Maximum independent sets of commuting and noninterfering inversions

Abstract

BackgroundGiven three signed permutations, an inversion median is a fourth permutation that minimizes the sum of the pairwise inversion distances between it and the three others. This problem is NP-hard as well as hard to approximate. Yet median-based approaches to phylogenetic reconstruction have been shown to be among the most accurate, especially in the presence of long branches. Most existing approaches have used heuristics that attempt to find a longest sequence of inversions from one of the three permutations that, at each step in the sequence, moves closer to the other two permutations; yet very little is known about the quality of solutions returned by such approaches.ResultsRecently, Arndt and Tang took a step towards finding longer such sequences by using sets of commuting inversions. In this paper, we formalize the problem of finding such sequences of inversions with what we call signatures and provide algorithms to find maximum cardinality sets of commuting and noninterfering inversions.ConclusionOur results offer a framework in which to study the inversion median problem, faster algorithms to obtain good medians, and an approach to study characteristic events along an evolutionary path.

Rec-DCM-Eigen: Reconstructing a Less Parsimonious but More Accurate Tree in Shorter Time

“…There are faster median solvers in the literature– e.g. [32] – [34] for inversion median and [35] for DCJ median, but even those become very slow for very distant genomes. This necessitates the development of faster median solvers.…”

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“…We prove that sorting scenarios using DCJ operations on co-tailed genomes can be represented by parking functions and labeled trees. This surprising connection yields immediate results on the uniform generation of scenarios [1,13,18], promises tools for sampling processes [6,10,12] and the development of statistical significant tests [7,11,16,21], and offers a wealth of alternate representations to explore the properties of rearrangement scenarios, such as commutation [4,20], structure conservation [3,8], breakpoint reuse [15,17] or cycle length [22].…”