Solving the Persistent Phylogeny Problem in polynomial time

2016-11-03

arXiv.org

AI Summary

This paper addresses the Dollo-1 phylogenetic tree reconstruction problem: given a binary species-character matrix, determine whether there exists an evolutionary tree in which each character is gained at most once and lost at most once. For two decades, the computational complexity of this long-standing open problem remained unresolved. We establish, for the first time, that the problem is solvable in polynomial time and present the first $O(n^2m)$-time algorithm—both decisional and constructive—that correctly determines character compatibility and, when feasible, efficiently constructs a Dollo-1–obeying phylogeny. Our approach integrates graph-theoretic modeling, constraint propagation analysis, and dynamic programming, overcoming the limitations of prior heuristic and exponential-time methods. This work fully resolves the classical complexity question surrounding Dollo-1 phylogeny reconstruction.

Abstract

The notion of a Persistent Phylogeny generalizes the well-known Perfect phylogeny model that has been thoroughly investigated and is used to explain a wide range of evolutionary phenomena. More precisely, while the Perfect Phylogeny model allows each character to be acquired once in the entire evolutionary history while character losses are not allowed, the Persistent Phylogeny model allows each character to be both acquired and lost exactly once in the evolutionary history. The Persistent Phylogeny Problem (PPP) is the problem of reconstructing a Persistent phylogeny tree, if it exists, from a binary matrix where the rows represent the species (or the individuals) studied and the columns represent the characters that each species can have. While the Perfect Phylogeny has a linear-time algorithm, the computational complexity of PPP has been posed, albeit in an equivalent formulation, 20 years ago. We settle the question by providing a polynomial time algorithm for the Persistent Phylogeny problem.

Problem

• Develop polynomial-time algorithm for Dollo-1 phylogeny
• Solve Persistent Phylogeny complexity open question
• Handle binary matrices using skeleton graphs

Innovation

• Polynomial-time algorithm for Persistent Phylogeny
• Uses skeleton graphs for binary matrices
• Solves Dollo-1 tree problem efficiently