Online Knapsack Problem and Budgeted Truthful Bipartite Matching

TL;DR

This paper introduces online algorithms for knapsack and truthful bipartite matching problems under the secretary model, achieving competitive ratios of 2e and 24, respectively, with applications in resource allocation and network communication.

Contribution

It presents nearly optimal online algorithms for two related problems, ensuring truthfulness in bipartite matching and broad applicability in various resource allocation scenarios.

Findings

Achieved a 2e-competitive algorithm for online knapsack.

Developed a 24-competitive truthful bipartite matching algorithm.

Demonstrated practical applications in ad allocation, crowdsourcing, and wireless networks.

Abstract

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.