The footprint sorting problem.

Phylogenetic footprints are short pieces of noncoding DNA sequence in the vicinity of a gene that are conserved between evolutionary distant species. A seemingly simple problem is to sort footprints in their order along the genomes. It is complicated by the fact that not all footprints are collinear: they may cross each other. The problem thus becomes the identification of the crossing footprints, the sorting of the remaining collinear cliques, and finally the insertion of the noncollinear ones at "reasonable" positions. We show that solving the footprint sorting problem requires the solution of the "Minimum Weight Vertex Feedback Set Problem", which is known to be NP-complete and APX-hard. Nevertheless good approximations can be obtained for data sets of interest. The remaining steps of the sorting process are straightforward: computation of the transitive closure of an acyclic graph, linear extension of the resulting partial order, and finally sorting w.r.t. the linear extension. Alternatively, the footprint sorting problem can be rephrased as a combinatorial optimization problem for which approximate solutions can be obtained by means of general purpose heuristics. Footprint sortings obtained with different methods can be compared using a version of multiple sequence alignment that allows the identification of unambiguously ordered sublists. As an application we show that the rat has a slightly increased insertion/deletion rate in comparison to the mouse genome.