On the complexity and approximation of syntenic distance

Abstract

The paper studies the computational complexity and approximation algorithms for a new evolutionary distance between multi-chromosomal genomes introduced recently by Ferretti, Nadeau and Sankoff. Here, a chromosome is represented as a set of genes and a genome is a collections of chromosomes. The syntenic distance between two genomes is defined as the minimum number of translocations, fusions and fissions required to transform one genome into the other. We prove that computing the syntenic distance is NP-hard and give a simple approximation algorithm with performance ratio 2. For the case when an upper bound d on the syntenic distance is known, we show that an optimal syntenic sequence can be found in O( nk + 2 o( d 2) ) time, where n and k are the number of chromosomes in the two given genomes. Next, we show that if the set of operations for transforming a genome is significantly restricted, we can nevertheless find a solution that performs at most O(log d) additional moves, where d is the number of moves performed by the unrestricted optimum. This result should help in the design of approximation algorithms. Finally, we investigate the median problem: Given three genomes, construct a genome minimizing the total syntenic distance to the three given genomes and compute the corresponding median distance. The problem has application in the inference of phytogenies based on the syntenic distance. We prove that the problem is NP-hard and design a polynomial time approximation algorithm with a performance ratio of 4+ ε for any constant ε > 0.