Mixed Integer Linear Programming for Maximum-Parsimony Phylogeny Inference

Abstract

Reconstruction of phylogenetic trees is a fundamental problem in computational biology. While excellent heuristic methods are available for many variants of this problem, new advances in phylogeny inference will be required if we are to be able to continue to make effective use of the rapidly growing stores of variation data now being gathered. In this paper, we present two integer linear programming (ILP) formulations to find the most parsimonious phylogenetic tree from a set of binary variation data. One method uses a flow-based formulation that can produce exponential numbers of variables and constraints in the worst case. The method has, however, proven extremely efficient in practice on datasets that are well beyond the reach of the available provably efficient methods, solving several large mtDNA and Y-chromosome instances within a few seconds and giving provably optimal results in times competitive with fast heuristics than cannot guarantee optimality. An alternative formulation establishes that the problem can be solved with a polynomial-sized ILP. We further present a web server developed based on the exponential-sized ILP that performs fast maximum parsimony inferences and serves as a front end to a database of precomputed phylogenies spanning the human genome.

A mixed integer linear programming model to reconstruct phylogenies from single nucleotide polymorphism haplotypes under the maximum parsimony criterion

“…Similarly, if the input haplotype set H satisfies the perfect phylogeny condition i.e., the requirement that each allele changes only once throughout the optimal phylogeny (see [19]), then the MPPEP-SNP can be still solved in polynomial time [26][27][28]. Unfortunately, it is possible to prove that in the general case the MPPEP-SNP is N P-hard (see [1,22]). In fact, the binary nature of the SNP haplotypes allows us to interpret a generic haplotype h i ∈ H as a vertex of a m-dimensional unit hypercube, its s-th allele as the s-th coordinate of the vertex h i , and the set H as the set of Steiner vertices of the unit hypercube.…”

“…Variables u i , i ∈ Q, do not need to be declared as their value will be always 1 any feasible solution to the problem. Finally, variables x s i , i ∈ V H , can be removed as their value is univocally assigned by the input haplotype set H. The reduction process can be further combined with the preprocessing strategies described in [1] to obtain even smaller formulations. Such strategies allow one to remove alleles from the input haplotype set H without altering the optimal solution to the problem.…”

“…Describing all the preprocessing techniques for shrinking the input haplotype set H is beyond the scope of the present article. The interested reader will find a systematic discussion of this topic in [1].…”

“…By applying the previously cited reduction strategies to Formulation 1 and denotingŜ as the set of alleles resulting from the application of the preprocessing strategies described in [1], w s as the number of alleles in S equal to the s-th, s ∈Ŝ, Z as the set for which variables z s ij are defined, R = {n + LB + 1, n + LB + 2, . .…”

“…As regards the MPPEP-SNP, the literature describes some (rare) circumstances for which it is possible to solve the problem in polynomial time (see Section Methods). Unfortunately, in the general case the MPPEP-SNP is N P-hard and solving provably to optimality therefore generally requires the use of exact approaches based on implicit enumeration algorithms, similar to the mixed integer programming strategies described in [1,2,22].…”