Online Knapsack Problem and Budgeted Truthful Bipartite Matching

Two related online problems: knapsack and truthful bipartite matching are considered. For these two problems, the common theme is how to `match' an arriving left vertex in an online fashion with any of the available right vertices, if at all, so as to maximize the sum of the value of the matched edges, subject to satisfying a sum-weight constraint on the matched left vertices. Assuming that the left vertices arrive in an uniformly random order (secretary model), two almost similar algorithms are proposed for the two problems, that are $2e$ competitive and $24$ competitive, respectively. The proposed online bipartite matching algorithm is also shown to be truthful: there is no incentive for any left vertex to misreport its bid/weight. Direct applications of these problems include job allocation with load balancing, generalized adwords, crowdsourcing auctions, and matching wireless users to cooperative relays in device-to-device communication enabled cellular network.