An electorate with fully-ranked innate preferences casts approval votes over a finite set of alternatives. As a result, only partial information about the true preferences is revealed to the voting authorities. In an effort to understand the nature of the true preferences given only partial information, one might ask whether the unknown innate preferences could possibly be single-crossing. The existence of a polynomial time algorithm to determine this has been asked as an outstanding problem in the works of Elkind and Lackner. We hereby give a polynomial time algorithm determining a single-crossing collection of fully-ranked preferences that could have induced the elicited approval ballots, or reporting the nonexistence thereof. Moreover, we consider the problem of identifying negative instances with a set of forbidden sub-ballots, showing that any such characterization requires infinitely many forbidden configurations.

翻译：具有完全排序内在偏好的选民群体对有限备选方案进行批准投票。因此，投票机构仅能获得关于真实偏好的部分信息。在仅凭部分信息理解真实偏好性质的过程中，人们可能提出疑问：未知的内在偏好是否可能具有单交叉性。Elkind与Lackner在其研究工作中将是否存在多项式时间算法判定该问题列为悬而未决的难题。本文给出了一种多项式时间算法，用于判定是否存在能够产生所获取批准投票的完全排序偏好单交叉集合，或在不存在时予以报告。此外，我们研究了通过一组禁止子投票识别负实例的问题，表明任何此类刻画都需要无穷多个禁止配置。