# Complexity and algorithms for copy-number evolution problems

Published in *Algorithms for Molecular Biology*, 2017

Recommended citation: M. El-Kebir, B.J. Raphael, R. Shamir, R. Sharan, S. Zaccaria, M. Zehavi, and R. Zeira. Complexity and algorithms for copy-number evolution problems. Algorithms for Molecular Biology, 12(1), 13. 2017 __https://almob.biomedcentral.com/articles/10.1186/s13015-017-0103-2__

# Abstract

## Background

Cancer is an evolutionary process characterized by the accumulation of somatic mutations in a population of cells that form a tumor. One frequent type of mutations is copy number aberrations, which alter the number of copies of genomic regions. The number of copies of each position along a chromosome constitutes the chromosome’s copy-number profile. Understanding how such profiles evolve in cancer can assist in both diagnosis and prognosis.

## Results

We model the evolution of a tumor by segmental deletions and amplifications, and gauge distance from profile a to b by the minimum number of events needed to transform a into b. Given two profiles, our first problem aims to find a parental profile that minimizes the sum of distances to its children. Given k profiles, the second, more general problem, seeks a phylogenetic tree, whose k leaves are labeled by the k given profiles and whose internal vertices are labeled by ancestral profiles such that the sum of edge distances is minimum.

## Conclusions

For the former problem we give a pseudo-polynomial dynamic programming algorithm that is linear in the profile length, and an integer linear program formulation. For the latter problem we show it is NP-hard and give an integer linear program formulation that scales to practical problem instance sizes. We assess the efficiency and quality of our algorithms on simulated instances.

## Availability

https://github.com/raphael-group/CNT-ILP