The zero exemplar distance problem

Given two genomes with duplicate genes, \textsc{Zero Exemplar Distance} is the problem of deciding whether the two genomes can be reduced to the same genome without duplicate genes by deleting all but one copy of each gene in each genome. Blin, Fertin, Sikora, and Vialette recently proved that \textsc{Zero Exemplar Distance} for monochromosomal genomes is NP-hard even if each gene appears at most two times in each genome, thereby settling an important open question on genome rearrangement in the exemplar model. In this paper, we give a very simple alternative proof of this result. We also study the problem \textsc{Zero Exemplar Distance} for multichromosomal genomes without gene order, and prove the analogous result that it is also NP-hard even if each gene appears at most two times in each genome. For the positive direction, we show that both variants of \textsc{Zero Exemplar Distance} admit polynomial-time algorithms if each gene appears exactly once in one genome and at least once in the other genome. In addition, we present a polynomial-time algorithm for the related problem \textsc{Exemplar Longest Common Subsequence} in the special case that each mandatory symbol appears exactly once in one input sequence and at least once in the other input sequence. This answers an open question of Bonizzoni et al. We also show that \textsc{Zero Exemplar Distance} for multichromosomal genomes without gene order is fixed-parameter tractable if the parameter is the maximum number of chromosomes in each genome.Comment: Strengthened and reorganize

Author manuscript, published in "Proc. 3rd Workshop on Algorithms and Computation (WALCOM 2009), Kolkata: Inde (2009)" The Exemplar Breakpoint Distance for non-trivial genomes cannot be approximated

Abstract. A promising and active field of comparative genomics consists in comparing two genomes by establishing a one-to-one correspondence (i.e., a matching) between their genes. This correspondence is usually chosen in order to optimize a predefined measure. One such problem is the Exemplar Breakpoint Distance problem (or EBD, for short), which asks, given two genomes modeled by signed sequences of characters, to keep and match exactly one occurrence of each character in the two genomes (a process called exemplarization), so as to minimize the number of breakpoints of the resulting genomes. Bryant [6] showed that EBD is NP-complete. In this paper, we close the study of the approximation of EBD by showing that no approximation factor can be derived for EBD considering non-trivial genomes – i.e. genomes that contain duplicated genes.