We present a method for finding envy-free prices in a combinatorial auction where the consumers' number $n$ coincides with that of distinct items for sale, each consumer can buy one single item and each item has only one unit available. This is a particular case of the {\it unit-demand envy-free pricing problem}, and was recently revisited by Arbib et al. (2019). These authors proved that using a Fibonacci heap for solving the maximum weight perfect matching and the Bellman-Ford algorithm for getting the envy-free prices, the overall time complexity for solving the problem is $O(n^3)$. We propose a method based on dynamic programming design strategy that seeks the optimal envy-free prices by increasing the consumers' utilities, which has the same cubic complexity time as the aforementioned approach, but whose theoretical and empirical results indicate that our method performs faster than the shortest paths strategy, obtaining an average time reduction in determining optimal envy-free prices of approximately 48\%.

翻译：我们提出了一种在组合拍卖中寻找无嫉妒价格的方法，其中消费者数量$n$与待售的不同商品数量相等，每位消费者只能购买一件商品，且每件商品仅有单件库存。这是\emph{单位需求无嫉妒定价问题}的一个特例，Arbib等人(2019)近期重新审视了该问题。这些作者证明：使用斐波那契堆求解最大权重完美匹配，并利用贝尔曼-福特算法获取无嫉妒价格时，求解该问题的整体时间复杂度为$O(n^3)$。我们提出一种基于动态规划设计策略的方法，通过提升消费者效用来寻找最优无嫉妒价格。该方法的复杂度与上述方法同为三次量级，但其理论分析与实验结果表明：相较于最短路径策略，我们的方法执行速度更快，在确定最优无嫉妒价格时平均耗时降低约48%。