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Documents authored by Habib, Michel

Document
DOI: 10.4230/LIPIcs.SAND.2026.5
Complexity Gaps Between Point and Interval Temporal Graphs for Some Reachability Problems

Authors: Guillaume Aubian, Filippo Brunelli, Feodor F. Dragan, Guillaume Ducoffe, Michel Habib, Allen Ibiapina, and Laurent Viennot

Published in: LIPIcs, Volume 373, 5th Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2026)

Abstract
Temporal graphs arise when modeling interactions that evolve over time. They usually come in several flavors, depending on the number of parameters used to describe the temporal aspects of the interactions: time of appearance, duration, delay of transmission. In the point model, edges appear at specific points in time, whereas in the more general interval model, edges can be present over specific time intervals. In both models, the delay for traversing an edge can change with each edge appearance. When time is discrete, the two models are equivalent in the sense that the presence of an edge during an interval is equivalent to a sequence of point-in-time occurrences of the edge. However, this transformation can drastically change the size of the input and has implications for complexity. Indeed, we show a gap between the two models with respect to the complexity of the classical problem of computing a fastest temporal path from a source vertex to a target vertex, i.e., a path where edges can be traversed one after another in time and such that the total duration from source to target is minimized. It can be solved in near-linear time in the point model, while we show that the interval model requires quadratic time under classical assumptions of fine-grained complexity. With respect to linear time, our lower bound implies a factor of the number of vertices, while the best known algorithm has a factor of the number of underlying edges. We also show a similar complexity gap for computing a shortest temporal path, i.e., a temporal path with a minimum number of edges. Here our lower bound matches known upper bounds up to a logarithmic factor. Interestingly, we show that near-linear time for fastest temporal path computation is possible in the interval model when it is restricted to uniform delay zero, i.e., when traversing an edge is instantaneous. However, this special case is not exempt from our lower bound for shortest temporal path computation. These two results should be contrasted with the computation of a foremost temporal path, i.e., a temporal path that arrives as early as possible. It is well known that this computation can be solved in near-linear time in both models. We also show that there is no gap in testing the all-to-all temporal connectivity of a temporal graph. We demonstrate a quadratic lower bound that applies to both the interval and point models and aligns with the existing upper bounds.
Cite as

Guillaume Aubian, Filippo Brunelli, Feodor F. Dragan, Guillaume Ducoffe, Michel Habib, Allen Ibiapina, and Laurent Viennot. Complexity Gaps Between Point and Interval Temporal Graphs for Some Reachability Problems. In 5th Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 373, pp. 5:1-5:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{aubian_et_al:LIPIcs.SAND.2026.5,
  author =	{Aubian, Guillaume and Brunelli, Filippo and Dragan, Feodor F. and Ducoffe, Guillaume and Habib, Michel and Ibiapina, Allen and Viennot, Laurent},
  title =	{{Complexity Gaps Between Point and Interval Temporal Graphs for Some Reachability Problems}},
  booktitle =	{5th Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2026)},
  pages =	{5:1--5:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-427-7},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{373},
  editor =	{Mertzios, George B. and Richa, Andr\'{e}a W.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAND.2026.5},
  URN =		{urn:nbn:de:0030-drops-262395},
  doi =		{10.4230/LIPIcs.SAND.2026.5},
  annote =	{Keywords: Temporal graphs, Dynamic networks, Time-dependent networks, Temporal connectivity, Foremost, Fastest or Shortest temporal path, Fine-grained complexity}
}
Document
DOI: 10.4230/LIPIcs.SAND.2026.7
Extending Ghouila-Houri’s Characterization of Comparability Graphs to Temporal Graphs

Authors: Pierre Charbit, Michel Habib, and Amalia Sorondo

Published in: LIPIcs, Volume 373, 5th Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2026)

Abstract
An orientation of a static graph is called transitive if for any three vertices a,b,c, the presence of arcs (a,b) and (b,c) forces the presence of arc (a,c). If only the presence of an arc between a and c is required, but its orientation is unconstrained, the orientation is called quasi-transitive. A fundamental result due to Ghouila-Houri [Ghouila-Houri, 1962] states that any static graph admitting a quasi-transitive orientation also admits a transitive orientation. In a seminal work [Mertzios et al., 2025], Mertzios et al. introduced the notion of temporal transitivity in order to model information flows in simple temporal networks. We revisit the model introduced by Mertzios et al. and propose an analogous to Ghouila-Houri’s characterization for the temporal scenario. We present a structural theorem that will allow us to express by a 2-SAT formula all the constraints imposed on a temporal graph for it to admit a temporal transitive orientation. The latter produces an efficient recognition algorithm for graphs admitting such orientations, that we will call comparability temporal graphs. Inspired by the lexicographic strategy presented by Hell and Huang in [Hell and Huang, 1995] to transitively orient static graphs, we then propose an algorithm for constructing a temporal transitive orientation of a YES instance. This algorithm is straightforward and has a running-time complexity of O(nm + min{kn,m²}), with n, m and k being respectively the number of vertices, edges and monolabel triangles, i.e., triangles having the same unique time-label on their edges, in the temporal graph. This represents an improvement compared to the algorithm presented in [Mertzios et al., 2025]. Additionally, we extend the temporal transitivity model to temporal graphs having multiple time-labels associated to their edges and claim that the previous results hold in the multilabel setting. Finally, we propose a characterization of comparability temporal graphs by forbidden temporal ordered patterns.
Cite as

Pierre Charbit, Michel Habib, and Amalia Sorondo. Extending Ghouila-Houri’s Characterization of Comparability Graphs to Temporal Graphs. In 5th Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 373, pp. 7:1-7:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{charbit_et_al:LIPIcs.SAND.2026.7,
  author =	{Charbit, Pierre and Habib, Michel and Sorondo, Amalia},
  title =	{{Extending Ghouila-Houri’s Characterization of Comparability Graphs to Temporal Graphs}},
  booktitle =	{5th Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2026)},
  pages =	{7:1--7:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-427-7},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{373},
  editor =	{Mertzios, George B. and Richa, Andr\'{e}a W.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAND.2026.7},
  URN =		{urn:nbn:de:0030-drops-262416},
  doi =		{10.4230/LIPIcs.SAND.2026.7},
  annote =	{Keywords: Temporal graphs, Transitive orientations, Graph algorithms}
}
Document
DOI: 10.4230/LIPIcs.STACS.2022.9
Subquadratic-Time Algorithm for the Diameter and All Eccentricities on Median Graphs

Authors: Pierre Bergé, Guillaume Ducoffe, and Michel Habib

Published in: LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)

Abstract
On sparse graphs, Roditty and Williams [2013] proved that no O(n^{2-ε})-time algorithm achieves an approximation factor smaller than 3/2 for the diameter problem unless SETH fails. We answer here an open question formulated in the literature: can we use the structural properties of median graphs to break this global quadratic barrier? We propose the first combinatorial algorithm computing exactly all eccentricities of a median graph in truly subquadratic time. Median graphs constitute the family of graphs which is the most studied in metric graph theory because their structure represents many other discrete and geometric concepts, such as CAT(0) cube complexes. Our result generalizes a recent one, stating that there is a linear-time algorithm for computing all eccentricities in median graphs with bounded dimension d, i.e. the dimension of the largest induced hypercube (note that 1-dimensional median graphs are exactly the forests). This prerequisite on d is not necessarily anymore to determine all eccentricities in subquadratic time. The execution time of our algorithm is O(n^{1.6456}log^{O(1)} n).
Cite as

Pierre Bergé, Guillaume Ducoffe, and Michel Habib. Subquadratic-Time Algorithm for the Diameter and All Eccentricities on Median Graphs. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 9:1-9:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{berge_et_al:LIPIcs.STACS.2022.9,
  author =	{Berg\'{e}, Pierre and Ducoffe, Guillaume and Habib, Michel},
  title =	{{Subquadratic-Time Algorithm for the Diameter and All Eccentricities on Median Graphs}},
  booktitle =	{39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
  pages =	{9:1--9:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-222-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{219},
  editor =	{Berenbrink, Petra and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.9},
  URN =		{urn:nbn:de:0030-drops-158192},
  doi =		{10.4230/LIPIcs.STACS.2022.9},
  annote =	{Keywords: Diameter, Eccentricities, Metric graph theory, Median graphs, Hypercubes}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2017.43
Maximum Induced Matching Algorithms via Vertex Ordering Characterizations

Authors: Michel Habib and Lalla Mouatadid

Published in: LIPIcs, Volume 92, 28th International Symposium on Algorithms and Computation (ISAAC 2017)

Abstract
We study the maximum induced matching problem on a graph G. Induced matchings correspond to independent sets in L^2(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 \mathcal{G} characterized by a vertex ordering, and a graph G=(V,E) \in \mathcal{G} with a corresponding vertex ordering \sigma of V, one can produce (in linear time in the size of G) an ordering on the vertices of L^2(G), that shows that L^2(G) \in \mathcal{G} - for a number of graph classes \mathcal{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 L^2(\cdot) for various graph families. Furthermore, these orderings on L^2(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 \mathcal{O}(mn) time algorithm to compute a maximum weighted induced matching on cocomparability graphs; an improvement from the best known \mathcal{O}(n^4) time algorithm for the unweighted case.
Cite as

Michel Habib and Lalla Mouatadid. Maximum Induced Matching Algorithms via Vertex Ordering Characterizations. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 43:1-43:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{habib_et_al:LIPIcs.ISAAC.2017.43,
  author =	{Habib, Michel and Mouatadid, Lalla},
  title =	{{Maximum Induced Matching Algorithms via Vertex Ordering Characterizations}},
  booktitle =	{28th International Symposium on Algorithms and Computation (ISAAC 2017)},
  pages =	{43:1--43:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-054-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{92},
  editor =	{Okamoto, Yoshio and Tokuyama, Takeshi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2017.43},
  URN =		{urn:nbn:de:0030-drops-82178},
  doi =		{10.4230/LIPIcs.ISAAC.2017.43},
  annote =	{Keywords: Maximum induced matching, Independent set, Vertex ordering charac- terization, Graph classes, Fast algorithms, Cocomparability graphs}
}
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