MUL-Tree Pruning for Consistency and Compatibility

Authors Christopher Hampson , Daniel J. Harvey, Costas S. Iliopoulos , Jesper Jansson , Zara Lim , Wing-Kin Sung

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Christopher Hampson
  • Department of Informatics, King’s College London, UK
Daniel J. Harvey
  • Graduate School of Informatics, Kyoto University, Japan
Costas S. Iliopoulos
  • Department of Informatics, King’s College London, UK
Jesper Jansson
  • Graduate School of Informatics, Kyoto University, Japan
Zara Lim
  • Department of Informatics, King’s College London, UK
Wing-Kin Sung
  • Department of Chemical Pathology, The Chinese University of Hong Kong, China
  • Hong Kong Genome Institute, Hong Kong Science Park, Shatin, China
  • Laboratory of Computational Genomics, Li Ka Shing Institute of Health Sciences, The Chinese University of Hong Kong, China

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Christopher Hampson, Daniel J. Harvey, Costas S. Iliopoulos, Jesper Jansson, Zara Lim, and Wing-Kin Sung. MUL-Tree Pruning for Consistency and Compatibility. In 34th Annual Symposium on Combinatorial Pattern Matching (CPM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 259, pp. 14:1-14:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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 (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. 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 MULSETPC is NP-complete even when there are only two MUL-trees, every leaf label is used at most twice, and every MUL-tree is either binary or has constant height. Furthermore, we introduce an extension of MULSETPC that we call MULSETPComp, which replaces the notion of consistency with compatibility, and prove that MULSETPComp is NP-complete even when there are only two MUL-trees, every leaf label is used at most thrice, and every MUL-tree has constant height. Finally, we present a polynomial-time algorithm for instances of MULSETPC with a constant number of binary MUL-trees, in the special case where every leaf label occurs exactly once in at least one MUL-tree.

Subject Classification

ACM Subject Classification
  • Theory of computation → Pattern matching
  • multi-labelled tree
  • phylogenetic tree
  • consistent
  • compatible
  • pruning
  • algorithm
  • NP-complete


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