The Spearman footrule is a voting rule that takes as input voter preferences expressed as rankings. It outputs a ranking that minimizes the sum of the absolute differences between the position of each candidate in the ranking and in the voters' preferences. In this paper, we study the computational complexity of two extensions of the Spearman footrule when the number of voters is a small constant. The first extension, introduced by Pascual et al. (2018), arises from the collective scheduling problem and treats candidates, referred to as tasks in their model, as having associated lengths. The second extension, proposed by Kumar and Vassilvitskii (2010), assigns weights to candidates; these weights serve both as lengths, as in the collective scheduling model, and as coefficients in the objective function to be minimized. Although computing a ranking under the standard Spearman footrule is polynomial-time solvable, we demonstrate that the first extension is NP-hard with as few as 3 voters, and the second extension is NP-hard with as few as 4 voters. Both extensions are polynomial-time solvable for 2 voters.

翻译：斯皮尔曼足距是一种投票规则，其输入为选民以排序形式表达的偏好。该规则输出一个排序，使得每个候选人在该排序中的位置与其在选民偏好中的位置之间的绝对差值之和最小。本文研究了当选民数量为小常数时，斯皮尔曼足距两种扩展的计算复杂度。第一种扩展由Pascual等人（2018）提出，源于集体调度问题，并将候选人（在其模型中称为任务）视为具有关联长度。第二种扩展由Kumar和Vassilvitskii（2010）提出，为候选人分配权重；这些权重既作为长度（如同集体调度模型），也作为待最小化目标函数中的系数。尽管在标准斯皮尔曼足距下计算排序是多项式时间可解的，但我们证明第一种扩展在仅有3位选民时即为NP难问题，而第二种扩展在仅有4位选民时即为NP难问题。两种扩展在仅有2位选民时均为多项式时间可解。