# Vincent Bouchitté's research while affiliated with Ecole normale supérieure de Lyon and other places

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## Publications (24)

We introduce a natural heuristic for approximating the treewidth of graphs. We prove that this heuristic gives a constant factor approximation for the treewidth of graphs with bounded asteroidal number. Using a different technique, we give a approximation algorithm for the treewidth of arbitrary graphs, where k is the treewidth of the input graph.

Robertson and Seymour conjectured that the treewidth of a planar graph and the treewidth of its geometric dual differ by at most one. Lapoire solved the conjecture in the affirmative, using algebraic techniques. We give here a much shorter proof of this result.

Using the specific structure of the minimal separators of AT-free graphs, we give a polynomial time algorithm that computes
a triangulation whose width is no more than twice the treewidth of the input graph.

Robertson and Seymour conjectured that the treewidth of a planar graph and the treewidth of its geometric dual differ by at most one. Lapoire solved the conjecture in the affirmative, using algebraic techniques. We give here a much shorter proof of this result.

We introduce a natural heuristic for approximating the treewidth of graphs. We prove that this heuristic gives a constant factor approximation for the treewidth of graphs with bounded asteroidal number. Using a different technique, we give a $O(\log k)$ approximation algorithm for the treewidth of arbitrary graphs, where $k$ is the treewidth of the...

We use the notion of potential maximal clique to characterize the maximal cliques appearing in minimal triangulations of a graph. We show that if these objects can be listed in polynomial time for a class of graphs, the treewidth and the minimum fill-in are polynomially tractable for these graphs. We prove that for all classes of graphs for which p...

Abstract A potential maximal clique of a graph is a vertex set that induces a maximal clique in some minimal triangulation of that graph. It is known that if these objects can be listed in polynomial time for a class of graphs, the treewidth and the minimum 5ll-in are polynomially tractable for these graphs. We show here that the potential maximal...

Using the specific structure of the minimal separators of AT-free graphs, we give a polynomial time algorithm that computes
a triangulation whose width is no more than twice the treewidth of the input graph.

A potential maximal clique of a graph is a vertex set that induces a maximal clique in some minimal triangulation of that graph. It is known that if these objects can be listed in polynomial time for a class of graphs, the treewidth and the minimum fill-in are polynomially tractable for these graphs. We show here that the potential maximal cliques...

We use the notion of potential maximal clique to characterize the maximal cliques appearing in minimal triangulations of a graph. We show that if these objects can be listed in polynomial time for a class of graphs, the treewidth and the minimum fill-in are polynomially tractable for these graphs. Finally we show how to compute in polynomial time t...

We give a characterization of minimal triangulation of graphs using the notion of “maximal set of neighbor separators”. We prove that if all the maximal sets of neighbor separators of some graphs can be computed in polynomial time, the treewidth of those graphs can be computed in polynomial time. This notion also unifies the already known algorithm...

Partially ordered sets appear as a basic tool in computer science and are particularly accurate for modeling dynamic behavior of complex systems. Motivated by considerations on the diagnosis of distributed computations a new kind of algorithm on posets has been developed and is now widely considered. In this paper we present this “on-line” algorith...

This paper deals with the problem of evaluating HPF style array expressions on massively parallel distributed-memory computers (DMPCs). This problem has been addressed by Chatterjee et al. under the strict hypothesis that computations and communications cannot overlap. As such a model appears to be unnecessarily restrictive for modeling state-of-th...

This chapter deals with the problem of evaluating high performance Fortran style array expressions on massively parallel distributed-memory computers (DMPCs). The difficulty of such an evaluation is to choose the placement of the evaluation order and location of the intermediate results. This problem has been addressed under the strict hypothesis t...

This paper is devoted to the study of contiguity orders i.e. orders having a linear extension L such that all upper (or lower) cover sets are intervals of L. This new class is a strict generalization of both interval orders and N--free orders, and is linearly recognizable. It is proved that computing the number of contiguity extensions is #P-comple...

In event structures, one of the classical models of parallelism, the concept of nice labelling occurs: this consists in attributing label to each event of the structure, in such a way that two different events may have the same label if either they are in temporal causality or they are not the initial occurrences of incompatible actions. The proble...

In this note we prove that any finite interval order P can be represented in the plane by using translations of line segments, each one having a single direction of motion. We state that |Succ(P)| different directions are sufficient.

Some orders can be represented by translating convex figures in the plane. It is proved thatN-free and interval orders admit such representations with an unbounded number of directions. Weak orders, tree-like orders and two-dimensional orders of height one are shown to be two- directional. In all cases line segments can be used as convex sets.

Certains ensembles ordonnes peuvent être représentés par des translations de figures convexes du plan. Il est prouvé, dans cette note, que les ordres sans N et les ordres d'intervalles admettent une telle représentation mais que le nombre de directions ne peut être borné. En revanche, les arbres sont représentables avec seulement deux directions. P...

In this paper we survey complexity results yielded by the calculation of untractable invariants of ordered sets dealing with the computation of linear extensions. The main invariants we consider here are the dimension, the jump number, and the number of linear extensions.
First we investigate recent NP-completeness results, and then consider some r...

Following the pioneering work of Kierstead, we present here some complexity results about the construction of depth-first greedy linear extensions. We prove that the recognition of Dilworth partially ordered sets of height 2, as defined by Syslo, is NP-complete. This last result yields a new proof of the NP-completeness of the jump number problem,...

Nous étudions le comportement des extensions linéaires au travers de deux invariants de comparabilité: le nombre de sauts et la dimension. La reconnaissance des ordres de Dilworth est montrée comme étant NP-complète, nous donnons des algorithmes polynomiaux pour résoudre ce problème sur deux sous-classes. Nous définissons les notions de dimension g...

This paper introduces a new concept of dimension for partially ordered sets. Dushnik and Miller in 1941 introduced the concept of dimension of a partial order P, as the minimum cardinality of a realizer, (i.e., a set of linear extensions of P whose intersection is P). Every poset has a greedy realizer (i.e., a realizer consisting of greedy linear e...

We prove that a bipartite graph is chordal if and only if it has an elimination scheme. This leads to a polynomial algorithm to recognize whether an ordered set is cycle-free.

## Citations

... Finally, S is a minimal separator if S is a u,v-minimal separator for some pair of vertices u and v. Minimal separators have a tremendous role in the design of graph algorithms, both directly, such as in the structural characterization of chordal graphs [5] but also indirectly in optimization algorithms for graph separation and routing problems (for example [17,16,21]). The theory of potential maximal cliques, developed by Bouchitté and Todinca [4] implies that a several fundamental graph problems, such as computing the treewidth and minimum fill in of a graph G can be done in time polynomial in the number of vertices of G and the number of minimal separators in G. Lokshtanov [15] showed that the same result holds for computing the tree-length of the graph G, while Fomin et al. [10] proved a general result that showed that a whole class of problems (including e.g. maximum independent set and minimum feedback vertex set) can be solved in time polynomial in the number of vertices and minimal separators of the graph. ...

... Complexité Dans [BH89] les auteurs conjecturent que le problème de comptage des extensions linéaires d'un ordre sans cycle est de complexité polynomiale. Nous avons montré que ce calcul pouvait se faire en évaluant une formule intégrale de taille linéaire en la taille de l'ordre en entrée. ...

... 18 V. Bouchitt e, A. Hilali, R. J egou, J.X. Rampon With Proposition 2 and condition (iv) of Theorem 10, we get a new proof of the lower{contiguity of interval orders stated in 4] . Moreover, adding condition (iii), we immediately have: P = (X; P ) is a lattice ii any two element subset of X has an innmum and a supremum in P. The innmum (resp. ...

Reference: Contiguity Orders

... We recall that if y x, then there exists a greedy linear extension that puts x before y [2]. An α-greedy balanced pair in P = (V, ≤) is a pair (x, y) of elements of V such that the ratio of greedy linear extensions of P that put x before y among all greedy linear extensions, denoted GP P (x < y), is in the real interval [α, 1 − α]. ...

... l Using the proof of the claim: "For any positive integer m, there is an ordered set with no m-directional representation" given in [S], we show in [2] that for every integer k there exists an interval order which is not k-directional. l These new notions which can model separability problems in computational geometry seem to be an attractive domain for further investigations. ...

Reference: On the Directionality of Interval Orders.

... , h k ; h 1 , . . . , h k } whose only edges are h i h j when there is an arc from h i to h j in H ; also see [1]. If H is transitively oriented, then [4] shows that H is weakly chordal if and only if its bipartite transform is chordal bipartite. ...

... Maximizing the jump number has many efficient, and some elegant, polynomial time algorithms (this is the bump number problem) [10]. Minimizing the number of jumps is called the jump number problem, and is NP-hard [11,1]. Exact algorithms for jump number have recently been found to run in O(1.824 n ) [5], slightly improving the previous bound of Kratsch and Kratsch's algorithm running in O(1.8638 n ) [6]; the latter also provide a O(1.7593 n )-time algorithm for interval posets. ...

... First, by leveraging on classic results from [39], we show that the problem of constructing a ρ-consistent path decomposition of approximately minimum width for the cocomparability graph G ρ of a given partial-order ρ is fixed-parameter tractable with respect to the pathwidth of G ρ . While it was known that the pathwidth and the ρ-consistent pathwidth of G ρ are always the same [3], and that there were fixed-parameter tractable algorithms for computing path decompositions of approximately minimum width due to structural properties of cocomparability graphs [8], the problem of computing such a decomposition satisfying the additional ρ-consistent requirement was open [3]. ...

... No hardness of approximation is known and not even the possibility of a polynomial-time approximation scheme for Treewidth has been ruled out. In many important special classes of graphs, such as planar graphs (Seymour & Thomas, 1994), asteroidal triple-free graphs (Bouchitté & Todinca, 2003), and H-minor-free graphs (Feige et al., 2005), constant factor approximations are known, but the general case has remained elusive. ...

... The second phase of the PMC framework has already been formulated for all of the problems that we consider [15,17,18,21,35], so our O * (3 cc ) time algorithms follow from an O * (3 cc ) time PMC enumeration algorithm that we give. This algorithm is based on the Bouchitté-Todinca algorithm [6]. We achieve the O * (3 cc ) bound by novel characterizations of minimal separators and PMCs with respect to an edge clique cover. ...