Kernel and Fast Algorithm for Dense Triplet Inconsistency

We study the parameterized complexity of inferring supertrees from sets of rooted triplets, an important problem in phylogenetics For a set L of labels and a dense set $\mathcal R$ of triplets distinctly leaf-labeled by 3-subsets of L we seek a tree distinctly leaf-labeled by L and containing all but at most p triplets from $\mathcal R$ as homeomorphic subtree Our results are the first polynomial kernel for this problem, with O(p2) labels, and a subexponential fixed-parameter algorithm running in time $2^{O(p^{1/3} \log p)} + O(n^4)$.

[1]  D. Bryant.  Building trees, hunting for trees, and comparing trees : theory and methods in phylogenetic analysis , 1997 .

[2]  Sylvain Guillemot,et al.  Fixed-Parameter Tractability of the Maximum Agreement Supertree Problem , 2007, CPM.

[3]  Kunihiko Sadakane,et al.  Rooted Maximum Agreement Supertrees , 2004, Algorithmica.

[4]  M. Steel.  The complexity of reconstructing trees from qualitative characters and subtrees , 1992 .

[5]  Jiong Guo,et al.  Problem Kernels for NP-Complete Edge Deletion Problems: Split and Related Graphs , 2007, ISAAC.

[6]  Noga Alon,et al.  Color-coding , 1995, JACM.

[7]  Jaroslaw Byrka,et al.  New Results on Optimizing Rooted Triplets Consistency , 2008, ISAAC.

[8]  Noga Alon,et al.  Fast Fast , 2009, ICALP.

[9]  Michael R. Fellows,et al.  On Problems without Polynomial Kernels (Extended Abstract) , 2008, ICALP.

[10]  Alfred V. Aho,et al.  Inferring a Tree from Lowest Common Ancestors with an Application to the Optimization of Relational Expressions , 1981, SIAM J. Comput..

[11]  Peter Rossmanith,et al.  New Fixed-Parameter Algorithms for the Minimum Quartet Inconsistency Problem , 2008, IWPEC.

[12]  Noga Alon,et al.  A Fast and Simple Randomized Parallel Algorithm for the Maximal Independent Set Problem , 1985, J. Algorithms.

[13]  A. Dress,et al.  Reconstructing the shape of a tree from observed dissimilarity data , 1986 .

[14]  Andrzej Lingas,et al.  On the Complexity of Constructing Evolutionary Trees , 1999, J. Comb. Optim..

[15]  François Nicolas,et al.  Maximum agreement and compatible supertrees , 2004, J. Discrete Algorithms.

[16]  Nir Ailon,et al.  Aggregating inconsistent information: Ranking and clustering , 2008 .

[17]  Michael R. Fellows,et al.  On problems without polynomial kernels , 2009, J. Comput. Syst. Sci..

[18]  Fedor V. Fomin,et al.  Kernels for feedback arc set in tournaments , 2009, J. Comput. Syst. Sci..

[19]  Jesper Jansson,et al.  On the Complexity of Inferring Rooted Evolutionary Trees , 2001, Electron. Notes Discret. Math..

[20]  Claire Mathieu,et al.  Electronic Colloquium on Computational Complexity, Report No. 144 (2006) How to rank with few errors A PTAS for Weighted Feedback Arc Set on Tournaments , 2006 .

[21]  Rolf Niedermeier,et al.  A fixed-parameter algorithm for minimum quartet inconsistency , 2003, J. Comput. Syst. Sci..