On the hardness of inclusion-wise minimal separators enumeration

TL;DR

This paper proves that enumerating inclusion-wise minimal separators in graphs cannot be done efficiently in output-polynomial time unless P equals NP, highlighting a fundamental computational complexity barrier.

Contribution

It establishes the first complexity hardness result for the enumeration of inclusion-wise minimal separators, resolving an open problem from 1998.

Findings

No output-polynomial time algorithm exists unless P=NP.

The result impacts practical algorithms for treedepth computation.

Enumeration of minimal separators is computationally hard.

Abstract

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.