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The Complexity of Finding Multiple Solutions to Betweenness and Quartet Compatibility
January/February 2012 (vol. 9 no. 1)
pp. 273-285
Maria Luisa Bonet, Universitat Politecnica de Catalunya (UPC), Barcelona
Simone Linz, University of Tübingen, Tübingen
Katherine St. John, City University of New York, Bronx
We show that two important problems that have applications in computational biology are ASP-complete, which implies that, given a solution to a problem, it is NP-complete to decide if another solution exists. We show first that a variation of Betweenness>, which is the underlying problem of questions related to radiation hybrid mapping, is ASP-complete. Subsequently, we use that result to show that Quartet Compatibility, a fundamental problem in phylogenetics that asks whether a set of quartets can be represented by a parent tree, is also ASP-complete. The latter result shows that Steel's Quartet Challenge, which asks whether a solution to Quartet Compatibility is unique, is coNP-complete.

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Index Terms:
ASP-complete, Betweenness, Quartet Challenge, phylogenetics, Quartet Compatibility, satisfiability.
Maria Luisa Bonet, Simone Linz, Katherine St. John, "The Complexity of Finding Multiple Solutions to Betweenness and Quartet Compatibility," IEEE/ACM Transactions on Computational Biology and Bioinformatics, vol. 9, no. 1, pp. 273-285, Jan.-Feb. 2012, doi:10.1109/TCBB.2011.108
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