Laurent Bulteau

Sorting by Transpositions Is Difficult

In comparative genomics, a transposition is an operation that exchanges two consecutive sequences of genes in a genome. The transposition distance between two genomes, that is, the minimum number of transpositions needed to transform a genome into another, is, according to numerous studies, a relevant evolutionary distance. The problem of computing this distance when genomes are represented by permutations is called the Sorting by Transpositions problem, and has been introduced by Bafna and Pevzner in [Proceedings of the Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, 1995, pp. 614--623]. It has naturally been the focus of a number of studies (see, for instance, [G. Fertin, A. Labarre, I. Rusu, E. Tannier, and S. Vialette, Combinatorics of Genome Rearrangements, The MIT Press, Cambridge, MA, 2009]), but the computational complexity of this problem has remained undetermined for 15 years. In this paper, we answer this long-standing open question by proving that the Sorting by Transpositions problem ...

Pancake flipping is hard

Pancake Flipping is the problem of sorting a stack of pancakes of different sizes (that is, a permutation), when the only allowed operation is to insert a spatula anywhere in the stack and to flip the pancakes above it (that is, to perform a prefix reversal). In the burnt variant, one side of each pancake is marked as burnt, and it is required to finish with all pancakes having the burnt side down. Computing the optimal scenario for any stack of pancakes and determining the worst-case stack for any stack size have been challenges over more than three decades. Beyond being an intriguing combinatorial problem in itself, it also yields applications, e.g. in parallel computing and computational biology. In this paper, we show that the Pancake Flipping problem, in its original (unburnt) variant, is NP-hard, thus answering the long-standing question of its computational complexity.

Pancake Flipping is hard

Pancake Flipping is the problem of sorting a stack of pancakes of different sizes (that is, a permutation), when the only allowed operation is to insert a spatula anywhere in the stack and to flip the pancakes above it (that is, to perform a prefix reversal). In the burnt variant, one side of each pancake is marked as burnt, and it is required to finish with all pancakes having the burnt side down. Computing the optimal scenario for any stack of pancakes and determining the worst-case stack for any stack size have been challenges for over more than three decades. Beyond being an intriguing combinatorial problem in itself, it also yields applications, e.g. in parallel computing and computational biology. In this paper, we show that the Pancake Flipping problem, in its original (unburnt) variant, is NP-hard, thus answering the long-standing question of its computational complexity.

Maximal strip recovery problem with gaps: Hardness and approximation algorithms

Given two comparative maps, that is two sequences of markers each representing a genome, the Maximal Strip Recovery problem (MSR) asks to extract a largest sequence of markers from each map such that the two extracted sequences are decomposable into non-intersecting strips (or synteny blocks). This aims at defining a robust set of synteny blocks between different species, which is a key to understand the evolution process since their last common ancestor. In this paper, we add a fundamental constraint to the initial problem, which expresses the biologically sustained need to bound the number of intermediate (non-selected) markers between two consecutive markers in a strip. We therefore introduce the problem @d-gap-MSR, where @d is a (usually small) non-negative integer that upper bounds the number of non-selected markers between two consecutive markers in a strip. We show that, if we restrict ourselves to comparative maps without duplicates, the problem is polynomial for @d=0, NP-complete for @d=1, and APX-hard for @d>=2. For comparative maps with duplicates, the problem is APX-hard for all @d>=0.

Sorting by transpositions is difficult

In comparative genomics, a transposition is an operation that exchanges two consecutive sequences of genes in a genome. The transposition distance, that is, the minimum number of transpositions needed to transform a genome into another, can be considered as a relevant evolutionary distance. The problem of computing this distance when genomes are represented by permutations, called the SORTING BY TRANSPOSITIONS problem (SBT), has been introduced by Bafna and Pevzner [3] in 1995. It has naturally been the focus of a number of studies, but the computational complexity of this problem has remained undetermined for 15 years. In this paper, we answer this long-standing open question by proving that the SORTING BY TRANSPOSITIONS problem is NP-hard. As a corollary of our result, we also prove that the following problem from [10] is NP-hard: given a permutation π, is it possible to sort π using db(π)/3 permutations, where db(π) is the number of breakpoints of π?

Maximal Strip Recovery Problem with Gaps: Hardness and Approximation Algorithms

Given two comparative maps, that is two sequences of markers each representing a genome, the Maximal Strip Recovery problem (MSR) asks to extract a largest sequence of markers from each map such that the two extracted sequences are decomposable into non-overlapping strips (or synteny blocks). This aims at defining a robust set of synteny blocks between different species, which is a key to understand the evolution process since their last common ancestor. In this paper, we add a fundamental constraint to the initial problem, which expresses the biologically sustained need to bound the number of intermediate (non-selected) markers between two consecutive markers in a strip. We therefore introduce the problem ?-gap-MSR, where ? is a (usually small) non-negative integer that upper bounds the number of non-selected markers between two consecutive markers in a strip. Depending on the nature of the comparative maps (i.e., with or without duplicates), we show that ?-gap-MSR is NP-complete for any ? ? 1, and even APX-hard for any ? ? 2. We also provide two approximation algorithms, with ratio 1.8 for ?= 1, and ratio 4 for ? ? 2.

A Fixed-Parameter Algorithm for Minimum Common String Partition with Few Duplications

Motivated by the study of genome rearrangements, the NP-hard Minimum Common String Partition problems asks, given two strings, to split both strings into an identical set of blocks. We consider an extension of this problem to unbalanced strings, so that some elements may not be covered by any block. We present an efficient fixed-parameter algorithm for the parameters number k of blocks and maximum occurrence d of a letter in either string. We then evaluate this algorithm on bacteria genomes and synthetic data.

Minimum common string partition parameterized by partition size is fixed-parameter tractable

The NP-hard Minimum Common String Partition problem asks whether two strings x and y can each be partitioned into at most k substrings such that both partitions use exactly the same substrings in a different order. We present the first fixed-parameter algorithm for Minimum Common String Partition using only parameter k.

Tractability and approximability of maximal strip recovery

An essential task in comparative genomics is to decompose two or more genomes into synteny blocks that are segments of chromosomes with similar contents. Given a set of d genomic maps each containing the same n markers without duplicates, the problem Maximal Strip Recovery (MSR) aims at finding a decomposition of the genomic maps into synteny blocks (strips) of the maximum total length @?, by deleting the minimum number k=n-@? of markers which are probably noise and ambiguities. In this paper, we present a collection of new or improved FPT and approximation algorithms for MSR and its variants. Our main results include a 2^O^(^d^@d^@?^)poly(nd) time FPT algorithm for @d-gap-MSR-d, a 2.36^kpoly(nd) time FPT algorithm for both CMSR-d and @d-gap-CMSR-d, and a (d+1.5)-approximation algorithm for both CMSR-d and @d-gap-CMSR-d.

Precedence-Constrained Scheduling Problems Parameterized by Partial Order Width

Negatively answering a question posed by Mnich and Wiese (Math. Program. 154(1-2):533-562), we show that P2|prec,