We study auction design in the celebrated interdependence model introduced by Milgrom and Weber [1982], where a mechanism designer allocates a good, maximizing the value of the agent who receives it, while inducing truthfulness using payments. In the lesser-studied procurement auctions, one allocates a chore, minimizing the cost incurred by the agent selected to perform it. Most of the past literature in theoretical computer science considers designing truthful mechanisms with constant approximation for the value setting, with restricted domains and monotone valuation functions. In this work, we study the general computational problems of optimizing the approximation ratio of truthful mechanism, for both value and cost, in the deterministic and randomized settings. Unlike most previous works, we remove the domain restriction and the monotonicity assumption imposed on value functions. We provide theoretical explanations for why some previously considered special cases are tractable, reducing them to classical combinatorial problems, and providing efficient algorithms and characterizations. We complement our positive results with hardness results for the general case, providing query complexity lower bounds, and proving the NP-Hardness of the general case.

翻译：我们研究了由Milgrom和Weber [1982]提出的著名相互依赖模型中的拍卖设计问题。在该模型中，机制设计者通过支付诱导真实性，分配一件商品以最大化获得该商品的代理人价值。在较少被研究的采购拍卖中，分配的是任务，需要最小化被选中执行任务的代理人成本。以往理论计算机科学领域的大多数文献，在价值设定下考虑设计具有恒定近似比的诚实机制，且局限于受限域和单调估值函数。本研究则系统研究了在确定性和随机环境下，针对价值与成本两种情形优化诚实机制近似比的一般计算问题。与以往多数研究不同，我们解除了对估值函数的域限制和单调性假设。我们为部分先前被视作特例的可解情形提供了理论解释，将其归约为经典组合优化问题，并给出了高效算法与刻画。同时，我们通过查询复杂度下界证明了一般情形的NP难度，用难度结论补充了正向结果。