Mechanism Design For Covering Problems

Algorithmic mechanism design deals with efficiently-computable algorithmic constructions in the presence of strategic players who hold the inputs to the problem and may misreport their input if doing so benefits them. Algorithmic mechanism design finds applications in a variety of internet settings such as resource allocation, facility location and e-commerce, such as sponsored search auctions. There is an extensive amount of work in algorithmic mechanism design on packing problems such as single-item auctions, multi-unit auctions and combinatorial auctions. But, surprisingly, covering problems, also called procurement auctions, have almost been completely unexplored, especially in the multidimensional setting. In this thesis, we systematically investigate multidimensional covering mechanismdesign problems, wherein there are m items that need to be covered and n players who provide covering objects, with each player i having a private cost for the covering objects he provides. A feasible solution to the covering problem is a collection of covering objects (obtained from the various players) that together cover all items. Two widely considered objectives in mechanism design are: (i) cost-minimization (CM) which aims to minimize the total cost incurred by the players and the mechanism designer; and (ii) payment minimization (PayM), which aims to minimize the payment to players. Covering mechanism design problems turn out to behave quite differently from packing mechanism design problems. In particular, various techniques utilized successfully for packing problems do not perform well for covering mechanism design problems, and this necessitates new approaches and solution concepts. In this thesis we devise various techniques for handling covering mechanism design problems, which yield a variety of results for both the CM and PayM objectives. In our investigation of the CM objective, we focus on two representative covering problems: uncapacitated facility location (UFL) and vertex cover. For multi-dimensional UFL, we give a black-box method to transform any Lagrangian-multiplier-preserving ρapproximation algorithm for UFL into a truthful-in-expectation, ρ-approximation mechanism. This yields the first result for multi-dimensional UFL, namely a truthful-in-expectation