Inferring Evolutionary History from DNA Sequences
SIAM Journal on Computing
A Fast Algorithm for the Computation and Enumeration of Perfect Phylogenies
SIAM Journal on Computing
Characterizations and algorithmic applications of chordal graph embeddings
Proceedings of the 4th Twente workshop on Graphs and combinatorial optimization
How to Use the Minimal Separators of a Graph for its Chordal Triangulation
ICALP '95 Proceedings of the 22nd International Colloquium on Automata, Languages and Programming
A Polynomial-Time Algorithm for Near-Perfect Phylogeny
ICALP '96 Proceedings of the 23rd International Colloquium on Automata, Languages and Programming
Generating All the Minimal Separators of a Graph
WG '99 Proceedings of the 25th International Workshop on Graph-Theoretic Concepts in Computer Science
Two Strikes Against Perfect Phylogeny
ICALP '92 Proceedings of the 19th International Colloquium on Automata, Languages and Programming
Incomplete Directed Perfect Phylogeny
SIAM Journal on Computing
COCOON'07 Proceedings of the 13th annual international conference on Computing and Combinatorics
Genome-wide compatible SNP intervals and their properties
Proceedings of the First ACM International Conference on Bioinformatics and Computational Biology
WABI'10 Proceedings of the 10th international conference on Algorithms in bioinformatics
ISBRA'11 Proceedings of the 7th international conference on Bioinformatics research and applications
Constructing perfect phylogenies and proper triangulations for three-state characters
WABI'11 Proceedings of the 11th international conference on Algorithms in bioinformatics
Efficiently solvable perfect phylogeny problems on binary and k-state data with missing values
WABI'11 Proceedings of the 11th international conference on Algorithms in bioinformatics
ISBRA'10 Proceedings of the 6th international conference on Bioinformatics Research and Applications
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The Multi-State Perfect Phylogeny Problem is an extension of the Binary Perfect Phylogeny Problem, allowing characters to take on more than two states. In this paper we consider three problems that extend the utility of the multi-state perfect phylogeny model: The Missing Data (MD) Problem where some entries in the input are missing and the question is whether (bounded) values for the missing data can be imputed so that the resulting data has a multi-state perfect phylogeny; The Character-Removal (CR) Problem where we want to minimize the number of characters to remove from the data so that the resulting data has a multi-state perfect phylogeny; and The Missing-Data Character-Removal (MDCR) Problem where the input has missing data and we want to impute values for the missing data to minimize the solution to the resulting Character-Removal Problem. We detail Integer Linear Programming (ILP) solutions to these problems for the special case of three permitted states per character and report on extensive empirical testing of these solutions. Then we develop a general theory to solve the MD problem for an arbitrary number of permitted states, using chordal graph theory and results on minimal triangulation of non-chordal graphs. This establishes new necessary and sufficient conditions for the existence of a perfect phylogeny with (or without) missing data. We implement the general theory using integer linear programming, although other optimization methods are possible. We extensively explore the empirical behavior of the general solution, showing that the methods are very practical for data of size and complexity that is characteristic of many current applications in phylogenetics. Some of the empirical results for the MD problem with an arbitrary number of permitted states are very surprising, suggesting the existence of additional combinatorial structure in multi-state perfect phylogenies.