In the envy-free perfect matching problem, $n$ items with unit supply are available to be sold to $n$ buyers with unit demand. The objective is to find allocation and prices such that both seller's revenue and buyers' surpluses are maximized -- given the buyers' valuations for the items -- and all items must be sold. A previous work has shown that this problem can be solved in cubic time, using maximum weight perfect matchings to find optimal envy-free allocations and shortest paths to find optimal envy-free prices. In this work, I consider that buyers have fixed budgets, the items have quality measures and so the valuations are defined by multiplying these two quantities. Under this approach, I prove that the valuation matrix have the inverse Monge property, thus simplifying the search for optimal envy-free allocations and, consequently, for optimal envy-free prices through a strategy based on dynamic programming. As result, I propose an algorithm that finds optimal solutions in quadratic time.

翻译：在免嫉妒完美匹配问题中，有 $n$ 件单位供给的商品待售给 $n$ 个具有单位需求的买家。目标是找到分配和价格，使得卖方收益和买方剩余均最大化——给定买家对商品的估值——且所有商品必须售出。先前的工作表明，该问题可在立方时间内求解，通过使用最大权重完美匹配寻找最优免嫉妒分配，以及使用最短路径寻找最优免嫉妒价格。在本工作中，我考虑买家具有固定预算，物品具有质量度量，因此估值定义为这两个量的乘积。在此方法下，我证明估值矩阵具有逆Monge性质，从而通过基于动态规划的策略简化了最优免嫉妒分配的搜索，进而简化了最优免嫉妒价格的搜索。结果，我提出了一种在二次时间内找到最优解的算法。