In 1979, Hylland and Zeckhauser \cite{hylland} gave a simple and general scheme for implementing a one-sided matching market using the power of a pricing mechanism. Their method has nice properties -- it is incentive compatible in the large and produces an allocation that is Pareto optimal -- and hence it provides an attractive, off-the-shelf method for running an application involving such a market. With matching markets becoming ever more prevalant and impactful, it is imperative to finally settle the computational complexity of this scheme. We present the following partial resolution: 1. A combinatorial, strongly polynomial time algorithm for the special case of $0/1$ utilities. 2. An example that has only irrational equilibria, hence proving that this problem is not in PPAD. Furthermore, its equilibria are disconnected, hence showing that the problem does not admit a convex programming formulation. 3. A proof of membership of the problem in the class FIXP. We leave open the (difficult) question of determining if the problem is FIXP-hard. Settling the status of the special case when utilities are in the set $\{0, {\frac 1 2}, 1 \}$ appears to be even more difficult.

中文翻译：

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单边匹配市场的 Hylland-Zeckhauser 方案的计算复杂性

1979 年，Hylland 和 Zeckhauser \cite{hylland} 给出了一个简单而通用的方案，用于利用定价机制的力量实现单边匹配市场。他们的方法具有很好的特性——它在大范围内是激励兼容的，并产生帕累托最优的分配——因此它提供了一种有吸引力的、现成的方法来运行涉及这样一个市场的应用程序。随着匹配市场变得越来越普遍和影响力，最终解决该方案的计算复杂性势在必行。我们提出了以下部分解决方案： 1. 针对 $0/1$ 实用程序的特殊情况的组合强多项式时间算法。2. 一个只有非理性均衡的例子，因此证明这个问题不在 PPAD 中。此外，它的平衡是断开的，因此表明该问题不承认凸规划公式。3. FIXP 类中问题的成员资格证明。我们留下确定问题是否是 FIXP 难的（困难的）问题。当实用程序在集合 $\{0, {\frac 1 2}, 1 \}$ 中时，解决特殊情况的状态似乎更加困难。