arXiv subjects

Michel Habib

Publications and source records attributed to Michel Habib.

At least 37 records · Page 2Linked to original sources

arXiv2018

Fast approximation of centrality and distances in hyperbolic graphs

We show that the eccentricities (and thus the centrality indices) of all vertices of a $δ$-hyperbolic graph $G=(V,E)$ can be computed in linear time with an additive one-sided error of at most $cδ$, i.e., after a linear time preprocessing, for every vertex $v$ of $G$ one can compute in $O(1)$ time an estimate $\hat{e}(v)$ of its eccentricity $ecc_G(v)$ such that $ecc_G(v)\leq \hat{e}(v)\leq ecc_G(v)+ cδ$ for a small constant $c$. We prove that every $δ$-hyperbolic graph $G$ has a shortest path tree, constructible in linear time, such that for every vertex $v$ of $G$, $ecc_G(v)\leq ecc_T(v)\leq ecc_G(v)+ cδ$. These results are based on an interesting monotonicity property of the eccentricity function of hyperbolic graphs: the closer a vertex is to the center of $G$, the smaller its eccentricity is. We also show that the distance matrix of $G$ with an additive one-sided error of at most $c'δ$ can be computed in $O(|V|^2\log^2|V|)$ time, where $c'< c$ is a small constant. Recent empirical studies show that many real-world graphs (including Internet application networks, web networks, collaboration networks, social networks, biological networks, and others) have small hyperbolicity. So, we analyze the performance of our algorithms for approximating centrality and distance matrix on a number of real-world networks. Our experimental results show that the obtained estimates are even better than the theoretical bounds.

cs.DS↗

arXiv2017

Maximum Induced Matching Algorithms via Vertex Ordering Characterizations

We study the maximum induced matching problem on a graph g. Induced matchings correspond to independent sets in L2(g), the square of the line graph of g. The problem is NP-complete on bipartite graphs. In this work, we show that for a number of graph families with forbidden vertex orderings, almost all forbidden patterns on three vertices are preserved when taking the square of the line graph. These orderings can be computed in linear time in the size of the input graph. In particular, given a graph class G characterized by a vertex ordering, and a graph g = (V, E) in G with a corresponding vertex ordering σof V , one can produce (in linear time in the size of g) an ordering on the vertices of L2(g), that shows that L2(g) in G - for a number of graph classes G - without computing the line graph or the square of the line graph of g. These results generalize and unify previous ones on showing closure under L2(.) for various graph families. Furthermore, these orderings on L2(g) can be exploited algorithmically to compute a maximum induced matching on G faster. We illustrate this latter fact in the second half of the paper where we focus on cocomparability graphs, a large graph class that includes interval, permutation, trapezoid graphs, and co-graphs, and we present the first O(mn) time algorithm to compute a maximum weighted induced matching on cocomparability graphs; an improvement from the best known O(n4) time algorithm for the unweighted case.

cs.DS↗

arXiv2016

Maximal cliques structure for cocomparability graphs and applications

A cocomparability graph is a graph whose complement admits a transitive orientation. An interval graph is the intersection graph of a family of intervals on the real line. In this paper we investigate the relationships between interval and cocomparability graphs. This study is motivated by recent results Corneil,Dalton, Habib (2013) and Dusart, Habib (2016) and that show that for some problems, the algorithm used on interval graphs can also be used with small modifications on cocomparability graphs. Many of these algorithms are based on graph searches that preserve cocomparability orderings. First we propose a characterization of cocomparability graphs via a lattice structure on the set of their maximal cliques. Using this characterization we can prove that every maximal interval subgraph of a cocomparability graph $G$ is also a maximal chordal subgraph of $G$. Although the size of this lattice of maximal cliques can be exponential in the size of the graph, it can be used as a framework to design and prove algorithms on cocomparability graphs. In particular we show that a new graph search, namely Local Maximal Neighborhood Search (LocalMNS) leads to an $O(n+mlogn)$ time algorithm to find a maximal interval subgraph of a cocomparability graph. Similarly we propose a linear time algorithm to compute all simplicial vertices in a cocomparability graph. In both cases we improve on the current state of knowledge.

cs.DM↗

arXiv2016

Read networks and k-laminar graphs

In this paper we introduce k-laminar graphs a new class of graphs which extends the idea of Asteroidal triple free graphs. Indeed a graph is k-laminar if it admits a diametral path that is k-dominating. This bio-inspired class of graphs was motivated by a biological application dealing with sequence similarity networks of reads (called hereafter read networks for short). We briefly develop the context of the biological application in which this graph class appeared and then we consider the relationships of this new graph class among known graph classes and then we study its recognition problem. For the recognition of k-laminar graphs, we develop polynomial algorithms when k is fixed. For k=1, our algorithm improves a Deogun and Krastch's algorithm (1999). We finish by an NP-completeness result when k is unbounded.

cs.DM↗

arXiv2015

A tie-break model for graph search

In this paper, we consider the problem of the recognition of various kinds of orderings produced by graph searches. To this aim, we introduce a new framework, the Tie-Breaking Label Search (TBLS), in order to handle a broad variety of searches. This new model is based on partial orders defined on the label set and it unifies the General Label Search (GLS) formalism of Krueger, Simonet and Berry (2011), and the "pattern-conditions" formalism of Corneil and Krueger (2008). It allows us to derive some general properties including new pattern-conditions (yielding memory-efficient certificates) for many usual searches, including BFS, DFS, LBFS and LDFS. Furthermore, the new model allows easy expression of multi-sweep uses of searches that depend on previous (search) orderings of the graph's vertex set.

cs.DS↗

arXiv2014

Into the Square - On the Complexity of Quadratic-Time Solvable Problems

This paper will analyze several quadratic-time solvable problems, and will classify them into two classes: problems that are solvable in truly subquadratic time (that is, in time $O(n^{2-ε})$ for some $ε>0$) and problems that are not, unless the well known Strong Exponential Time Hypothesis (SETH) is false. In particular, we will prove that some quadratic-time solvable problems are indeed easier than expected. We will provide an algorithm that computes the transitive closure of a directed graph in time $O(mn^{\frac{ω+1}{4}})$, where $m$ denotes the number of edges in the transitive closure and $ω$ is the exponent for matrix multiplication. As a side effect, we will prove that our algorithm runs in time $O(n^{\frac{5}{3}})$ if the transitive closure is sparse. The same time bounds hold if we want to check whether a graph is transitive, by replacing m with the number of edges in the graph itself. As far as we know, this is the fastest algorithm for sparse transitive digraph recognition. Finally, we will apply our algorithm to the comparability graph recognition problem (dating back to 1941), obtaining the first truly subquadratic algorithm. The second part of the paper deals with hardness results. Starting from an artificial quadratic-time solvable variation of the k-SAT problem, we will construct a graph of Karp reductions, proving that a truly subquadratic-time algorithm for any of the problems in the graph falsifies SETH. The analyzed problems are the following: computing the subset graph, finding dominating sets, computing the betweenness centrality of a vertex, computing the minimum closeness centrality, and computing the hyperbolicity of a pair of vertices. We will also be able to include in our framework three proofs already appeared in the literature, concerning the graph diameter computation, local alignment of strings and orthogonality of vectors.

cs.CC↗

arXiv2013

Algorithmic Aspects of Switch Cographs

This paper introduces the notion of involution module, the first generalization of the modular decomposition of 2-structure which has a unique linear-sized decomposition tree. We derive an O(n^2) decomposition algorithm and we take advantage of the involution modular decomposition tree to state several algorithmic results. Cographs are the graphs that are totally decomposable w.r.t modular decomposition. In a similar way, we introduce the class of switch cographs, the class of graphs that are totally decomposable w.r.t involution modular decomposition. This class generalizes the class of cographs and is exactly the class of (Bull, Gem, Co-Gem, C_5)-free graphs. We use our new decomposition tool to design three practical algorithms for the maximum cut, vertex cover and vertex separator problems. The complexity of these problems was still unknown for this class of graphs. This paper also improves the complexity of the maximum clique, the maximum independant set, the chromatic number and the maximum clique cover problems by giving efficient algorithms, thanks to the decomposition tree. Eventually, we show that this class of graphs has Clique-Width at most 4 and that a Clique-Width expression can be computed in linear time.

cs.DM↗

arXiv2013

A general method for common intervals

Given an elementary chain of vertex set V, seen as a labelling of V by the set {1, ...,n=|V|}, and another discrete structure over $V$, say a graph G, the problem of common intervals is to compute the induced subgraphs G[I], such that $I$ is an interval of [1, n] and G[I] satisfies some property Pi (as for example Pi= "being connected"). This kind of problems comes from comparative genomic in bioinformatics, mainly when the graph $G$ is a chain or a tree (Heber and Stoye 2001, Heber and Savage 2005, Bergeron et al 2008). When the family of intervals is closed under intersection, we present here the combination of two approaches, namely the idea of potential beginning developed in Uno, Yagiura 2000 and Bui-Xuan et al 2005 and the notion of generator as defined in Bergeron et al 2008. This yields a very simple generic algorithm to compute all common intervals, which gives optimal algorithms in various applications. For example in the case where $G$ is a tree, our framework yields the first linear time algorithms for the two properties: "being connected" and "being a path". In the case where $G$ is a chain, the problem is known as: common intervals of two permutations (Uno and Yagiura 2000), our algorithm provides not only the set of all common intervals but also with some easy modifications a tree structure that represents this set.

cs.DS↗

arXiv2013

Detecting 2-joins faster

2-joins are edge cutsets that naturally appear in the decomposition of several classes of graphs closed under taking induced subgraphs, such as balanced bipartite graphs, even-hole-free graphs, perfect graphs and claw-free graphs. Their detection is needed in several algorithms, and is the slowest step for some of them. The classical method to detect a 2-join takes $O(n^3m)$ time where $n$ is the number of vertices of the input graph and $m$ the number of its edges. To detect \emph{non-path} 2-joins (special kinds of 2-joins that are needed in all of the known algorithms that use 2-joins), the fastest known method takes time $O(n^4m)$. Here, we give an $O(n^2m)$-time algorithm for both of these problems. A consequence is a speed up of several known algorithms.

cs.DS↗

arXiv2011

Structure and Recognition of 3,4-leaf Powers of Galled Phylogenetic Networks in Polynomial Time

A graph is a $k$-leaf power of a tree $T$ if its vertices are leaves of $T$ and two vertices are adjacent in $T$ if and only if their distance in $T$ is at most $k$. Then $T$ is a $k$-leaf root of $G$. This notion was introduced by Nishimura, Ragde, and Thilikos [2002] motivated by the search for underlying phylogenetic trees. We study here an extension of the $k$-leaf power graph recognition problem. This extension is motivated by a new biological question for the evaluation of the latteral gene transfer on a population of viruses. We allow the host graph to slightly differs from a tree and allow some cycles. In fact we study phylogenetic galled networks in which cycles are pairwise vertex disjoint. We show some structural results and propose polynomial algorithms for the cases $k=3$ and $k=4$. As a consequence, squares of galled networks can also be recognized in polynomial time.

cs.DM↗

arXiv2011

Constructing a Minimum-Level Phylogenetic Network from a Dense Triplet Set in Polynomial Time

For a given set $\mathcal{L}$ of species and a set $\mathcal{T}$ of triplets on $\mathcal{L}$, one wants to construct a phylogenetic network which is consistent with $\mathcal{T}$, i.e which represents all triplets of $\mathcal{T}$. The level of a network is defined as the maximum number of hybrid vertices in its biconnected components. When $\mathcal{T}$ is dense, there exist polynomial time algorithms to construct level-$0,1,2$ networks (Aho et al. 81, Jansson et al. 04, Iersel et al. 08). For higher levels, partial answers were obtained by Iersel et al. 2008 with a polynomial time algorithm for simple networks. In this paper, we detail the first complete answer for the general case, solving a problem proposed by Jansson et al. 2004: for any $k$ fixed, it is possible to construct a minimum level-$k$ network consistent with $\mathcal{T}$, if there is any, in time $O(|\mathcal{T}|^{k+1}n^{\lfloor\frac{4k}{3}\rfloor+1})$. This is an improved result of our preliminary version presented at CPM'2009.

q-bio.PE↗

arXiv2011

On a conjecture of compatibility of multi-states characters

Perfect phylogeny consisting of determining the compatibility of a set of characters is known to be NP-complete. We propose in this article a conjecture on the necessary and sufficient conditions of compatibility: Given a set $\mathcal{C}$ of $r$-states full characters, there exists a function $f(r)$ such that $\mathcal{C}$ is compatible iff every set of $f(r)$ characters of $\mathcal{C}$ is compatible. Some previous work showed that $f(2)=2$, $f(3)=3$ and $f(r) \ge r-1$. Gusfield et al. 09 conjectured that $f(r) = r$ for any $r \ge 2$. In this paper, we present an example showing that $f(4) \ge 5$ and then a closure operation for chordal sandwich graphs. The later problem is a common approach of perfect phylogeny. This operation can be the first step to simplify the problem before solving some particular cases $f(4), f(5), ... $, and determining the function $f(r)$.

cs.DS↗

arXiv2010

Unique perfect phylogeny is NP-hard

We answer, in the affirmative, the following question proposed by Mike Steel as a $100 challenge: "Is the following problem NP-hard? Given a ternary phylogenetic X-tree T and a collection Q of quartet subtrees on X, is T the only tree that displays Q ?"

q-bio.PE↗

arXiv2009

A survey on algorithmic aspects of modular decomposition

The modular decomposition is a technique that applies but is not restricted to graphs. The notion of 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 70's, 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.

cs.DM↗

arXiv2009

Fast Computation of Empirically Tight Bounds for the Diameter of Massive Graphs

The diameter of a graph is among its most basic parameters. Since a few years, it moreover became a key issue to compute it for massive graphs in the context of complex network analysis. However, known algorithms, including the ones producing approximate values, have too high a time and/or space complexity to be used in such cases. We propose here a new approach relying on very simple and fast algorithms that compute (upper and lower) bounds for the diameter. We show empirically that, on various real-world cases representative of complex networks studied in the literature, the obtained bounds are very tight (and even equal in some cases). This leads to rigorous and very accurate estimations of the actual diameter in cases which were previously untractable in practice.

cs.DS↗

arXiv2009

On some simplicial elimination schemes for chordal graphs

We present here some results on particular elimination schemes for chordal graphs, namely we show that for any chordal graph we can construct in linear time a simplicial elimination scheme starting with a pending maximal clique attached via a minimal separator maximal (resp. minimal) under inclusion among all minimal separators.

cs.DS↗

arXiv2009

Level-k Phylogenetic Network can be Constructed from a Dense Triplet Set in Polynomial Time

Given a dense triplet set $\mathcal{T}$, there arise two interesting questions: Does there exists any phylogenetic network consistent with $\mathcal{T}$? And if so, can we find an effective algorithm to construct one? For cases of networks of levels $k=0$ or 1 or 2, these questions were answered with effective polynomial algorithms. For higher levels $k$, partial answers were recently obtained with an $O(|\mathcal{T}|^{k+1})$ time algorithm for simple networks. In this paper we give a complete answer to the general case. The main idea is to use a special property of SN-sets in a level-k network. As a consequence, we can also find the level-k network with the minimum number of reticulations in polynomial time.

q-bio.PE↗

arXiv2007

A Note On Computing Set Overlap Classes

Let ${\cal V}$ be a finite set of $n$ elements and ${\cal F}=\{X_1,X_2, >..., X_m\}$ a family of $m$ subsets of ${\cal V}.$ Two sets $X_i$ and $X_j$ of ${\cal F}$ overlap if $X_i \cap X_j \neq \emptyset,$ $X_j \setminus X_i \neq \emptyset,$ and $X_i \setminus X_j \neq \emptyset.$ Two sets $X,Y\in {\cal F}$ are in the same overlap class if there is a series $X=X_1,X_2, ..., X_k=Y$ of sets of ${\cal F}$ in which each $X_iX_{i+1}$ overlaps. In this note, we focus on efficiently identifying all overlap classes in $O(n+\sum_{i=1}^m |X_i|)$ time. We thus revisit the clever algorithm of Dahlhaus of which we give a clear presentation and that we simplify to make it practical and implementable in its real worst case complexity. An useful variant of Dahlhaus's approach is also explained.

cs.DS↗