Algorithms as Mechanisms: The Price of Anarchy of Relax-and-Round

The seminal result of Borodin and Lucier [SODA 2010] shows that combining a greedy algorithm that is an α-approximation algorithm with a pay-your-bid payment rule yields a mechanism whose Price of Anarchy is O(α). In this paper we significantly extend the class of algorithms for which such a result is available by showing that this close connection between approximation ratio on the one hand and Price of Anarchy on the other also holds for the design principle of relaxation and rounding provided that the relaxation is smooth and the rounding is oblivious.

We demonstrate the far-reaching consequences of our result by showing its implications for sparse packing integer programs, such as multi-unit auctions and generalized matching, for the maximum traveling salesman problem, for combinatorial auctions, and for single source unsplittable flow problems. In all these problems our approach leads to novel simple, near-optimal mechanisms whose Price of Anarchy either matches or beats the performance guarantees of known mechanisms.