On price-induced minmax matchings

We study a natural combinatorial pricing problem for sequentially arriving buyers with equal budgets. Each buyer is interested in exactly one pair of items and purchases this pair if and only if, upon arrival, both items are still available and the sum of the item prices does not exceed the budget. The goal of the seller is to set prices to the items such that the number of transactions is maximized when buyers arrive in adversarial order. Formally, we are given an undirected graph where vertices represent items and edges represent buyers. Once prices are set to the vertices, edges with a total price exceeding the buyers' budgets are evicted. Any arrival order of the buyers leads to a set of transactions that forms a maximal matching in this subgraph, and an adversarial arrival order results in a minimum maximal matching. In order to measure the performance of a pricing strategy, we compare the size of such a matching to the size of a maximum matching in the original graph. It was shown by Correa et al. [IPCO 2022] that the best ratio any pricing strategy can guarantee lies within $[1/2, 2/3]$. Our contribution to the problem is two-fold: First, we provide several characterizations of subgraphs that may result from pricing schemes. Second, building upon these, we show an improved upper bound of $3/5$ and a lower bound of $1/2 + 2/n$, where $n$ is the number of items.