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Ubiquitous Parameterization  Invitation to FixedParameter Algorithms
 In Proc. 29th MFCS, volume 3153 of LNCS
, 2004
"... Problem parameters are ubiquitous. In every area of computer science, we find all kinds of "special aspects" to the problems encountered. ..."
Abstract

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Problem parameters are ubiquitous. In every area of computer science, we find all kinds of "special aspects" to the problems encountered.
Breakpoint Distance and PQTrees
"... Abstract. The PQtree is a fundamental data structure that can encode large sets of permutations. It has recently been used in comparative genomics to model ancestral genomes with some uncertainty: given a phylogeny for some species, extant genomes are represented by permutations on the leaves of th ..."
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Abstract. The PQtree is a fundamental data structure that can encode large sets of permutations. It has recently been used in comparative genomics to model ancestral genomes with some uncertainty: given a phylogeny for some species, extant genomes are represented by permutations on the leaves of the tree, and each internal node in the phylogenetic tree represents an extinct ancestral genome, represented by a PQtree. An open problem related to this approach is then to quantify the evolution between genomes represented by PQtrees. In this paper we present results for two problems of PQtree comparison motivated by this application. First, we show that the problem of comparing two PQtrees by computing the minimum breakpoint distance among all pairs of permutations generated respectively by the two considered PQtrees is NPcomplete for unsigned permutations. Next, we consider a generalization of the classical Breakpoint Median problem, where an ancestral genome is represented by a PQtree and p permutations are given, with p ≥ 1, and we want to compute a permutation generated by the PQtree that minimizes the sum of the breakpoint distances to the p permutations. We show that this problem is FixedParameter Tractable with respect to the breakpoint distance value. This last result applies both on signed and unsigned permutations, and to unichromosomal and multichromosomal permutations. 1