We analyze the (parameterized) computational complexity of “fair” variants of bipartite many-to-one matching, where each vertex from the “left” side is matched to exactly one vertex and each vertex from the “right” side may be matched to multiple vertices. We want to find a “fair” matching, in which each vertex from the right side is matched to a “fair” set of vertices. Assuming that each vertex from the left side has one color modeling its “attribute”, we study two fairness criteria. For instance, in one of them, we deem a vertex set fair if for any two colors, the difference between the numbers of their occurrences does not exceed a given threshold. Fairness is, for instance, relevant when finding many-to-one matchings between students and colleges, voters and constituencies, and applicants and firms. Here colors may model sociodemographic attributes, party memberships, and qualifications, respectively.

We show that finding a fair many-to-one matching is NP-hard even for three colors and maximum degree five. Our main contribution is the design of fixed-parameter tractable algorithms with respect to the number of vertices on the right side. Our algorithms make use of a variety of techniques including color coding. At the core lie integer linear programs encoding Hall like conditions. We establish the correctness of our integer programs, based on Frank’s separation theorem [Frank, Discrete Math. 1982]. We further obtain complete complexity dichotomies regarding the number of colors and the maximum degree of each side.

我们分析了二分多对一匹配的“公平”变体的（参数化）计算复杂度，其中来自“左侧”的每个顶点恰好匹配一个顶点，来自“右侧”的每个顶点可以匹配到多个顶点。我们想要找到一个“公平”的匹配，其中右侧的每个顶点都与一组“公平”的顶点匹配。假设左侧的每个顶点都有一种颜色来建模其“属性”，我们研究两个公平标准。例如，在其中一种颜色中，如果对于任何两种颜色，它们出现次数之间的差异不超过给定的阈值，我们就认为顶点集是公平的。例如，在寻找学生和大学、选民和选区、申请人和公司之间的多对一匹配时，公平性就很重要。这里，颜色可以分别模拟社会人口统计属性、党员身份和资格。

我们证明，即使对于三种颜色和最大程度为五，找到公平的多对一匹配也是 NP 困难的。我们的主要贡献是设计关于右侧顶点数量的固定参数易处理算法。我们的算法利用了多种技术，包括颜色编码。其核心在于编码类似霍尔条件的整数线性程序。我们根据弗兰克分离定理[弗兰克，离散数学。 1982]。我们进一步获得关于颜色数量和每条边的最大程度的完整复杂性二分法。