Clustering is an unsupervised learning task that aims to partition data into a set of clusters. In many applications, these clusters correspond to real-world constructs (e.g. electoral districts) whose benefit can only be attained by groups when they reach a minimum level of representation (e.g. 50\% to elect their desired candidate). This paper considers the problem of performing k-means clustering while ensuring groups (e.g. demographic groups) have that minimum level of representation in a specified number of clusters. We show that the popular $k$-means algorithm, Lloyd's algorithm, can result in unfair outcomes where certain groups lack sufficient representation past the minimum threshold in a proportional number of clusters. We formulate the problem through a mixed-integer optimization framework and present a variant of Lloyd's algorithm, called MiniReL, that directly incorporates the fairness constraints. We show that incorporating the fairness criteria leads to a NP-Hard sub-problem within Lloyd's algorithm, but we provide computational approaches that make the problem tractable for even large datasets. Numerical results show that the approach is able to create fairer clusters with practically no increase in the k-means clustering cost across standard benchmark datasets.

翻译：集群是一个无人监督的学习任务,目的是将数据分成一组集群。在许多应用中,这些集群与真实世界结构(如选区)相对应,只有群体达到最低代表水平(如50 ⁇ )才能从中得益(如50 ⁇ ),本文考虑了在确保群体(如人口群体)在特定数量组群中拥有最低代表水平的同时,执行k means集群的问题。我们表明,流行的 $- means 运算法,即劳埃德的算法,如果某些群体在一定比例组群中没有足够代表超过最低门槛,则会产生不公平的结果。我们通过混合整数优化框架来提出问题,并提出了劳埃德算法的变式,称为MiniReL,直接纳入了公平性限制。我们表明,纳入公平性标准标准标准可以在劳埃德的算法中引入NP-Hard 子标值,但我们提供计算方法,使问题甚至大数据集可以定位。数字结果显示,该方法能够建立更公平的组合群集,实际上无法在标准的千兆位上增加成本。