We study the complexity of the classic Hylland-Zeckhauser scheme [HZ'79] for one-sided matching markets. We show that the problem of finding an $\epsilon$-approximate equilibrium in the HZ scheme is PPAD-hard, and this holds even when $\epsilon$ is polynomially small and when each agent has no more than four distinct utility values. Our hardness result, when combined with the PPAD membership result of [VY'21], resolves the approximation complexity of the HZ scheme. We also show that the problem of approximating the optimal social welfare (the weight of the matching) achievable by HZ equilibria within a certain constant factor is NP-hard.

翻译：我们研究了经典的Hylland-Zeckhauser计划[HZ'79]对单方匹配市场的复杂性。我们表明,在HZ计划中找到美元近似平衡的问题是PPAD硬的,即使$-Epsilon是多元的,而且每个代理商的公用事业价值不超过四个不同的价值,这也仍然存在。当我们与PPPAD成员结果[VY'21]相结合时,我们的强硬结果解决了HZ计划的近似复杂性。我们还表明,HZe equilibria在一定不变因素中实现的最佳社会福利(匹配权的权重)的近似性问题是NP硬的。