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：具有完全排他内在偏好的选民群体对有限备选方案集进行批准投票。因此，投票机构仅能获取关于真实偏好的部分信息。为理解仅凭部分信息下真实偏好的性质，自然产生一个疑问：未知的内在偏好是否可能具有单交叉性。Elkind与Lackner在其研究中将是否存在多项式时间算法判定该性质列为未解决的难题。本文给出一个多项式时间算法，用于确定是否存在一组能够导出所获得批准投票的完全排位单交叉偏好集合，或在不存在时报告无解。此外，我们通过一组禁止子投票来识别否定实例的问题，并证明任何此类刻画都需要无限多个禁止构型。