Sponsored search auctions are commonly modeled as an assignment of a fixed set of slots (positions) to a set of advertisers, with welfare maximization being reducible to a standard matching problem. Motivated by modern ad formats, we study a richer variant of the classical position auctions model, in which ads have heterogeneous sizes and the platform must jointly select and assign a subset of ads to positions subject to a global space constraint. We formulate this as a matching problem with a capacity constraint, and propose an algorithmic technique that goes beyond simple greedy methods while achieving constant factor approximation guarantees. Our allocation rule augments density-based ordering with capacity-aware local improvements, which allow for re-allocations that improve welfare, while respecting the capacity constraint. Applied in the context of position auctions, we analyze this mechanism under the assumption of single-parameter agents and position-dependent click-through-rates (CTRs). We show that a minor modification to our approach yields a universally truthful randomized mechanism with a constant factor approximation guarantee. To the best of our knowledge, this is the first truthful constant-approximation mechanism for this variant of capacity-constrained matching.

翻译：赞助搜索拍卖通常被建模为将固定的一组广告位（位置）分配给一组广告商，其福利最大化可简化为标准匹配问题。受现代广告形式启发，我们研究经典位置拍卖模型的一种更丰富的变体，其中广告具有异质性尺寸，且平台必须在全局空间约束下联合选择并分配广告子集至各位置。我们将此问题形式化为具有容量约束的匹配问题，并提出一种超越简单贪心方法、同时实现常数因子近似保证的算法技术。我们的分配规则在基于密度的排序中融合了容量感知的局部改进，允许在遵守容量约束的前提下进行能够提升福利的重新分配。在位置拍卖场景中应用时，我们在单一参数代理和位置依赖点击率（CTRs）假设下分析该机制。研究表明，对方法的轻微修改可产生具有常数因子近似保证的通用真随机机制。据我们所知，这是针对此类带容量约束匹配变体的首个具有真随机近似保障的机制。