A note on improving the search of optimal prices in envy-free perfect matchings

TL;DR

This paper introduces a dynamic programming approach to find envy-free prices in a specific combinatorial auction setting, achieving similar theoretical complexity but with significant empirical speed improvements over existing shortest path methods.

Contribution

The authors propose a novel dynamic programming method for computing envy-free prices that outperforms previous shortest path algorithms in practice.

Findings

The new method has the same $O(n^3)$ theoretical complexity as previous approaches.

Empirical results show approximately 48% faster performance on average.

The approach effectively increases consumers' utilities to find optimal prices.

Abstract

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\%.