Sponsored search auctions are commonly modeled as an assignment of a fixed set of positions to a set of advertisers, with welfare maximization reducible to a standard matching problem. Motivated by modern advertising environments, we study a richer variant of the classical position auctions model, in which ads may have heterogeneous sizes and the platform must jointly select a subset of ads and assign them to positions subject to a global capacity constraint. The size of an ad can capture the physical space it occupies on a page or its consumption of a scarce platform resource, such as communication bandwidth or computational capacity. Thus, size measures the resource cost imposed by an ad on the platform, which need not be related to the value generated by displaying it. We begin by studying a general formulation of the matching with a capacity constraint problem and propose an algorithmic technique that goes beyond known greedy methods while achieving constant-factor approximation guarantees for social welfare. 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 algorithm under the standard assumptions of single-parameter agents, publicly known ad sizes, and position-dependent click-through rates (CTRs). In this setting, the size of an ad determines the amount of capacity it consumes, while the CTR depends only on the position to which it is assigned. We show that a minor modification to our algorithmic approach yields a universally truthful mechanism that achieves a 4-approximation of the optimal social welfare, and we show that this guarantee is tight. To the best of our knowledge, this is the first such mechanism for this variant of the capacity-constrained matching.

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