Improved Truthful Rank Approximation for Rank-Maximal Matchings

In this work, we study truthful mechanisms for the rank-maximal matching problem, in the view of approximation. Our result reduces the gap from both the upper and lower bound sides. We propose a lexicographically truthful (LT) and nearly Pareto optimal (PO) randomized mechanism with an approximation ratio \(\frac{2 \sqrt{e}-1}{2 \sqrt{e}-2} \approx 1.77\) . The previous best result is 2. The crucial and novel ingredients of our algorithm are preservation lemmas, which allow us to utilize techniques from online algorithms to analyze the new ratio. We also provide several hardness results in variant settings to complement our upper bound. In particular, we improve the lower bound of the approximation ratio for our LT and PO mechanism to \(18/13\approx 1.38\) . To the best of our knowledge, it is the first time to obtain a lower bound by utilizing the linear programming approach along this research line.