Motivated by many practical applications, in this paper we study {\em budget feasible mechanisms} where the goal is to procure independent sets from matroids. More specifically, we are given a matroid $\mathcal{M}=(E,\mathcal{I})$ where each ground (indivisible) element is a selfish agent. The cost of each element (i.e., for selling the item or performing a service) is only known to the element itself. There is a buyer with a budget having additive valuations over the set of elements $E$. The goal is to design an incentive compatible (truthful) budget feasible mechanism which procures an independent set of the matroid under the given budget that yields the largest value possible to the buyer. Our result is a deterministic, polynomial-time, individually rational, truthful and budget feasible mechanism with $4$-approximation to the optimal independent set. Then, we extend our mechanism to the setting of matroid intersections in which the goal is to procure common independent sets from multiple matroids. We show that, given a polynomial time deterministic blackbox that returns $\alpha-$approximation solutions to the matroid intersection problem, there exists a deterministic, polynomial time, individually rational, truthful and budget feasible mechanism with $(3\alpha +1)-$approximation to the optimal common independent set.
翻译:在很多实际应用的推动下,我们在本文中研究“预算可行的机制 ”, 目标是从类中采购独立的成品。 更具体地说, 我们得到的是一个以每块(不可分的)元素为自私的代理。 每个元素( 不可分的) 的成本( 即出售物品或提供服务) 只为元素本身所知道。 有一个购买方的预算对一组元素具有添加价值的附加价值 $E。 目标是设计一个符合( 坚固的) 预算可行的激励机制, 在给定预算下购买一套独立的成品, 给买主带来最大的价值。 我们的结果是一个确定性、 多元性、 个别合理、 真实和预算可行的机制, 使用4美元 的成品, 只能由该元素本身自己知道。 然后, 我们把我们的机制扩大到一个可行的配料交叉点, 其目标就是从多个类中采购通用的独立成品。 我们展示了一个解决方案, 以一个固定的固定时间框 。