Christophe Paul

University of Paris-Est, Centre, France

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Publications (115)27.22 Total impact

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    ABSTRACT: The Hadwiger number of a graph G is the largest integer h such that G has the complete graph K_h as a minor. We show that the problem of determining the Hadwiger number of a graph is NP-hard on co-bipartite graphs, but can be solved in polynomial time on cographs and on bipartite permutation graphs. We also consider a natural generalization of this problem that asks for the largest integer h such that G has a minor with h vertices and diameter at most $s$. We show that this problem can be solved in polynomial time on AT-free graphs when s>=2, but is NP-hard on chordal graphs for every fixed s>=2.
    06/2014;
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    ABSTRACT: The Outerplanar Diameter Improvement problem asks, given a graph $G$ and an integer $D$, whether it is possible to add edges to $G$ in a way that the resulting graph is outerplanar and has diameter at most $D$. We provide a dynamic programming algorithm that solves this problem in polynomial time. Outerplanar Diameter Improvement demonstrates several structural analogues to the celebrated and challenging Planar Diameter Improvement problem, where the resulting graph should, instead, be planar. The complexity status of this latter problem is open.
    03/2014;
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    ABSTRACT: Several algorithmic meta-theorems on kernelization have appeared in the last years, starting with the result of Bodlaender et al. [FOCS 2009] on graphs of bounded genus, then generalized by Fomin et al. [SODA 2010] to graphs excluding a fixed minor, and by Kim et al. [ICALP 2013] to graphs excluding a fixed topological minor. Typically, these results guarantee the existence of linear or polynomial kernels on sparse graph classes for problems satisfying some generic conditions but, mainly due to their generality, it is not clear how to derive from them constructive kernels with explicit constants. In this paper we make a step toward a fully constructive meta-kernelization theory on sparse graphs. Our approach is based on a more explicit protrusion replacement machinery that, instead of expressibility in CMSO logic, uses dynamic programming, which allows us to find an explicit upper bound on the size of the derived kernels. We demonstrate the usefulness of our techniques by providing the first explicit linear kernels for r-Dominating Set and r-Scattered Set on apex-minor-free graphs, and for Planar-F-Deletion on graphs excluding a fixed (topological) minor in the case where all the graphs in F are connected.
    12/2013;
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    Philippe Gambette, Vincent Berry, Christophe Paul
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    ABSTRACT: Phylogenetic networks were introduced to describe evolution in the presence of exchanges of genetic material between coexisting species or individuals. Split networks in particular were introduced as a special kind of abstract network to visualize conflicts between phylogenetic trees which may correspond to such exchanges. More recently, methods were designed to reconstruct explicit phylogenetic networks (whose vertices can be interpreted as biological events) from triplet data. In this article, we link abstract and explicit networks through their combinatorial properties, by introducing the unrooted analog of level-k networks. In particular, we give an equivalence theorem between circular split systems and unrooted level-1 networks. We also show how to adapt to quartets some existing results on triplets, in order to reconstruct unrooted level-k phylogenetic networks. These results give an interesting perspective on the combinatorics of phylogenetic networks and also raise algorithmic and combinatorial questions.
    Journal of Bioinformatics and Computational Biology 08/2012; 10(4):1250004. · 0.93 Impact Factor
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    ABSTRACT: A \emph{$t$-treewidth-modulator} of a graph $G$ is a set $X \subseteq V(G)$ such that the treewidth of $G-X$ is at most some constant $t-1$. In this paper, we present a novel algorithm to compute a decomposition scheme for graphs $G$ that come equipped with a $t$-treewidth-modulator. This decomposition, called a \emph{protrusion decomposition}, is the cornerstone in obtaining the following two main results. We first show that any parameterized graph problem (with parameter $k$) that has \emph{finite integer index} and is \emph{treewidth-bounding} admits a linear kernel on $H$-topological-minor-free graphs, where $H$ is some arbitrary but fixed graph. A parameterized graph problem is called treewidth-bounding if all positive instances have a $t$-treewidth-modulator of size $O(k)$, for some constant $t$. This result partially extends previous meta-theorems on the existence of linear kernels on graphs of bounded genus [Bodlaender et al., FOCS 2009] and $H$-minor-free graphs [Fomin et al., SODA 2010]. Our second application concerns the Planar-$\mathcal{F}$-Deletion problem. Let $\mathcal{F}$ be a fixed finite family of graphs containing at least one planar graph. Given an $n$-vertex graph $G$ and a non-negative integer $k$, Planar-$\mathcal{F}$-Deletion asks whether $G$ has a set $X\subseteq V(G)$ such that $|X|\leq k$ and $G-X$ is $H$-minor-free for every $H\in \mathcal{F}$. Very recently, an algorithm for Planar-$\mathcal{F}$-Deletion with running time $2^{O(k)} n \log^2 n$ (such an algorithm is called \emph{single-exponential}) has been presented in [Fomin et al., FOCS 2012] under the condition that every graph in $\mathcal{F}$ is connected. Using our algorithm to construct protrusion decompositions as a building block, we get rid of this connectivity constraint and present an algorithm for the general Planar-$\mathcal{F}$-Deletion problem running in time $2^{O(k)} n^2$.
    07/2012;
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    ABSTRACT: A circle graph is the intersection graph of a set of chords in a circle. Keil [Discrete Applied Mathematics, 42(1):51-63, 1993] proved that Dominating Set, Connected Dominating Set, and Total Dominating Set are NP-complete in circle graphs. To the best of our knowledge, nothing was known about the parameterized complexity of these problems in circle graphs. In this paper we prove the following results, which contribute in this direction: - Dominating Set, Independent Dominating Set, Connected Dominating Set, Total Dominating Set, and Acyclic Dominating Set are W[1]-hard in circle graphs, parameterized by the size of the solution. - Whereas both Connected Dominating Set and Acyclic Dominating Set are W[1]-hard in circle graphs, it turns out that Connected Acyclic Dominating Set is polynomial-time solvable in circle graphs. - If T is a given tree, deciding whether a circle graph has a dominating set isomorphic to T is NP-complete when T is in the input, and FPT when parameterized by |V(T)|. We prove that the FPT algorithm is subexponential.
    Theory of Computing Systems 05/2012; · 0.48 Impact Factor
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    Eun Jung Kim, Christophe Paul, Geevarghese Philip
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    ABSTRACT: Given an input graph G and an integer k, the parameterized K_4-minor cover problem asks whether there is a set S of at most k vertices whose deletion results in a K_4-minor-free graph, or equivalently in a graph of treewidth at most 2. This problem is inspired by two well-studied parameterized vertex deletion problems, Vertex Cover and Feedback Vertex Set, which can also be expressed as Treewidth-t Vertex Deletion problems: t=0 for Vertex Cover and t=1 for Feedback Vertex Set. While a single-exponential FPT algorithm has been known for a long time for \textsc{Vertex Cover}, such an algorithm for Feedback Vertex Set was devised comparatively recently. While it is known to be unlikely that Treewidth-t Vertex Deletion can be solved in time c^{o(k)}.n^{O(1)}, it was open whether the K_4-minor cover problem could be solved in single-exponential FPT time, i.e. in c^k.n^{O(1)} time. This paper answers this question in the affirmative.
    Journal of Computer and System Sciences 04/2012; · 1.00 Impact Factor
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    ABSTRACT: Vertex deletion and edge deletion problems play a central role in Parameterized Complexity. Examples include classical problems like Feedback Vertex Set, Odd Cycle Transversal, and Chordal Deletion. Interestingly, the study of edge contraction problems of this type from a parameterized perspective has so far been left largely un-explored. We consider two basic edge contraction problems, which we call Path-Contractibility and Tree-Contractibility. Both prob-lems take an undirected graph G and an integer k as input, and the task is to determine whether we can obtain a path or an acyclic graph, respectively, by contracting at most k edges of G. Our main contribu-tion is an algorithm with running time 4 k+O(log 2 k) + n O(1) for Path-Contractibility and an algorithm with running time 4.88 k n O(1) for Tree-Contractibility, based on a novel application of the color cod-ing technique of Alon, Yuster and Zwick. Furthermore, we show that Path-Contractibility has a kernel with at most 5k + 3 vertices, while Tree-Contractibility does not have a polynomial kernel unless coNP ⊆ NP/poly. We find the latter result surprising, because of the strong connection between Tree-Contractibility and Feedback Vertex Set, which is known to have a vertex kernel with size O(k 2).
    Algorithmica 05/2011; · 0.49 Impact Factor
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    ABSTRACT: The "c-pumpkin" is the graph with two vertices linked by c>0 parallel edges. A c-pumpkin-model in a graph G is a pair A,B of disjoint subsets of vertices of G, each inducing a connected subgraph of G, such that there are at least c edges in G between A and B. We focus on covering and packing c-pumpkin-models in a given graph: On the one hand, we provide an FPT algorithm running in time 2^O(k) n^O(1) deciding, for any fixed c>0, whether all c-pumpkin-models can be covered by at most k vertices. This generalizes known single-exponential FPT algorithms for Vertex Cover and Feedback Vertex Set, which correspond to the cases c=1,2 respectively. On the other hand, we present a O(log n)-approximation algorithm for both the problems of covering all c-pumpkin-models with a smallest number of vertices, and packing a maximum number of vertex-disjoint c-pumpkin-models.
    Computing Research Repository - CORR. 05/2011;
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    ABSTRACT: Split decomposition of graphs was introduced by Cunningham (under the name join decomposition) as a generalization of the modular decomposition. This paper undertakes an investigation into the algorithmic properties of split decomposition. We do so in the context of graph-labelled trees (GLTs), a new combinatorial object designed to simplify its consideration. GLTs are used to derive an incremental characterization of split decomposition, with a simple combinatorial description, and to explore its properties with respect to Lexicographic Breadth-First Search (LBFS). Applying the incremental characterization to an LBFS ordering results in a split decomposition algorithm that runs in time $O(n+m)\alpha(n+m)$, where $\alpha$ is the inverse Ackermann function, whose value is smaller than 4 for any practical graph. Compared to Dahlhaus' linear-time split decomposition algorithm [Dahlhaus'00], which does not rely on an incremental construction, our algorithm is just as fast in all but the asymptotic sense and full implementation details are given in this paper. Also, our algorithm extends to circle graph recognition, whereas no such extension is known for Dahlhaus' algorithm. The companion paper [Gioan et al.] uses our algorithm to derive the first sub-quadratic circle graph recognition algorithm.
    Algorithmica 04/2011; · 0.49 Impact Factor
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    ABSTRACT: Circle graphs are the intersection graphs of chords in a circle. This paper presents the first sub-quadratic recognition algorithm for the class of circle graphs. Our algorithm is O(n + m) times the inverse Ackermann function, {\alpha}(n + m), whose value is smaller than 4 for any practical graph. The algorithm is based on a new incremental Lexicographic Breadth-First Search characterization of circle graphs, and a new efficient data-structure for circle graphs, both developed in the paper. The algorithm is an extension of a Split Decomposition algorithm with the same running time developed by the authors in a companion paper.
    Algorithmica 04/2011; · 0.49 Impact Factor
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    ABSTRACT: We initiate the study of the Bipartite Contraction problem from the perspective of parameterized complexity. In this problem we are given a graph $G$ and an integer $k$, and the task is to determine whether we can obtain a bipartite graph from $G$ by a sequence of at most $k$ edge contractions. Our main result is an $f(k) n^{O(1)}$ time algorithm for Bipartite Contraction. Despite a strong resemblance between Bipartite Contraction and the classical Odd Cycle Transversal (OCT) problem, the methods developed to tackle OCT do not seem to be directly applicable to Bipartite Contraction. Our algorithm is based on a novel combination of the irrelevant vertex technique, introduced by Robertson and Seymour, and the concept of important separators. Both techniques have previously been used as key components of algorithms for fundamental problems in parameterized complexity. However, to the best of our knowledge, this is the first time the two techniques are applied in unison.
    SIAM Journal on Discrete Mathematics 02/2011; · 0.66 Impact Factor
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    Christophe Paul, Anthony Perez, Stéphan Thomassé
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    ABSTRACT: We develop a technique that we call Conflict Packing in the context of kernelization. We illustrate this technique on several well-studied problems: Feedback Arc Set in Tournaments, Dense Rooted Triplet Inconsistency and Betweenness in Tournaments. For the former, one is given a tournament T = (V,A) and seeks a set of at most k arcs whose reversal in T leads to an acyclic tournament. While a linear vertex-kernel is already known for this problem [6], using the Conflict Packing allows us to find a so-called safe partition, the central tool of the kernelization algorithm in [6], with simpler arguments. Regarding the Dense Rooted Triplet Inconsistency problem, one is given a set of vertices V and a dense collection R of rooted binary trees over three vertices of V and seeks a rooted tree over V containing all but at most k triplets from R. Using again the Conflict Packing, we prove that the Dense Rooted Triplet Inconsistency problem admits a linear vertex-kernel. This result improves the best known bound of O(k^2) vertices for this problem [19]. Finally, we use this technique to obtain a linear vertex-kernel for Betweenness in Tournaments, where one is given a set of vertices V and a dense collection R of betweenness triplets and seeks an ordering containing all but at most k triplets from R. To the best of our knowledge this result constitutes the rst polynomial kernel for the problem.
    Computing Research Repository - CORR. 01/2011;
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    ABSTRACT: Some of the most well studied problems in algorithmic graph theory deal with modifying a graph into an acyclic graph or into a path, using as few operations as possible. In Feedback Vertex Set and Longest Induced Path, the only allowed operation is vertex deletion, and in Spanning Tree and Longest Path, only edge deletions are permitted. We study the edge contraction variant of these problems: given a graph G and an integer k, decide whether G can be transformed into an acyclic graph or into a path using at most k edge contractions. Both problems are known to be NP-complete in general. We show that on chordal graphs these problems can be solved in O(n+m) and O(nm) time, respectively. On the negative side, both problems remain NP-complete when restricted to bipartite input graphs.
    Discrete Applied Mathematics 01/2011; 37:87-92. · 0.72 Impact Factor
  • Christophe Paul, Anthony Perez, Stéphan Thomassé
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    ABSTRACT: We develop a technique that we call Conflict Packing in the context of kernelization [7]. We illustrate this technique on several well-studied problems: Feedback Arc Set in Tournaments, Dense Rooted Triplet Inconsistency and Betweenness in Tournaments. For the former, one is given a tournament T = (V,A) and seeks a set of at most k arcs whose reversal in T results in an acyclic tournament. While a linear vertex-kernel is already known for this problem [6], using the Conflict Packing allows us to find a so-called safe partition, the central tool of the kernelization algorithm in [6], with simpler arguments. Regarding the Dense Rooted Triplet Inconsistency problem, one is given a set of vertices V and a dense collection R\mathcal{R} of rooted binary trees over three vertices of V and seeks a rooted tree over V containing all but at most k triplets from R\mathcal{R}. Using again the Conflict Packing, we prove that the Dense Rooted Triplet Inconsistency problem admits a linear vertex-kernel. This result improves the best known bound of O(k 2) vertices for this problem [16]. Finally, we use this technique to obtain a linear vertex-kernel for Betweenness in Tournaments, where one is given a set of vertices V and a dense collection R\mathcal{R} of betweenness triplets and seeks an ordering containing all but at most k triplets from R\mathcal{R}. To the best of our knowledge this result constitutes the first polynomial kernel for the problem.
    Mathematical Foundations of Computer Science 2011 - 36th International Symposium, MFCS 2011, Warsaw, Poland, August 22-26, 2011. Proceedings; 01/2011
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    Stéphane Bessy, Christophe Paul, Anthony Perez
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    ABSTRACT: A graph G=(V,E) is a 3-leaf power iff there exists a tree T the leaf set of which is V and such that uv∈E iff u and v are at distance at most 3 in T. The 3-leaf power graph edge modification problems, i.e. edition (also known as the closest 3-leaf power), completion and edge-deletion are FPT when parameterized by the size of the edge set modification. However, polynomial kernels were known for none of these three problems. For each of them, we provide kernels with O(k3) vertices that can be computed in linear time. We thereby answer an open problem first mentioned by Dom et al. (2004) [8].
    Discrete Applied Mathematics 08/2010; · 0.72 Impact Factor
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    ABSTRACT: Cluster Editing is a classical graph theoretic approach to tackle the problem of data set clustering: it consists of modifying a similarity graph into a disjoint union of cliques, i.e, clusters. As pointed out in a number of recent papers, the cluster editing model is too rigid to capture common features of real data sets. Several generalizations have thereby been proposed. In this paper, we introduce (p, q)-cluster graphs, where each cluster misses at most p edges to be a clique, and there are at most q edges between a cluster and other clusters. Our generalization is the first one that allows a large number of false positives and negatives in total, while bounding the number of these locally for each cluster by p and q. We show that recognizing (p, q)-cluster graphs is NP-complete when p and q are input. On the positive side, we show that (0, q)-cluster, (p, 1)-cluster, (p, 2)-cluster, and (1, 3)-cluster graphs can be recognized in polynomial time.
    WG'10: International Workshop on Graph Theoretical Concepts in Computer Science. 06/2010;
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    Christophe Paul, Christophe Crespelle
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    ABSTRACT: This paper considers the problem of maintaining a compact representation (O(n) space) of permutation graphs under vertex and edge modifications (insertion or deletion). That representation allows us to answer adjacency queries in O(1) time. The approach is based on a fully dynamic modular decomposition algorithm for permutation graphs that works in O(n) time per edge and vertex modification. We thereby obtain a fully dynamic algorithm for the recognition of permutation graphs.
    Algorithmica 01/2010; · 0.49 Impact Factor
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    Sylvain Guillemot, Christophe Paul, Anthony Perez
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    ABSTRACT: Given a graph G = (V,E) and an integer k, an edge modification problem for a graph property P{\it \Pi} consists in deciding whether there exists a set of edges F of size at most k such that the graph H=(V,E\vartriangle F)H=(V,E\vartriangle F) satisfies the property P{\it \Pi}. In the P{\it \Pi} edge-completion problem, the set F of edges is constrained to be disjoint from E; in the P{\it \Pi} edge-deletion problem, F is a subset of E; no constraint is imposed on F in the P{\it \Pi} edge-editing problem. A number of optimization problems can be expressed in terms of graph modification problems which have been extensively studied in the context of parameterized complexity. When parameterized by the size k of the edge set F, it has been proved that if P{\it \Pi} is an hereditary property characterized by a finite set of forbidden induced subgraphs, then the three P{\it \Pi} edge-modification problems are FPT [4]. It was then natural to ask [4] whether these problems also admit a polynomial size kernel. Using recent lower bound techniques, Kratsch and Wahlström answered this question negatively [15]. However, the problem remains open on many natural graph classes characterized by forbidden induced subgraphs. Kratsch and Wahlström asked whether the result holds when the forbidden subgraphs are paths and pointed out that the problem is already open in the case of P 4-free graphs (i.e. cographs). This paper provides positive and negative results in that line of research. We prove that parameterized cograph edge modification problems have cubic vertex kernels whereas polynomial kernels are unlikely to exist for P l -free and C l -free edge deletion problems for large enough l.
    Parameterized and Exact Computation - 5th International Symposium, IPEC 2010, Chennai, India, December 13-15, 2010. Proceedings; 01/2010
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    Sylvain Guillemot, Christophe Paul, Anthony Perez
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    ABSTRACT: Given a graph G = (V , E) and an integer k, an edge modification problem for a graph property Π consists in deciding whether there exists a set of edges F of size at most k such that the graph H = (V , E △ F ) satisfies the property Π. In the Π edge-completion problem, the set F of edges is constrained to be disjoint from E; in the Π edge-deletion problem, F is a subset of E; no constraint is imposed on F in the Π edge-edition problem. A number of optimization problems can be expressed in terms of graph modification problems which have been extensively studied in the context of parameterized complexity. When parameterized by the size k of the edge set F , It has been proved that (Cai, IPL:58(4)-1996) if Π is an hereditary property characterized by a finite set of forbidden induced subgraph, then the three Π edge-modification problems are fixed-parameter tractable. It was then natural to ask (Cai, IWPEC 2006) whether these Π edge-modification problems also admit a polynomial size kernel (i.e. any instance (G, k) can be reduced in polynomial time to an equivalent instance (G′ , k′ ) with size bounded by a polynomial in k). Using recent lower bound techniques, Kratsch and Wahlstr¨om (IWPEC 2009) answered this question negatively. However, the problem remains open on many natural graph classes characterized by forbidden induced subgraph. It is a challenging question to characterized for which type of graph properties, the parameterized edge-modification problems have polynomial kernels. Kratsch and Wahlstr¨om asked whether the result holds when the forbidden subgraphs are paths and pointed out that the problem is already open in the case of P4 -free graphs (i.e. cographs). This paper provides positive and negative results in that line of research. We prove that parameterized cograph edge modification problems have cubic vertex kernels whereas polynomial kernels are unlikely to exist for Pl -free graphs and Cl -free graph for large enough l.
    Algorithmica 01/2010; · 0.49 Impact Factor

Publication Stats

1k Citations
27.22 Total Impact Points

Institutions

  • 2012
    • University of Paris-Est
      Centre, France
  • 2005–2011
    • University of Bergen
      • Department of Informatics
      Bergen, Hordaland Fylke, Norway
  • 2001–2011
    • Université de Montpellier 1
      Montpelhièr, Languedoc-Roussillon, France
    • University of Toronto
      • Department of Computer Science
      Toronto, Ontario, Canada
  • 2008–2010
    • Paris Diderot University
      Lutetia Parisorum, Île-de-France, France
  • 2009
    • University of Tuebingen
      • Center for Bioinformatics
      Tübingen, Baden-Wuerttemberg, Germany
  • 2007
    • French National Institute for Agricultural Research
      • Département de Mathématiques et Informatiques Appliquées
      Avignon, Provence-Alpes-Cote d'Azur, France
    • McGill University
      • School of Computer Science
      Montréal, Quebec, Canada
    • University of Manitoba
      Winnipeg, Manitoba, Canada
  • 2001–2003
    • University of Bordeaux
      Burdeos, Aquitaine, France
  • 2000
    • National Institute for Research in Computer Science and Control
      Le Chesney, Île-de-France, France