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：枚举问题常作为精确计算图参数（如色数、树宽或树深）的关键子程序出现。在树深计算中，包含意义下极小分隔符的枚举扮演着至关重要的角色。然而令人惊讶的是，自Kloks与Kratsch于1998年将其列为开放方向以来，该问题的复杂性状态尚未解决。近期在专注于树深计算的PACE 2020竞赛中，求解器通过罗列所有极小$a$-$b$分隔符并过滤掉非包含意义下极小者来规避该问题，但以牺牲效率为代价。自然，拥有列举包含意义下极小分隔符的高效算法将极大改进此类实际算法。然而本文指出，从输出敏感性角度出发，高效算法不可期——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。