This paper develops algorithms to solve strong-substitutes product-mix auctions. That is, it finds competitive equilibrium prices and quantities for agents who use this auction's bidding language to truthfully express their strong-substitutes preferences over an arbitrary number of goods, each of which is available in multiple discrete units. (Strong substitutes preferences are also known, in other literatures, as $M^\natural$-concave, matroidal and well-layered maps, and valuated matroids). Our use of the bidding language, and the information it provides, contrasts with existing algorithms that rely on access to a valuation or demand oracle to find equilibrium. We compute market-clearing prices using algorithms that apply existing submodular minimisation methods. Allocating the supply among the bidders at these prices then requires solving a novel constrained matching problem. Our algorithm iteratively simplifies the allocation problem, perturbing bids and prices in a way that resolves tie-breaking choices created by bids that can be accepted on more than one good. We provide practical running time bounds on both price-finding and allocation, and illustrate experimentally that our allocation mechanism is practical.

翻译：本文开发了求解强替代产品组合拍卖的算法。即，为使用该拍卖竞价语言真实表达对任意数量商品（每种商品均有多个离散单位）的强替代偏好的参与者，找到竞争均衡价格与数量。（在其他文献中，强替代偏好也被称为$M^\natural$-凹函数、拟阵分层映射及估值拟阵）。我们对该竞价语言及其所提供信息的运用方式，与现有依赖估值或需求预言机来寻找均衡的算法形成对比。我们利用现有次模最小化方法，通过算法计算市场出清价格。在以此价格向竞拍者分配供给时，需要求解一个新型受限匹配问题。我们的算法通过迭代方式简化分配问题，对竞价和价格进行扰动，以解决由可在多个商品上被接受的竞价所产生的打破平局选择。我们给出了价格发现与分配两方面的实用运行时间复杂度边界，并通过实验证明我们的分配机制具有实用性。