# Two strikes against perfect phylogeny

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## Abstract

One of the major efforts in molecular biology is the computation of phytogenies for species sets. A longstanding open problem in this area is called the Perfect Phylogeny problem. For almost two decades the complexity of this problem remained open, with progress limited to polynomial time algorithms for a few special cases, and many relaxations of the problem shown to be NP-Complete. From an applications point of view, the problem is of interest both in its general form, where the number of characters may vary, and in its fixed-parameter form. The Perfect Phylogeny problem has been shown to be equivalent to the problem of triangulating colored graphs[30]. It has also been shown recently that for a given fixed number of characters the yes-instances have bounded treewidth[45], opening the possibility of applying methodologies for bounded treewidth to the fixed-parameter form of the problem. We show that the Perfect Phylogeny problem is difficult in two different ways. We show that the general problem is NP-Complete, and we show that the various finite-state approaches for bounded treewidth cannot be applied to the fixed-parameter forms of the problem.

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### References

- [1]S. Arnborg. Efficient algorithms for combinatorial problems on graphs with bounded decomposability — A survey.
*BIT*, 25:2–23, 1985.Google Scholar - [2]S. Arnborg, D. Corneil, and A. Proskurowski. Complexity of finding embeddings in a k-tree.
*SIAM J. Alg. Discr. Meth.*, 8:277–284, 1987.Google Scholar - [3]S. Arnborg, B. Courcelle, A. Proskurowski, and D. Seese. An algebraic theory of graph reduction. Technical Report 90-02, Laboratoire Bordelais de Recherche en Informatique, Bordeaux, 1990. To appear in Proceedings 4th Workshop on Graph Grammars and Their Applications to Computer Science.Google Scholar
- [4]S. Arnborg, J. Lagergren, and D. Seese. Easy problems for tree-decomposable graphs
*J. Algorithms*, 12:308–340, 1991.CrossRefGoogle Scholar - [5]S. Arnborg and A. Proskurowski. Linear time algorithms for NP-hard problems restricted to partial
*k*-trees.*Disc. Appl. Math.*, 23:11–24, 1989.CrossRefGoogle Scholar - [6]H. L. Bodlaender. Dynamic programming algorithms on graphs with bounded tree-width. In
*Proceedings of the 15'th International Colloquium on Automata, Languages and Programming*, pages 105–119. Springer Verlag, Lecture Notes in Computer Science volume 317, 1988.Google Scholar - [7]H. L. Bodlaender and T. Kloks.
*A simple linear time algorithm for triangulating three-colored graphs*. Technical Report RUU-CS-91-13, Department of Computer Science, Utrecht University, the Netherlands, 1991. To appear in: Proceedings STACS'92.Google Scholar - [8]H.L. Bodlaender and T. Kloks. Better algorithms for the pathwidth and treewidth of graphs. In
*Proceedings 18'th International Colloquium on Automata, Languages and Programming*, pages 544–555. Springer Verlag, Lecture Notes in Computer Science volume 510, 1991.Google Scholar - [9]H. L. Bodlaender and R. H. Möhring, The pathwidth and treewidth of cographs. In
*Proceedings 2nd Scandinavian Workshop on Algorithm Theory*, pages 301–309. Springer Verlag Lecture Notes in Computer Science volume 447, 1990.Google Scholar - [10]K. Booth and G. Lueker. Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms.
*J. Comp. Syst. Sc.*, 13:335–370, 1976.Google Scholar - [11]R. B. Borie, R. G. Parker, and C. A. Tovey. Automatic generation of linear algorithms from predicate calculus descriptions of problems on recursive constructed graph families. Manuscript, 1988.Google Scholar
- [12]P. Buneman. A characterization of rigid circuit graphs.
*Discrete Math.*9:205–212, 1974.CrossRefGoogle Scholar - [13]J. Camin and R. Sokal,
*A method for deducing branching sequences in phylogeny*, Evolution 19, (1965), pp. 311–326.Google Scholar - [14]B. Courcelle. The monadic second-order logic of graphs I: Recognizable sets of finite graphs.
*Information and Computation*, 85:12–75, 1990.CrossRefGoogle Scholar - [15]G. A. Dirac. On rigid circuit graphs.
*Abh. Math. Sem. Univ. Hamburg*, 25: 71–76, 1961.Google Scholar - [16]G.F. Estabrook,
*Cladistic Methodology: a discussion of the theoretical basis for the induction of evolutionary history*, Annu. Rev. Evol. Syst., 3 (1972), pp. 427–456.CrossRefGoogle Scholar - [17]G.F. Estabrook, C.S. Johnson, Jr. and F.R. McMorris,
*An idealized concept of the true cladistic character*, Math. Biosci. 23, 1975, pp. 263–272.CrossRefGoogle Scholar - [18]G.F. Estabrook, C.S. Johnson, Jr., and F.R. McMorris,
*An algebraic analysis of cladistic characters*, Discrete Math., 16, 1976, pp. 141–147.CrossRefGoogle Scholar - [19]G.F. Estabrook, C.S. Johnson, Jr., and F.R. McMorris,
*A mathematical foundation for the analysis of cladistic character compatibility*, Math. Biosci., 29, 1976, pp. 181–187.CrossRefGoogle Scholar - [20]M. R. Fellows and K. Abrahamson,
*Cutset-Regularity Beats Well-Quasi-Ordering for Bounded Treewidth*. Manuscript, Nov. 1989.Google Scholar - [21]J. Felsenstein. Numerical methods for inferring evolutionary trees.
*The Quaterly Review of Biology*, Vol. 57, No. 4, Dec. 1982.Google Scholar - [22]W. M. Fitch and E. Margoliash. The construction of phylogenetic trees.
*Science*, 155, 1967.Google Scholar - [23]L. R. Foulds, and R. L. Graham, The Steiner problem in phytogeny is NP-Complete.
*Advances in Applied Mathematics*, 3:43–49, 1982.CrossRefGoogle Scholar - [24]D. R. Fulkerson and O. A. Gross. Incidence matrices and interval graphs.
*Pacific J. Mathematics*, 15:835–855, 1965.Google Scholar - [25]F. Gavril. The intersection graphs of subtrees in trees are exactly the chordal graphs.
*J. Combinatorial Theory series B*, 16:47–56, 1974.CrossRefGoogle Scholar - [26]M. C. Golumbic.
*Algorithmic Graph Theory and Perfect Graphs*. Academic Press, New York, 1980.Google Scholar - [27]D. Gusfield.
*The Steiner tree problem in phylogeny*. Technical Report 332, Department of Computer Science, Yale University, Sept. 1984.Google Scholar - [28]D. Gusfield. Efficient algorithms for inferring evolutionary trees.
*Networks*, 21:19–28, 1991.Google Scholar - [29]A. Habel.
*Hyperedge Replacement: Grammars and Languages*. PhD thesis, Univ. Bremen, 1988.Google Scholar - [30]S. Kannan and T. Warnow. Triangulating three-colored graphs. In
*Proceedings Second Annual ACMSIAM Symp. on Discrete Algorithms*, pages 337–343, San Francisco, Jan. 1991. Also to appear in SIAM J. on Discrete Mathematics.Google Scholar - [31]S. Kannan and T. Warnow. Inferring evolutionary history from DNA sequences. In
*Proceedings 31st Annual Symposium on the Foundations of Computer Science*, pages 362–371, St. Louis, Missouri, 1990.Google Scholar - [32]J. Lagergren.
*Algorithms and Minimal Forbidden Minors for Tree-decomposable Graphs*. PhD thesis, Royal Institute of Technology, Stockholm, Sweden, 1991.Google Scholar - [33]C. Lautemann. Efficient algorithms on context-free graph languages. In
*Proceedings of the 15th International Colloquium on Automata, Languages and Programming*, pages 362–378, 1988. Springer Verlag Lectures Notes in Computer Science volume 317.Google Scholar - [34]C. G. Lekkerkerker and J. Ch. Boland. Representations of a finite graph by a set of intervals on the real line,
*Fund. Math.*51:45–64, 1962.Google Scholar - [35]W. J. LeQuesne. The uniquely evolved character concept and its cladistic application,
*Syst. Zool.*, 23:513–517, 1974.Google Scholar - [36]W. J. LeQuesne. The uniquely evolved character concept.
*Syst. Zool.*, 26:218–223, 1977.Google Scholar - [37]W.J. LeQuesne,
*A method of selection of characters in numerical taxonomy*, Syst. Zool., 18, pp. 201–205, 1969.Google Scholar - [38]W.J. LeQuesne,
*Further studies on the uniquely derived character concept*, Syst. Zool., 21, pp. 281–288, 1972.Google Scholar - [39]W.J. LeQuesne,
*The uniquely evolved character concept and its cladistic application*, Syst. Zool., 23, pp. 513–517, 1974.Google Scholar - [40]W.J. LeQuesne,
*Discussion of preceeding papers*, In G.F. Estabrook (ed.),*Proc. Eighth International Conference on Numerical Taxonomy*, pp. 416–429. W.H. Freeman, San Francisco, 1975.Google Scholar - [41]W.J. LeQuesne,
*The uniquely evolved character concept*, Syst. Zool., 26, pp. 218–223, 1977.Google Scholar - [42]F. R. McMorris. Compatibility criteria for cladistic and qualitative taxonomic characters. In
*Proceedings 8th Internatinal Conference on Numerical Taxonomy*, G.F. Estrabrook, ed., pp. 339–415. W.H. Freeman, San Francisco, 1975.Google Scholar - [43]F. R. McMorris. On the compatibility of binary qualitative taxonomic characters.
*Bull. Math. Biol.*, 39:133–138, 1977.PubMedGoogle Scholar - [44]F. R. McMorris and C. A. Meacham. Partition intersection graphs.
*Ars Combinatorica*, 16-B:135–138, 1983.Google Scholar - [45]F. R. McMorris, T. Warnow, and T. Wimer.
*Triangulating colored graphs*. Submitted to Information Processing Letters.Google Scholar - [46]C. A. Meacham and G. F. Estabrook. Compatibility methods in systematics.
*Annual Review of Ecology and Systematics*, 16:431–446, 1985.CrossRefGoogle Scholar - [47]C. A. Meacham. Evaluating characters by character compatibility analysis. In: T. Duncan and T. F. Stuessy (eds.),
*Cladistics: Perspectives on the estimation of evolutionary history*, pp. 152–165. Columbia Univ. Press: New York, 1984.Google Scholar - [48]C. A. Meacham. Theoretical and computational considerations of the compatibility of qualitative taxonomic characters. In: J. Felsenstein (ed.),
*Numerical Taxonomy*, pages 304–314. NATO ASI Series, volume G1. Springer-Verlag: Berlin, Heidelberg, 1983.Google Scholar - [49]A. Proskurowski. Separating Subgraphs in k-trees: Cables and Caterpillars.
*Discrete Math.*, 49:275–285, 1984.CrossRefGoogle Scholar - [50]B. Reed. Finding approximate separators and computing treewidth quickly. Manuscript, 1992. To appear in: Proceedings of the 24'th Annual Symposium on Theory of Computing STOC'92.Google Scholar
- [51]N. Robertson and P. D. Seymour.
*Graph minors XIII: The disjoint path problem*. Manuscript, September 1986.Google Scholar - [52]D. J. Rose. Triangulated graphs and the elimination process.
*J. Math. Anal. Appl.*, 32:597–609, 1970.CrossRefGoogle Scholar - [53]D. J. Rose. On simple characterization of k-trees.
*Discrete Math.*, 7:317–322, 1974.CrossRefGoogle Scholar - [54]P. Scheffler. Linear-time algorithms for NP-complete problems restricted to partial k-trees. Report R-MATH-03/87, Karl-Weierstrass-Institut Für Mathematik, Berlin, GDR, 1987.Google Scholar
- [55]R. R. Sokal and P. H. A. Sneath.
*Principles of Numerical Taxonomy*. W.H. Freeman, San Francisco, 1963.Google Scholar - [56]R. E. Tarjan.
*Data Structures and Network Algorithms*. Society for Industrial and Applied Mathematics, Philadelphia, 1983.Google Scholar - [57]J. R. Walter.
*Representations of Rigid Circuit Graphs*. Ph.D. thesis, Wayne State University.Google Scholar - [58]E. O. Wilson. A Consistency Test for Phylogenies Based upon Contemporaneous Species.
*Systematic Zoology*, 14:214–220.Google Scholar - [59]T. V. Wimer.
*Linear algorithms on k-terminal graphs*. PhD thesis, Dept. of Computer Science, Clemson University, 1987.Google Scholar