent applications of secretary problems to online auctions. We also show a family of linear programs that characterize all algorithms that are allowed to choose J candidates and gain profit from the K best candidates. We believe that a linear programming based approach may be very helpful in the context of other variants of the secretary problem.
@cite_23: We consider the generalizations of two classical problems, online bipartite matching and ski rental, in the field of online algorithms, and establish a novel connection between them.
@cite_9 considered the edge-weighted with random arrivals, and proposed a @math -competitive algorithm. The competitive ratio is tight as it matches the lower bound on the classical secretary problem @cite_1 . Wang and Wong @cite_23 considered a different model of with both sides of vertices arriving online (in an arbitrary order): A vertex can only actively match other vertices at its arrival; if it fails to match at its arrival, it may still get matched passively by other vertices later. They showed a @math -competitive algorithm for a fractional version of the problem.