arXiv subjects
Publications and source records attributed to Nicolas Bousquet.
In this short note, we prove that every twin-free graph on $n$ vertices contains a locating-dominating set of size at most $\lceil\frac{5}{8}n\rceil$. This improves the earlier bound of $\lfloor\frac{2}{3}n\rfloor$ due to Foucaud, Henning, Löwenstein and Sasse from 2016, and makes some progress towards the well-studied locating-dominating conjecture of Garijo, González and Márquez.
Detecting specific structures in a network has been a very active theme of research in distributed computing for at least a decade. In this paper, we start the study of subgraph detection from the perspective of local certification. Remember that a local certification is a distributed mechanism enabling the nodes of a network to check the correctness of the current configuration, thanks to small pieces of information called certificates. Our main question is: For a given graph $H$, what is the minimum certificate size that allows checking that the network does not contain $H$ as a (possibly induced) subgraph? We show a variety of lower and upper bounds, uncovering an interesting interplay between the optimal certificate size, the size of the forbidden subgraph, and the locality of the verification. Along the way we introduce several new technical tools, in particular what we call the \emph{layered map}, which is not specific to forbidden subgraphs and that we expect to be useful for certifying many other properties.
Performing an additive decomposition of arbitrary functions of random elements is paramount for global sensitivity analysis and, therefore, the interpretation of black-box models. The well-known seminal work of Hoeffding characterized the summands in such a decomposition in the particular case of mutually independent inputs. Going beyond the framework of independent inputs has been an ongoing challenge in the literature. Existing solutions have so far required constraining assumptions or suffer from a lack of interpretability. In this paper, we generalize Hoeffding's decomposition for dependent inputs under very mild conditions. For that purpose, we propose a novel framework to handle dependencies based on probability theory, functional analysis, and combinatorics. It allows for characterizing two reasonable assumptions on the dependence structure of the inputs: non-perfect functional dependence and non-degenerate stochastic dependence. We then show that any square-integrable, real-valued function of random elements respecting these two assumptions can be uniquely additively decomposed and offer a characterization of the summands using oblique projections. We then introduce and discuss the theoretical properties and practical benefits of the sensitivity indices that ensue from this decomposition. Finally, the decomposition is analytically illustrated on bivariate functions of Bernoulli inputs.
In this paper, we investigate how local rules enforced at every node can influence the topology of a network. More precisely, we establish several results on the diameter of trees as a function of the number of nodes, as listed below. These results have important consequences on the landscape of locally checkable labelings (LCL) on \emph{unbounded} degree graphs, a case in which our lack of knowledge is in striking contrast with that of \emph{bounded degree graphs}, that has been intensively studied recently. [See paper for full abstract.]
Input variables in numerical models are often subject to several levels of uncertainty, usually modeled by probability distributions. In the context of uncertainty quantification applied to these models, studying the robustness of output quantities with respect to the input distributions requires: (a) defining variational classes for these distributions; (b) calculating boundary values for the output quantities of interest with respect to these variational classes. The latter should be defined in a way that is consistent with the information structure defined by the ``baseline'' choice of input distributions. Considering parametric families, the variational classes are defined using the geodesic distance in their Riemannian manifold, a generic approach to such problems. Theoretical results and application tools are provided to justify and facilitate the implementation of such robustness studies, in concrete situations where these distributions are truncated -- a setting frequently encountered in applications. The feasibility of our approach is illustrated in a simplified industrial case study.
Dominating sets in graphs are often used to model some monitoring of the graph: guards are posted on the vertices of the dominating set, and they can thus react to attacks occurring on the unguarded vertices by moving there (yielding a new set of guards, which may not be dominating anymore). A dominating set is eternal if it can endlessly resist to attacks. From the attacker's perspective, if we are given a non-eternal dominating set, the question is to determine how fast can we provoke an attack that cannot be handled by a neighboring guard. We investigate this question from a computational complexity point of view, by showing that this question is PSPACE-hard, even for graph classes where finding a minimum eternal dominating set is in P. We then complement this result by giving polynomial time algorithms for cographs and trees, and showing a connection with tree-depth for the latter. We also investigate the problem from a parameterized complexity perspective, mainly considering two parameters: the number of guards and the number of steps.
An st-shortest path, or st-path for short, in a graph G is a shortest (induced) path from s to t in G. Two st-paths are said to be adjacent if they differ on exactly one vertex. A reconfiguration sequence between two st-paths P and Q is a sequence of adjacent st-paths starting from P and ending at Q. Deciding whether there exists a reconfiguration sequence between two given $st$-paths is known to be PSPACE-complete, even on restricted classes of graphs such as graphs of bounded bandwidth (hence pathwidth). On the positive side, and rather surprisingly, the problem is polynomial-time solvable on planar graphs. In this paper, we study the parameterized complexity of the Shortest Path Reconfiguration (SPR) problem. We show that SPR is W[1]-hard parameterized by k + \ell, even when restricted to graphs of bounded (constant) degeneracy; here k denotes the number of edges on an st-path, and \ell denotes the length of a reconfiguration sequence from P to Q. We complement our hardness result by establishing the fixed-parameter tractability of SPR parameterized by \ell and restricted to nowhere-dense classes of graphs. Additionally, we establish fixed-parameter tractability of SPR when parameterized by the treedepth, by the cluster-deletion number, or by the modular-width of the input graph.
This article provides a critical review of the main methods used to produce conservative estimators of probabilities of rare events, or critical failures, for reliability and certification studies in the broadest sense. These probabilities must theoretically be calculated from simulations of (certified) numerical models, but which typically suffer from prohibitive computational costs. This occurs frequently, for instance, for complex and critical industrial systems. We focus therefore in adapting the common use of surrogates to replace these numerical models, the aim being to offer a high level of confidence in the results. We suggest avenues of research to improve the guarantees currently reachable.
We show that every $3$-connected $K_{2,\ell}$-minor free graph with minimum degree at least $4$ has maximum degree at most $7\ell$. As a consequence, we show that every 3-connected $K_{2,\ell}$-minor free graph with minimum degree at least $5$ and no twins of degree $5$ has bounded size. Our proofs use Steiner trees and nested cuts; in particular, they do not rely on Ding's characterization of $K_{2,\ell}$-minor free graphs.
In this work, we provide an upper bound for global certification of graph homomorphism, a generalization of graph coloring. In certification, the nodes of a network should decide if the network satisfies a given property, thanks to small pieces of information called certificates. Here, there is only one global certificate which is shared by all the nodes, and the property we want to certify is the existence of a graph homomorphism to a given graph. For bipartiteness, a special case of graph homomorphism, Feuilloley and Hirvonen proved in~\cite{FeuilloleyH18} some upper and lower bounds on the size of the optimal certificate, and made the conjecture that their lower bound could be improved to match their upper bound. We prove that this conjecture is false: their lower bound was in fact optimal, and we prove it by providing the matching upper bound using a known result of perfect hashing.
Given a graph $G$ and two independent sets of $G$, the independent set reconfiguration problem asks whether one independent set can be transformed into the other by moving a single vertex at a time, such that at each intermediate step we have an independent set of $G$. We study the complexity of this problem for $H$-free graphs under the token sliding and token jumping rule. Our contribution is twofold. First, we prove a reconfiguration analogue of Alekseev's theorem, showing that the problem is PSPACE-complete unless $H$ is a path or a subdivision of the claw. We then show that under the token sliding rule, the problem admits a polynomial-time algorithm if the input graph is fork-free.
We say that a graph is $k$-mixing if it is possible to transform any $k$-coloring into any other via a sequence of single vertex recolorings keeping a proper coloring all along. Cereceda, van den Heuvel and Johnson proved that deciding if a graph is $3$-mixing is co-NP-complete and left open the case $k \ge 4$. We prove that for every $k \ge 4$, $k$-mixing is co-NP-hard.
Local certification is a distributed mechanism enabling the nodes of a network to check the correctness of the current configuration, thanks to small pieces of information called certificates. For many classic global properties, like checking the acyclicity of the network, the optimal size of the certificates depends on the size of the network, $n$. In this paper, we focus on properties for which the size of the certificates does not depend on $n$ but on other parameters. We focus on three such important properties and prove tight bounds for all of them. Namely, we prove that the optimal certification size is: $Θ(\log k)$ for $k$-colorability (and even exactly $\lceil \log k \rceil$ bits in the anonymous model while previous works had only proved a $2$-bit lower bound); $(1/2)\log t+o(\log t)$ for dominating sets at distance $t$ (an unexpected and tighter-than-usual bound) ; and $Θ(\log Δ)$ for perfect matching in graphs of maximum degree $Δ$ (the first non-trivial bound parameterized by $Δ$). We also prove some surprising upper bounds, for example, certifying the existence of a perfect matching in a planar graph can be done with only two bits. In addition, we explore various specific cases for these properties, in particular improving our understanding of the trade-off between locality of the verification and certificate size.
A non-crossing spanning tree of a set of points in the plane is a spanning tree whose edges pairwise do not cross. Avis and Fukuda in 1996 proved that there always exists a flip sequence of length at most $2n-4$ between any pair of non-crossing spanning trees (where $n$ denotes the number of points). Hernando et al. proved that the length of a minimal flip sequence can be of length at least $\frac 32 n$. Two recent results of Aichholzer et al. and Bousquet et al. improved the Avis and Fukuda upper bound by proving that there always exists a flip sequence of length respectively at most $2n - \log n$ and $2n - \sqrt{n}$. We improve the upper bound by a linear factor for the first time in 25 years by proving that there always exists a flip sequence between any pair of non-crossing spanning trees $T_1,T_2$ of length at most $c n$ where $c \approx 1.95$. Our result is actually stronger since we prove that, for any two trees $T_1,T_2$, there exists a flip sequence from $T_1$ to $T_2$ of length at most $c |T_1 \setminus T_2|$. We also improve the best lower bound in terms of the symmetric difference by proving that there exists a pair of trees $T_1,T_2$ such that a minimal flip sequence has length $\frac 53 |T_1 \setminus T_2|$, improving the lower bound of Hernando et al. by considering the symmetric difference instead of the number of vertices. We generalize this lower bound construction to non-crossing flips (where we close the gap between upper and lower bounds) and rotations.
Given two $k$-dicolourings of a digraph $D$, we prove that it is PSPACE-complete to decide whether we can transform one into the other by recolouring one vertex at each step while maintaining a dicolouring at any step even for $k=2$ and for digraphs with maximum degree $5$ or oriented planar graphs with maximum degree $6$. A digraph is said to be $k$-mixing if there exists a transformation between any pair of $k$-colourings. We show that every digraph $D$ is $k$-mixing for all $k\geq δ^*_{\min}(D)+2$, generalizing a result due to Dyer et al. We also prove that every oriented graph $\vec{G}$ is $k$-mixing for all $k\geq δ^*_{\max}(\vec{G}) +1$ and for all $k\geq δ^*_{\rm avg}(\vec{G})+1$. We conjecture that, for every digraph $D$, the dicolouring graph of $D$ on $k\geq δ_{\min}^*(D)+2$ colours has diameter at most $O(|V(D)|^2)$ and give some evidences. We first prove that the dicolouring graph of any digraph $D$ on $k\geq 2δ_{\min}^*(D) + 2$ colours has linear diameter, extending a result from Bousquet and Perarnau. We also prove that the conjecture is true when $k\geq \frac{3}{2}(δ_{\min}^*(D)+1)$. Restricted to the special case of oriented graphs, we prove that the dicolouring graph of any subcubic oriented graph on $k\geq 2$ colours is connected and has diameter at most $2n$. We conjecture that every non $2$-mixing oriented graph has maximum average degree at least $4$, and we provide some support for this conjecture by proving it on the special case of $2$-freezable oriented graphs. More generally, we show that every $k$-freezable oriented graph on $n$ vertices must contain at least $kn + k(k-2)$ arcs, and we give a family of $k$-freezable oriented graphs that reach this bound. In the general case, we prove as a partial result that every non $2$-mixing oriented graph has maximum average degree at least $\frac{7}{2}$.
Recoloring a graph is about finding a sequence of proper colorings of this graph from an initial coloring $σ$ to a target coloring $η$. Adding the constraint that each pair of consecutive colorings must differ on exactly one vertex, one asks: Is there a sequence of colorings from $σ$ to $η$? If yes, how short can it be? In this paper, we focus on $(Δ+1)$-colorings of graphs of maximum degree $Δ$. Feghali, Johnson and Paulusma proved that, if both colorings are non-frozen (i.e. we can change the color of a least one vertex), then a quadratic recoloring sequence always exists. We improve their result by proving that there actually exists a linear transformation (assuming that $Δ$ is a constant). In addition, we prove that the core of our algorithm can be performed locally. Informally, this means that after some preprocessing, the color changes that a given node has to perform only depend on the colors of the vertices in a constant size neighborhood. We make this precise by designing of an efficient recoloring algorithm in the LOCAL model of distributed computing.
A coloring of a digraph is a partition of its vertex set such that each class induces a digraph with no directed cycles. A digraph is $k$-chromatic if $k$ is the minimum number of classes in such partition, and a digraph is oriented if there is at most one arc between each pair of vertices. Clearly, the smallest $k$-chromatic digraph is the complete digraph on $k$ vertices, but determining the order of the smallest $k$-chromatic oriented graphs is a challenging problem. It is known that the smallest $2$-, $3$- and $4$-chromatic oriented graphs have $3$, $7$ and $11$ vertices, respectively. In 1994, Neumann-Lara conjectured that a smallest $5$-chromatic oriented graph has $17$ vertices. We solve this conjecture and show that the correct order is $19$.
The metric dimension has been introduced independently by Harary, Melter and Slater in 1975 to identify vertices of a graph G using its distances to a subset of vertices of G. A resolving set X of a graph G is a subset of vertices such that, for every pair (u,v) of vertices of G, there is a vertex x in X such that the distance between x and u and the distance between x and v are distinct. The metric dimension of the graph is the minimum size of a resolving set. Computing the metric dimension of a graph is NP-hard even on split graphs and interval graphs. Bonnet and Purohit proved that the metric dimension problem is W[1]-hard parameterized by treewidth. Li and Pilipczuk strenghtened this result by showing that it is NP-hard for graphs of treewidth. In this article, we prove that that metric dimension is FPT parameterized by treewidth in chordal graphs.