Christophe Paul

Lex-BFS and partition refinement, with applications to transitive orientation, interval graph recognition and consecutive ones testing

By making use of lexicographic breadth rst search (Lex-BFS) and partition renement with pivots, we obtain very simple algorithms for some well-known problems in graph theory. We give a O(n+mlogn) algorithm for transitive orientation of a comparability graph, and simple linear algorithms to recognize interval graphs, convex graphs, Y-semichordal graphs and matrices that have the consecutive ones property. Previous approaches to these problems used dicult preprocessing steps, such as computing PQ-trees or modular decomposition. The algorithms we give are easy to understand and straightforward to prove. They do not make use of sophisticated data structures, and the complexity analysis is straightforward. c 2000 Elsevier Science B.V. All rights reserved.

Survey: A survey of the algorithmic aspects of modular decomposition

Modular decomposition is a technique that applies to (but is not restricted to) graphs. The notion of a module naturally appears in the proofs of many graph theoretical theorems. Computing the modular decomposition tree is an important preprocessing step to solve a large number of combinatorial optimization problems. Since the first polynomial time algorithm in the early 1970s, the algorithmic of the modular decomposition has known an important development. This paper survey the ideas and techniques that arose from this line of research.

Approximate Distance Labeling Schemes

We consider the problem of labeling the nodes of an n-node graph G with short labels in such a way that the distance between any two nodes u, v of G can be approximated efficiently (in constant time) by merely inspecting the labels of u and v, without using any other information. We develop such constant approximate distance labeling schemes for the classes of trees, bounded treewidth graphs, planar graphs, k-chordal graphs, and graphs with a dominating pair (including for instance interval, permutation, and AT-free graphs). We also establish lower bounds, and prove that most of our schemes are optimal in terms of the length of the labels generated and the quality of the approximation.

Simpler Linear-Time Modular Decomposition Via Recursive Factorizing Permutations

Modular decomposition is fundamental for many important problems in algorithmic graph theory including transitive orientation, the recognition of several classes of graphs, and certain combinatorial optimization problems. Accordingly, there has been a drive towards a practical, linear-time algorithm for the problem. This paper posits such an algorithm; we present a linear-time modular decomposition algorithm that proceeds in four straightforward steps. This is achieved by introducing the notion of factorizing permutations to an earlier recursive approach. The only data structure used is an ordered list of trees, and each of the four steps amounts to simple traversals of these trees. Previous algorithms were either exceedingly complicated or resorted to impractical data-structures.

PARTITION REFINEMENT TECHNIQUES: AN INTERESTING ALGORITHMIC TOOL KIT

Partition refinement techniques lead to simple and efficient algorithms for various applications: automaton minimization, string sorting … and also for algorithms on graphs. A generic algorithm that can be used for all these applications is presented and briefly discussed. Such an approach is interesting in an algorithmic tool kit perspective. New instances of the generic algorithm are presented: O(n+mlog n) very simple and practical algorithms for cographs recognition and for modular decomposition are designed. Although these algorithms are not linear, they are very easy to implement (as an instanciation of a generic procedure) and based on new ideas.

Perfect Sorting by Reversals Is Not Always Difficult

We propose new algorithms for computing pairwise rearrangement scenarios that conserve the combinatorial structure of genomes. More precisely, we investigate the problem of sorting signed permutations by reversals without breaking common intervals. We describe a combinatorial framework for this problem that allows us to characterize classes of signed permutations for which one can compute, in polynomial time, a shortest reversal scenario that conserves all common intervals. In particular, we define a class of permutations for which this computation can be done in linear time with a very simple algorithm that does not rely on the classical Hannenhalli-Pevzner theory for sorting by reversals. We apply these methods to the computation of rearrangement scenarios between permutations obtained from 16 synteny blocks of the X chromosomes of the human, mouse, and rat

Computing galled networks from real data

Motivation: Developing methods for computing phylogenetic networks from biological data is an important problem posed by molecular evolution and much work is currently being undertaken in this area. Although promising approaches exist, there are no tools available that biologists could easily and routinely use to compute rooted phylogenetic networks on real datasets containing tens or hundreds of taxa. Biologists are interested in clades, i.e. groups of monophyletic taxa, and these are usually represented by clusters in a rooted phylogenetic tree. The problem of computing an optimal rooted phylogenetic network from a set of clusters, is hard, in general. Indeed, even the problem of just determining whether a given network contains a given cluster is hard. Hence, some researchers have focused on topologically restricted classes of networks, such as galled trees and level-k networks, that are more tractable, but have the practical draw-back that a given set of clusters will usually not possess such a representation. Results: In this article, we argue that galled networks (a generalization of galled trees) provide a good trade-off between level of generality and tractability. Any set of clusters can be represented by some galled network and the question whether a cluster is contained in such a network is easy to solve. Although the computation of an optimal galled network involves successively solving instances of two different NP-complete problems, in practice our algorithm solves this problem exactly on large datasets containing hundreds of taxa and many reticulations in seconds, as illustrated by a dataset containing 279 prokaryotes. Availability: We provide a fast, robust and easy-to-use implementation of this work in version 2.0 of our tree-handling software Dendroscope, freely available from http://www.dendroscope.org. Contact: huson@informatik.uni-tuebingen.de

A simple paradigm for graph recognition: application to cographs and distance hereditary graphs

An easy way for graph recognition algorithms is to use a two-step process: first, compute a characteristic feature as if the graph belongs to that class; second, check whether the computed characteristic feature as if the graph belongs to that class; second, check whether the computed separating them may yield new and much more easily understood algorithms. In this paper we apply that paradigm to the cograph and distance hereditary graph recognition problems.

Chordal Graphs and Their Clique Graphs

In the first part of this paper, a new structure for chordal graph is introduced, namely the clique graph. This structure is shown to be optimal with regard to the set of clique trees. The greedy aspect of the recognition algorithms of chordal graphs is studied. A new greedy algorithm that generalizes both Maximal cardinality Search (MCS) and Lexicographic Breadth first search is presented. The trace of an execution of MCS is defined and used in two linear time and space algorithms: one builds a clique tree of a chordal graph and the other is a simple recognition procedure of chordal graphs.

Linear kernels and single-exponential algorithms via protrusion decompositions

We present a linear-time algorithm to compute a decomposition scheme for graphs G that have a set X⊆V(G), called a treewidth-modulator, such that the treewidth of G−X is bounded by a constant. Our decomposition, called a protrusion decomposition, is the cornerstone in obtaining the following two main results. Our first result is that any parameterized graph problem (with parameter k) that has finite integer index and such that positive instances have a treewidth-modulator of size O(k) admits a linear kernel on the class of H-topological-minor-free graphs, for any fixed graph H. This result partially extends previous meta-theorems on the existence of linear kernels on graphs of bounded genus and H-minor-free graphs. Let