Abstract
Interpreting threeleaf binary trees or rooted triples as constraints yields an entailment relation, whereby binary trees satisfying some rooted triples must also thus satisfy others, and thence a closure operator, which is known to be polynomialtime computable. This is extended to inconsistent triple sets by defining that a triple is entailed by such a set if it is entailed by any consistent subset of it.
Determining whether the closure of an inconsistent rooted triple set can be computed in polynomial time was posed as an open problem in the Isaac Newton Institute's "Phylogenetics" program in 2007. It appears (as NC4) in a collection of such open problems maintained by Mike Steel, and it is the last of that collection's five problems concerning computational complexity to have remained open. We resolve the complexity of computing this closure, proving that its decision version is NPComplete.
In the process, we also prove that detecting the existence of any acyclic Bhyperpath (from specified source to destination) is NPComplete, in a significantly narrower special case than the version whose minimization problem was recently proven NPhard by Ritz et al. This implies it is NPhard to approximate (our special case of) their minimization problem to within any factor.
BibTeX  Entry
@InProceedings{johnson:LIPIcs:2018:9960,
author = {Matthew P. Johnson},
title = {{Deciding the Closure of Inconsistent Rooted Triples Is NPComplete}},
booktitle = {29th International Symposium on Algorithms and Computation (ISAAC 2018)},
pages = {12:112:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770941},
ISSN = {18688969},
year = {2018},
volume = {123},
editor = {WenLian Hsu and DerTsai Lee and ChungShou Liao},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/9960},
URN = {urn:nbn:de:0030drops99600},
doi = {10.4230/LIPIcs.ISAAC.2018.12},
annote = {Keywords: phylogenetic trees, rooted triple entailment, NPCompleteness, directed hypergraphs, acyclic induced subgraphs, computational complexity}
}
Keywords: 

phylogenetic trees, rooted triple entailment, NPCompleteness, directed hypergraphs, acyclic induced subgraphs, computational complexity 
Seminar: 

29th International Symposium on Algorithms and Computation (ISAAC 2018) 
Issue Date: 

2018 
Date of publication: 

27.11.2018 