Enumeration problems are often encountered as key subroutines in the exact computation of graph parameters such as chromatic number, treewidth, or treedepth. In the case of treedepth computation, the enumeration of inclusion-wise minimal separators plays a crucial role. However and quite surprisingly, the complexity status of this problem has not been settled since it has been posed as an open direction by Kloks and Kratsch in 1998. Recently at the PACE 2020 competition dedicated to treedepth computation, solvers have been circumventing that by listing all minimal $a$-$b$ separators and filtering out those that are not inclusion-wise minimal, at the cost of efficiency. Naturally, having an efficient algorithm for listing inclusion-wise minimal separators would drastically improve such practical algorithms. In this note, however, we show that no efficient algorithm is to be expected from an output-sensitive perspective, namely, we prove that there is no output-polynomial time algorithm for inclusion-wise minimal separators enumeration unless P = NP.

翻译：枚举问题常常作为精确计算图参数（如色数、树宽或树深度）的关键子程序出现。在树深度计算中，包含意义下极小分隔符的枚举起着至关重要的作用。然而令人惊讶的是，自1998年Kloks和Kratsch将其列为开放方向以来，该问题的复杂性状态一直未得到解决。最近在专攻树深度计算的PACE 2020竞赛中，求解器通过列出所有极小$a$-$b$分隔符并过滤非包含意义下极小分隔符的方式规避该难题，但代价是效率低下。自然地，设计高效算法枚举包含意义下极小分隔符将极大改善此类实用算法。然而在本报告中，我们从输出敏感视角证明不应期待存在高效算法——即我们证明除非P = NP，否则不存在输出多项式时间算法用于枚举包含意义下极小分隔符。