Christophe Paul

Université de Montpellier, Montpelhièr, Languedoc-Roussillon, France

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Publications (123)42.44 Total impact

  • Eunjung Kim · Christophe Paul · Ignasi Sau · Dimitrios M. Thilikos
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    ABSTRACT: In this paper we design {\sf FPT}-algorithms for two parameterized problems. The first is \textsc{List Digraph Homomorphism}: given two digraphs $G$ and $H$ and a list of allowed vertices of $H$ for every vertex of $G$, the question is whether there exists a homomorphism from $G$ to $H$ respecting the list constraints. The second problem is a variant of \textsc{Multiway Cut}, namely \textsc{Min-Max Multiway Cut}: given a graph $G$, a non-negative integer $\ell$, and a set $T$ of $r$ terminals, the question is whether we can partition the vertices of $G$ into $r$ parts such that (a) each part contains one terminal and (b) there are at most $\ell$ edges with only one endpoint in this part. We parameterize \textsc{List Digraph Homomorphism} by the number $w$ of edges of $G$ that are mapped to non-loop edges of $H$ and we give a time $2^{O(\ell\cdot\log h+\ell^2\cdot \log \ell)}\cdot n^{4}\cdot \log n$ algorithm, where $h$ is the order of the host graph $H$. We also prove that \textsc{Min-Max Multiway Cut} can be solved in time $2^{O((\ell r)^2\log \ell r)}\cdot n^{4}\cdot \log n$. Our approach introduces a general problem, called {\sc List Allocation}, whose expressive power permits the design of parameterized reductions of both aforementioned problems to it. Then our results are based on an {\sf FPT}-algorithm for the {\sc List Allocation} problem that is designed using a suitable adaptation of the {\em randomized contractions} technique (introduced by [Chitnis, Cygan, Hajiaghayi, Pilipczuk, and Pilipczuk, FOCS 2012]).
    No preview · Article · Sep 2015
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    ABSTRACT: The tree-cut width of a graph is a graph parameter defined by Wollan [J. Comb. Theory, Ser. B, 110:47-66, 2015] with the help of tree-cut decompositions. In certain cases, tree-cut width appears to be more adequate than treewidth as an invariant that, when bounded, can accelerate the resolution of intractable problems. While designing algorithms for problems with bounded tree-cut width, it is important to have a parametrically tractable way to compute the exact value of this parameter or, at least, some constant approximation of it. In this paper we give a parameterized 2-approximation algorithm for the computation of tree-cut width; for an input $n$-vertex graph $G$ and an integer $w$, our algorithm either confirms that the tree-cut width of $G$ is more than $w$ or returns a tree-cut decomposition of $G$ certifying that its tree-cut width is at most $2w$, in time $2^{O(w^2\log w)} \cdot n^2$. Prior to this work, no constructive parameterized algorithms, even approximated ones, existed for computing the tree-cut width of a graph. As a consequence of the Graph Minors series by Robertson and Seymour, only the existence of a decision algorithm was known.
    No preview · Article · Sep 2015
  • Christophe Paul · Anthony Perez · Stéphan Thomassé
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    ABSTRACT: We develop a technique, that we call conflict packing, to obtain (and improve) polynomial kernels for several well-studied editing problems. We first illustrate our technique on Feedback Arc Set in Tournaments (k -FAST) yielding an alternative and simple proof of a linear kernel for this problem. The technique is then applied to obtain the first linear kernel for the. Dense Rooted Triplet Inconsistency (k -dense-RTI) problem. A linear kernel for Betweenness in Tournaments (k -BIT) is also proved. All these problems share common features. First, they can be expressed as modification problems on a dense set of constant-arity constraints. Also the consistency of the set of constraints can be characterized by means of a bounded size obstructions. The conflict packing technique basically consists of computing a maximal set of small obstructions allowing us either to bound the size of the input or to reduce the input.
    No preview · Article · Sep 2015 · Journal of Computer and System Sciences
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    ABSTRACT: We consider the class of graphs having linear rank-width one, also known as thread graphs, and investigate a related graph modification problem called the Thread Vertex Deletion. In this problem, given an $n$ vertex graph $G$ and a positive integer $k$, we want to decide whether there is a set of at most $k$ vertices whose removal turns $G$ into a thread graph and if one exists, find such a vertex set. While the meta-theorem of Courcelle, Makowsky, Rotics implies that Thread Vertex Deletion can be computed in time $f(k)\cdot n^3$, it is not clear whether this problem allows a runtime with a modest exponential function. We establish that Thread Vertex Deletion can be solved in time $8^k\cdot n^{\mathcal{O}(1)}$. The major obstacle to this end is how to handle a long induced cycle as an obstruction. To fix this issue, we define a graph class called the necklace graphs and investigate its structural properties. We also show that the Thread Vertex Deletion has a polynomial kernel.
    No preview · Article · Apr 2015
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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.
    Full-text · Article · Jun 2014 · SIAM Journal on Discrete Mathematics
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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.
    Preview · Article · Mar 2014
  • Christophe Crespelle · Emeric Gioan · Christophe Paul

    No preview · Article · Jan 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.
    Preview · Article · Dec 2013 · SIAM Journal on Discrete Mathematics

  • No preview · Book · Apr 2013
  • Eunjung Kim · Christophe Paul · Geevarghese Philip

    No preview · Article · Mar 2013
  • Emeric Gioan · Christophe Paul · Marc Tedder · Derek Corneil

    No preview · Article · Jan 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.
    Full-text · Article · Aug 2012 · Journal of Bioinformatics and Computational Biology
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    ABSTRACT: 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 \(\mathcal{F}\) 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 ⊆ V(G) such that \(|X|\leqslant k\) and G − X is H-minor-free for every \(H\in \mathcal{F}\). As our second application, we present the first single-exponential algorithm to solve Planar \(\mathcal{F}\)- Deletion. Namely, our algorithm runs in time 2O(k)·n 2, which is asymptotically optimal with respect to k. So far, single-exponential algorithms were only known for special cases of the family \(\mathcal{F}\).
    Full-text · Article · Jul 2012 · ACM Transactions on Algorithms
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    ABSTRACT: A circle graph is the intersection graph of a set of chords in a circle. Keil [Discrete Appl. Math., 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 G has a dominating set inducing a graph isomorphic to T is NP-complete when T is in the input, and FPT when parameterized by t=|V(T)|. We prove that the FPT algorithm runs in subexponential time, namely \(2^{\mathcal{O}(t \cdot\frac{\log\log t}{\log t})} \cdot n^{\mathcal{O}(1)}\), where n=|V(G)|.
    Full-text · Article · May 2012 · Theory of Computing Systems
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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.
    Full-text · Article · Apr 2012 · Journal of Computer and System Sciences
  • 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.
    No preview · Conference Paper · Nov 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.
    Full-text · Article · Aug 2011 · Discrete Applied Mathematics
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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. The study of analogous edge contraction problems has so far been left largely unexplored from a parameterized perspective. We consider two basic problems of this type: Tree Contraction and Path Contraction. These two problems take as input an undirected graph G on n vertices and an integer k, and the task is to determine whether we can obtain an acyclic graph or a path, respectively, by a sequence of at most k edge contractions in G. We present an algorithm with running time 4.98k n O(1) for Tree Contraction, based on a variant of the color coding technique of Alon, Yuster and Zwick, and an algorithm with running time 2k + o(k) + n O(1) for Path Contraction. Furthermore, we show that Path Contraction has a kernel with at most 5k + 3 vertices, while Tree Contraction does not have a polynomial kernel unless NP ⊆ coNP/poly. We find the latter result surprising, because of the connection between Tree Contraction and Feedback Vertex Set, which is known to have a kernel with 4k 2 vertices.
    Full-text · Article · May 2011 · Algorithmica
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    ABSTRACT: The c-pumpkin is the graph with two vertices linked by c ≥ 1 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 hitting and packing c-pumpkin-models in a given graph: On the one hand, we provide an FPT algorithm running in time \(2^{\mathcal{O}(k)} n^{\mathcal{O}(1)}\) deciding, for any fixed c ≥ 1, whether all c-pumpkin-models can be hit by at most k vertices. This generalizes the single-exponential FPT algorithms for Vertex Cover and Feedback Vertex Set, which correspond to the cases c = 1,2 respectively. For this, we use a combination of iterative compression and a kernelization-like technique. On the other hand, we present an \(\mathcal{O}(\log n)\)-approximation algorithm for both the problems of hitting all c-pumpkin-models with a smallest number of vertices, and packing a maximum number of vertex-disjoint c-pumpkin-models. Our main ingredient here is a combinatorial lemma saying that any properly reduced n-vertex graph has a c-pumpkin-model of size at most f(c) logn, for a function f depending only on c.
    Preview · Article · May 2011 · SIAM Journal on Discrete Mathematics
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    Emeric Gioan · Christophe Paul · Marc Tedder · Derek Corneil
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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.
    Preview · Article · Apr 2011 · Algorithmica

Publication Stats

2k Citations
42.44 Total Impact Points

Institutions

  • 2003-2015
    • Université de Montpellier
      Montpelhièr, Languedoc-Roussillon, France
    • Université Bordeaux 1
      Talence, Aquitaine, France
  • 2004-2014
    • French National Centre for Scientific Research
      • Laboratoire de Biométrie et Biologie Évolutive (LBBE)
      Lutetia Parisorum, Île-de-France, France
  • 1999-2011
    • Laboratoire d'Informatique, de Robotique et de Microélectronique de Montpellier (LIRMM)
      Montpelhièr, Languedoc-Roussillon, France
  • 2009
    • University of Tuebingen
      • Center for Bioinformatics
      Tübingen, Baden-Württemberg, Germany
  • 2007-2009
    • McGill University
      • School of Computer Science
      Montréal, Quebec, Canada
  • 2008
    • Université Montpellier 2 Sciences et Techniques
      Montpelhièr, Languedoc-Roussillon, France
  • 2000
    • National Institute for Research in Computer Science and Control
      Le Chesney, Île-de-France, France