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We study the PSPACE-complete $k$-Canadian Traveller Problem, where a weighted graph $G=(V,E,\omega)$ with a source $s\in V$ and a target $t\in V$ are given. This problem also has a hidden input $E_* \subsetneq E$ of cardinality at most $k$ representing blocked edges. The objective is to travel from $s$ to $t$ with the minimum distance. At the beginning of the walk, the blockages $E_*$ are unknown: the traveller discovers that an edge is blocked when visiting one of its endpoints. Online algorithms, also called strategies, have been proposed for this problem and assessed with the competitive ratio, i.e. the ratio between the distance actually traversed by the traveller divided by the distance we would have traversed knowing the blockages in advance. Even though the optimal competitive ratio is $2k+1$ even on unit-weighted planar graphs of treewidth 2, we design a polynomial-time strategy achieving competitive ratio $9$ on unit-weighted outerplanar graphs. This value $9$ also stands as a lower bound for this family of graphs as we prove that, for any $\varepsilon > 0$, no strategy can achieve a competitive ratio $9-\varepsilon$. Finally, we show that it is not possible to achieve a constant competitive ratio (independent of $G$ and $k$) on weighted outerplanar graphs.

We consider here the MultiBot problem for the scheduling and the resource parametrization of jobs related to the production or the transportation of different products inside a given time horizon. Those jobs must meet known in advance demands. The time horizon is divided into several discrete identical periods representing each the time needed to proceed a job. The objective is to find a parametrization and a schedule for the jobs in such a way they require as less resources as possible. Though this problem derived from the applicative context of reconfigurable robots, we focus here on fundamental issues. We show that the resulting strongly NP-hard Multibot problem may be handled in a greedy way with an approximation ratio of $\frac{4}{3}$., Comment: 26 pages, 4 figures

For any $\varepsilon > 0$, we give a polynomial-time $n^\varepsilon$-approximation algorithm for Max Independent Set in graphs of bounded twin-width given with an $O(1)$-sequence. This result is derived from the following time-approximation trade-off: We establish an $O(1)^{2^q-1}$-approximation algorithm running in time $\exp(O_q(n^{2^{-q}}))$, for every integer $q \geqslant 0$. Guided by the same framework, we obtain similar approximation algorithms for Min Coloring and Max Induced Matching. In general graphs, all these problems are known to be highly inapproximable: for any $\varepsilon > 0$, a polynomial-time $n^{1-\varepsilon}$-approximation for any of them would imply that P$=$NP [Hastad, FOCS '96; Zuckerman, ToC '07; Chalermsook et al., SODA '13]. We generalize the algorithms for Max Independent Set and Max Induced Matching to the independent (induced) packing of any fixed connected graph $H$. In contrast, we show that such approximation guarantees on graphs of bounded twin-width given with an $O(1)$-sequence are very unlikely for Min Independent Dominating Set, and somewhat unlikely for Longest Path and Longest Induced Path. Regarding the existence of better approximation algorithms, there is a (very) light evidence that the obtained approximation factor of $n^\varepsilon$ for Max Independent Set may be best possible. This is the first in-depth study of the approximability of problems in graphs of bounded twin-width. Prior to this paper, essentially the only such result was a~polynomial-time $O(1)$-approximation algorithm for Min Dominating Set [Bonnet et al., ICALP '21]., Comment: 32 pages, 3 figures, 1 table

In the 70s, Berge introduced 1-extendable graphs (also called B-graphs), which are graphs where every vertex belongs to a maximum independent set. Motivated by an application in the design of wireless networks, we study the computational complexity of 1-extendability, the problem of deciding whether a graph is 1-extendable. We show that, in general, 1-extendability cannot be solved in $2^{o(n)}$ time assuming the Exponential Time Hypothesis, where $n$ is the number of vertices of the input graph, and that it remains NP-hard in subcubic planar graphs and in unit disk graphs (which is a natural model for wireless networks). Although 1-extendability seems to be very close to the problem of finding an independent set of maximum size (a.k.a. Maximum Independent Set), we show that, interestingly, there exist 1-extendable graphs for which Maximum Independent Set is NP-hard. Finally, we investigate a parameterized version of 1-extendability., Comment: Extended abstract in IWOCA 2022

We show that determining if an $n$-vertex graph has twin-width at most 4 is NP-complete, and requires time $2^{\Omega(n/\log n)}$ unless the Exponential-Time Hypothesis fails. Along the way, we give an elementary proof that $n$-vertex graphs subdivided at least $2 \log n$ times have twin-width at most 4. We also show how to encode trigraphs $H$ (2-edge colored graphs involved in the definition of twin-width) into graphs $G$, in the sense that every $d$-sequence (sequence of vertex contractions witnessing that the twin-width is at most $d$) of $G$ inevitably creates $H$ as an induced subtrigraph, whereas there exists a partial $d$-sequence that actually goes from $G$ to $H$. We believe that these facts and their proofs can be of independent interest., Comment: 30 pages, 17 figures

On sparse graphs, Roditty and Williams [2013] proved that no $O(n^{2-\varepsilon})$-time algorithm achieves an approximation factor smaller than $\frac{3}{2}$ for the diameter problem unless SETH fails. In this article, we solve 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 combinatiorial 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 all eccentricities in median graphs with bounded dimension $d$, i.e. the dimension of the largest induced hypercube. This prerequisite on $d$ is not necessarily anymore to determine all eccentricities in subquadratic time. The execution time of our algorithm is $O(n^{1.6408}\log^{O(1)} n)$. We provide also some satellite outcomes related to this general result. In particular, restricted to simplex graphs, this algorithm enumerates all eccentricities with a quasilinear running time. Moreover, an algorithm is proposed to compute exactly all reach centralities in time $O(2^{3d}n\log^{O(1)}n)$., Comment: 43 pages, extended abstract in STACS 2022

Median graphs form the class of graphs which is the most studied in metric graph theory. Recently, B\'en\'eteau et al. [2019] designed a linear-time algorithm computing both the $\Theta$-classes and the median set of median graphs. A natural question emerges: is there a linear-time algorithm computing the diameter and the radius for median graphs? We answer positively to this question for median graphs $G$ with constant dimension $d$, i.e. the dimension of the largest induced hypercube of $G$. We propose a combinatorial algorithm computing all eccentricities of median graphs with running time $O(2^{O(d\log d)}n)$. As a consequence, this provides us with a linear-time algorithm determining both the diameter and the radius of median graphs with $d = O(1)$, such as cube-free median graphs. As the hypercube of dimension 4 is not planar, it shows also that all eccentricities of planar median graphs can be computed in $O(n)$., Comment: 22 pages, an extended abstract of this paper will appear in the proceedings of LAGOS 2021

The parameterized complexity of counting minimum cuts stands as a natural question because Ball and Provan showed its #P-completeness. For any undirected graph $G=(V,E)$ and two disjoint sets of its vertices $S,T$, we design a fixed-parameter tractable algorithm which counts minimum edge $(S,T)$-cuts parameterized by their size $p$. Our algorithm operates on a transformed graph instance. This transformation, called drainage, reveals a collection of at most $n=\left| V \right|$ successive minimum $(S,T)$-cuts $Z_i$. We prove that any minimum $(S,T)$-cut $X$ contains edges of at least one cut $Z_i$. This observation, together with Menger's theorem, allows us to build the algorithm counting all minimum $(S,T)$-cuts with running time $2^{O(p^2)}n^{O(1)}$. Initially dedicated to counting minimum cuts, it can be modified to obtain an FPT sampling of minimum edge $(S,T)$-cuts., Comment: 13 pages, 10 figures, full version of the paper accepted in WG 2019

Constraints such as separation-of-duty are widely used to specify requirements that supplement basic authorization policies. However, the existence of constraints (and authorization policies) may mean that a user is unable to fulfill her/his organizational duties because access to resources has been denied. In short, there is a tension between the need to protect resources (using policies and constraints) and the availability of resources. Recent work on workflow satisfiability and resiliency in access control asks whether this tension compromises the ability of an organization to achieve its objectives. In this paper, we develop a new method of specifying constraints which subsumes much related work and allows a wider range of constraints to be specified. The use of such constraints leads naturally to a range of questions related to "policy existence", where a positive answer means that an organization's objectives can be realized. We analyze the complexity of these policy existence questions and, for particular sub-classes of constraints defined by our language, develop fixed-parameter tractable algorithms to solve them.

The goal of our research was to enhance local search heuristics used to construct Latin Hypercube Designs. First, we introduce the \textit{1D-move} perturbation to improve the space exploration performed by these algorithms. Second, we propose a new evaluation function $\psi_{p,\sigma}$ specifically targeting the Maximin criterion. Exhaustive series of experiments with Simulated Annealing, which we used as a typically well-behaving local search heuristics, confirm that our goal was reached as the result we obtained surpasses the best scores reported in the literature. Furthermore, the $\psi_{p,\sigma}$ function seems very promising for a wide spectrum of optimization problems through the Maximin criterion., Comment: extended version of ACM GECCO 2016 paper entitled "Search Space Exploration and an Optimization Criterion for Hard Design Problems"