On the Approximability of Budget Feasible Mechanisms

Budget feasible mechanisms, recently initiated by Singer (FOCS 2010), extend algorithmic mechanism design problems to a realistic setting with a budget constraint. We consider the problem of designing truthful budget feasible mechanisms for general submodular functions: we give a randomized mechanism with approximation ratio $7.91$ (improving the previous best-known result 112), and a deterministic mechanism with approximation ratio $8.34$. Further we study the knapsack problem, which is special submodular function, give a $2+\sqrt{2}$ approximation deterministic mechanism (improving the previous best-known result 6), and a 3 approximation randomized mechanism. We provide a similar result for an extended knapsack problem with heterogeneous items, where items are divided into groups and one can pick at most one item from each group. Finally we show a lower bound of approximation ratio of $1+\sqrt{2}$ for deterministic mechanisms and 2 for randomized mechanisms for knapsack, as well as the general submodular functions. Our lower bounds are unconditional, which do not rely on any computational or complexity assumptions.