A multi-labelled tree (or MUL-tree) is a rooted tree leaf-labelled by a set of labels, where each label may appear more than once in the tree. We consider the MUL-tree Set Pruning for Consistency problem ( \(\textsf{MULSETPC}\) ), which takes as input a set of MUL-trees and asks whether there exists a perfect pruning of each MUL-tree that results in a consistent set of single-labelled trees. \(\textsf{MULSETPC}\) was proven to be NP-complete by Gascon et al. when the MUL-trees are binary, each leaf label is used at most three times, and the number of MUL-trees is unbounded. To determine the computational complexity of the problem when the number of MUL-trees is constant was left as an open problem. Here, we resolve this question by proving a much stronger result, namely that \(\textsf{MULSETPC}\) is NP-complete even when there are only two MUL-trees, every leaf label is used at most twice, and either every MUL-tree is binary or every MUL-tree has constant height. Furthermore, we introduce an extension of \(\textsf{MULSETPC}\) that we call \(\textsf{MULSETPComp}\) , which replaces the notion of consistency with compatibility. We prove that \(\textsf{MULSETPComp}\) is NP-complete even in two highly restricted cases – when there are only two MUL-trees, every MUL-tree has constant height and either every leaf label is used at most thrice or the instance contains a single-labelled tree containing all leaf labels. Finally, we present a polynomial-time algorithm for a special case of \(\textsf{MULSETPC}\) with a constant number of binary MUL-trees where every leaf label appears exactly once in at least one MUL-tree.
