Budget Feasible Mechanism Design via Random Sampling

Budget feasible mechanism considers algorithmic mechanism design questions where there is a budget constraint on the total payment of the mechanism. An important question in the field is that under which valuation domains there exist budget feasible mechanisms that admit ‘small’ approximations (compared to a socially optimal solution). Singer [20] showed that additive and submodular functions admit a constant approximation mechanism. Recently, Dobzinski, Papadimitriou, and Singer [10] gave an O(log n) approximation mechanism for subadditive functions and remarked that: “A fundamental question is whether, regardless of computational constraints, a constant-factor budget feasible mechanism exists for subadditive function.” In this paper, we give the first attempt to this question. We give a polynomial time O( logn log logn ) sub-logarithmic approximation ratio mechanism for subadditive functions, improving the best known ratio O(log n). Further, we connect budget feasible mechanism design to the concept of approximate core in cooperative game theory, and show that there is a mechanism for subadditive functions whose approximation is, via a characterization of the integrality gap of a linear program, linear to the largest value to which an approximate core exists. Our result implies in particular that the class of XOS functions, which is a superclass of submodular functions, admits a constant approximation mechanism. We believe that our work could be a solid step towards solving the above fundamental problem eventually, and possibly, with an affirmative answer. Tsinghua University, China. Email: [email protected]. Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore. Email: [email protected], [email protected]. Microsoft Research Asia. Email: [email protected].