We introduce and study isomorphic distances between ordinal elections (with the same numbers of candidates and voters). The main feature of these distances is that they are invariant to renaming the candidates and voters, and two elections are at distance zero if and only if they are isomorphic. Specifically, we consider isomorphic extensions of distances between preference orders: Given such a distance d, we extend it to distance d-ID between elections by unifying candidate names and finding a matching between the votes, so that the sum of the d-distances between the matched votes is as small as possible. We show that testing isomorphism of two elections can be done in polynomial time so, in principle, such distances can be tractable. Yet, we show that two very natural isomorphic distances are NP-complete and hard to approximate. We attempt to rectify the situation by showing FPT algorithms for several natural parameterizations.

翻译：我们引入并研究了序数选举（具有相同候选人和选民数量）之间的同构距离。这些距离的主要特征是它们对候选人和选民的重命名具有不变性，且两个选举的距离为零当且仅当它们是同构的。具体而言，我们考虑偏好序之间距离的同构扩展：给定一个距离d，我们通过统一候选人名称并寻找投票之间的匹配，将其扩展为选举间的距离d-ID，使得匹配投票之间的d距离之和尽可能小。我们证明，测试两个选举的同构性可以在多项式时间内完成，因此原则上这类距离可以是可计算的。然而，我们证明两种非常自然的同构距离是NP完全且难以近似的。我们尝试通过展示针对几种自然参数化的FPT算法来改善这一状况。