Near optimal algorithms for online weighted bipartite matching in adversary model

Abstract

Bipartite matching is an important problem in graph theory. With the prosperity of electronic commerce, such as online auction and AdWords allocation, bipartite matching problem has been extensively studied under online circumstances. In this work, we study the online weighted bipartite matching problem in adversary model, that is, there is a weighted bipartite graph \(G=(L,R,E)\) and the left side L is known as input, while the vertices in R come one by one in an order arranged by the adversary. When each vertex in R comes, its adjacent edges and relative weights are revealed. The algorithm should irreversibly decide whether to match this vertex to an unmatched neighbor in L with the objective to maximize the total weight of the obtained matching. When the weights are unbounded, the best algorithm can only achieve a competitive ratio \(\varTheta \left( \frac{1}{n}\right) \), where n is the number of vertices coming online. Thus, we mainly deal with two variants: the bounded weight problem in which all weights are in the range \([\alpha , \beta ]\), and the normalized summation problem in which each vertex in one side has the same total weights. We design algorithms for both variants with competitive ratio \(\varTheta \left( \max \left\{ \frac{1}{\log \frac{\beta }{\alpha }},\frac{1}{n}\right\} \right) \) and \(\varTheta \left( \frac{1}{\log n}\right) \) respectively. Furthermore, we show these two competitive ratios are tight by providing the corresponding hardness results.