arXiv subjects

Julien Baste

Publications and source records attributed to Julien Baste.

At least 19 recordsLinked to original sources

arXiv2026

On the diversity problems parameterized by the size of the solutions

Diversity optimization seeks multiple high-quality solutions that are sufficiently different from one another, providing a richer representation of the solution space than a single optimum while avoiding the prohibitive cost of complete enumeration. In this work, we introduce the notion of natural diversity, a general condition that connects a combinatorial problem $\Pi$ and a diversity measure $\texttt{dist}$. We show that if a pair $(\Pi,\texttt{dist})$ is naturally diverse and we have in hand a completion algorithm that, from a partial solution of $\Pi$, can complete it into a solution of $\Pi$, then we can produce a parameterized algorithm solving the diversity variant of $\Pi$ with regard to the distance $\texttt{dist}$ when parameterized by the size of the solutions and the number of expected solutions. Furthermore, we show that several widely used diversity measures, including pairwise disjointness, Hamming distance, Jaccard distance, and the Otsuka-Ochiai coefficient (under both minimum and sum aggregation), satisfy the natural diversity property for problems in which all the solutions have the same size. Finally, we demonstrate the applicability of our framework by deriving fixed-parameter algorithms for diverse variants of Minimum Vertex Cover and Minimum Steiner Tree. Our results broaden the scope of parameterized diversity algorithms by accommodating natural solution-size parameterizations and a wider class of diversity measures.

cs.DS↗

arXiv2026

Impact of diversity on bounded archives for multi-objective local search

This work tackles two critical challenges related to the development of metaheuristics for Multi-Objective Optimization Problems (MOOPs): the exponential growth of non-dominated solutions and the tendency of metaheuristics to disproportionately concentrate their search on a subset of the Pareto Front. To counteract the first, bounded archives are employed as a strategic mechanism for effectively managing the increasing number of non-dominated solutions. Addressing the second challenge involves an in-depth exploration of solution diversity algorithms found in existing literature. Upon recognizing that current approaches predominantly center on diversity within the objective space, this research introduces innovative methods specifically designed to enhance diversity in the solution space. Results demonstrate the efficacy of the Hamming Distance Archiving Algorithm, one of the newly proposed algorithms for multi-objective local search, surpassing the performance of the Adaptive Grid Archiving and the Hypervolume Archiving, both drawn from the literature. This outcome suggests a promising avenue for enhancing the overall efficiency of metaheuristics employed for solving MOOPs.

cs.NE↗

arXiv2025

A polynomial bound on the pathwidth of graphs edge-coverable by $k$ shortest paths

Dumas, Foucaud, Perez and Todinca (2024) recently proved that every graph whose edges can be covered by $k$ shortest paths has pathwidth at most $O(3^k)$. In this paper, we improve this upper bound on the pathwidth to a polynomial one; namely, we show that every graph whose edge set can be covered by $k$ shortest paths has pathwidth $O(k^4)$, answering a question from the same paper. Moreover, we prove that when $k\leq 3$, every such graph has pathwidth at most $k$ (and this bound is tight). Finally, we show that even though there exist graphs with arbitrarily large treewidth whose vertex set can be covered by $2$ isometric trees, every graph whose set of edges can be covered by $2$ isometric trees has treewidth at most $2$.

math.CO↗

arXiv2023

The Leafed Induced Subtree in chordal and bounded treewidth graphs

In the Fully Leafed Induced Subtrees, one is given a graph $G$ and two integers $a$ and $b$ and the question is to find an induced subtree of $G$ with $a$ vertices and at least $b$ leaves. This problem is known to be NP-complete even when the input graph is $4$-regular. Polynomial algorithms are known when the input graph is restricted to be a tree or series-parallel. In this paper we generalize these results by providing an FPT algorithm parameterized by treewidth. We also provide a polynomial algorithm when the input graph is restricted to be a chordal graph.

cs.DS↗

arXiv2022

Contraction Bidimensionality of Geometric Intersection Graphs

Given a graph $G$, we define ${\bf bcg}(G)$ as the minimum $k$ for which $G$ can be contracted to the uniformly triangulated grid $Γ_{k}$. A graph class ${\cal G}$ has the SQG${\bf C}$ property if every graph $G\in{\cal G}$ has treewidth $\mathcal{O}({\bf bcg}(G)^{c})$ for some $1\leq c<2$. The SQG${\bf C}$ property is important for algorithm design as it defines the applicability horizon of a series of meta-algorithmic results, in the framework of bidimensionality theory, related to fast parameterized algorithms, kernelization, and approximation schemes. These results apply to a wide family of problems, namely problems that are contraction-bidimensional. Our main combinatorial result reveals a wide family of graph classes that satisfy the SQG${\bf C}$ property. This family includes, in particular, bounded-degree string graphs. This considerably extends the applicability of bidimensionality theory for contraction bidimensional problems.

math.CO↗

arXiv2022

Non-monotone target sets for threshold values restricted to $0$, $1$, and the vertex degree

We consider a non-monotone activation process $(X_t)_{t\in\{ 0,1,2,\ldots\}}$ on a graph $G$, where $X_0\subseteq V(G)$, $X_t=\{ u\in V(G):|N_G(u)\cap X_{t-1}|\geq τ(u)\}$ for every positive integer $t$, and $τ:V(G)\to \mathbb{Z}$ is a threshold function. The set $X_0$ is a so-called non-monotone target set for $(G,τ)$ if there is some $t_0$ such that $X_t=V(G)$ for every $t\geq t_0$. Ben-Zwi, Hermelin, Lokshtanov, and Newman [Discrete Optimization 8 (2011) 87-96] asked whether a target set of minimum order can be determined efficiently if $G$ is a tree. We answer their question in the affirmative for threshold functions $τ$ satisfying $τ(u)\in \{ 0,1,d_G(u)\}$ for every vertex~$u$. For such restricted threshold functions, we give a characterization of target sets that allows to show that the minimum target set problem remains NP-hard for planar graphs of maximum degree $3$ but is efficiently solvable for graphs of bounded treewidth.

math.CO↗

arXiv2021

Hitting minors on bounded treewidth graphs. I. General upper bounds

For a finite collection of graphs ${\cal F}$, the ${\cal F}$-M-DELETION problem consists in, given a graph $G$ and an integer $k$, deciding whether there exists $S \subseteq V(G)$ with $|S| \leq k$ such that $G \setminus S$ does not contain any of the graphs in ${\cal F}$ as a minor. We are interested in the parameterized complexity of ${\cal F}$-M-DELETION when the parameter is the treewidth of $G$, denoted by $tw$. Our objective is to determine, for a fixed ${\cal F}$, the smallest function $f_{\cal F}$ such that {${\cal F}$-M-DELETION can be solved in time $f_{\cal F}(tw) \cdot n^{O(1)}$ on $n$-vertex graphs. We prove that $f_{\cal F}(tw) = 2^{2^{O(tw \cdot\log tw)}}$ for every collection ${\cal F}$, that $f_{\cal F}(tw) = 2^{O(tw \cdot\log tw)}$ if ${\cal F}$ contains a planar graph, and that $f_{\cal F}(tw) = 2^{O(tw)}$ if in addition the input graph $G$ is planar or embedded in a surface. We also consider the version of the problem where the graphs in ${\cal F}$ are forbidden as topological minors, called ${\cal F}$-TM-DELETION. We prove similar results for this problem, except that in the last two algorithms, instead of requiring ${\cal F}$ to contain a planar graph, we need it to contain a subcubic planar graph. This is the first of a series of articles on this topic.

cs.DS↗

arXiv2021

Hitting minors on bounded treewidth graphs. IV. An optimal algorithm

For a fixed finite collection of graphs ${\cal F}$, the ${\cal F}$-M-DELETION problem asks, given an $n$-vertex input graph $G,$ for the minimum number of vertices that intersect all minor models in $G$ of the graphs in ${\cal F}$. by Courcelle Theorem, this problem can be solved in time $f_{\cal F}(tw)\cdot n^{O(1)},$ where $tw$ is the treewidth of $G$, for some function $f_{\cal F}$ depending on ${\cal F}$ In a recent series of articles, we have initiated the programme of optimizing asymptotically the function $f_{\cal F}$. Here we provide an algorithm showing that $f_{\cal F}(tw) = 2^{O(tw\cdot \log tw)}$ for every collection ${\cal F}$. Prior to this work, the best known function $f_{\cal F}$ was double-exponential in $tw$. In particular, our algorithm vastly extends the results of Jansen et al. [SODA 2014] for the particular case ${\cal F}=\{K_5,K_{3,3}\}$ and of Kociumaka and Pilipczuk [Algorithmica 2019] for graphs of bounded genus, and answers an open problem posed by Cygan et al. [Inf Comput 2017]. We combine several ingredients such as the machinery of boundaried graphs in dynamic programming via representatives, the Flat Wall Theorem, Bidimensionality, the irrelevant vertex technique, treewidth modulators, and protrusion replacement. Together with our previous results providing single-exponential algorithms for particular collections ${\cal F}$ [Theor Comput Sci 2020] and general lower bounds [J Comput Syst Sci 2020], our algorithm yields the following complexity dichotomy when ${\cal F} = \{H\}$ contains a single connected graph $H,$ assuming the Exponential Time Hypothesis: $f_H(tw)=2^{Θ(tw)}$ if $H$ is a contraction of the chair or the banner, and $f_H(tw)=2^{Θ(tw\cdot \log tw)}$ otherwise.

cs.DS↗

arXiv2021

Hitting minors on bounded treewidth graphs. II. Single-exponential algorithms

For a finite collection of graphs ${\cal F}$, the ${\cal F}$-M-DELETION (resp. ${\cal F}$-TM-DELETION) problem consists in, given a graph $G$ and an integer $k$, decide whether there exists $S \subseteq V(G)$ with $|S| \leq k$ such that $G \setminus S$ does not contain any of the graphs in ${\cal F}$ as a minor (resp. topological minor). We are interested in the parameterized complexity of both problems when the parameter is the treewidth of $G$, denoted by $tw$, and specifically in the cases where ${\cal F}$ contains a single connected planar graph $H$. We present algorithms running in time $2^{O(tw)} \cdot n^{O(1)}$, called single-exponential, when $H$ is either $P_3$, $P_4$, $C_4$, the paw, the chair, and the banner for both $\{H\}$-M-DELETION and $\{H\}$-TM-DELETION, and when $H=K_{1,i}$, with $i \geq 1$, for $\{H\}$-TM-DELETION. Some of these algorithms use the rank-based approach introduced by Bodlaender et al. [Inform Comput, 2015]. This is the second of a series of articles on this topic, and the results given here together with other ones allow us, in particular, to provide a tight dichotomy on the complexity of $\{H\}$-M-DELETION in terms of $H$.

cs.DS↗

arXiv2021

Hitting minors on bounded treewidth graphs. III. Lower bounds

For a finite collection of graphs ${\cal F}$, the ${\cal F}$-M-DELETION problem consists in, given a graph $G$ and an integer $k$, decide whether there exists $S \subseteq V(G)$ with $|S| \leq k$ such that $G \setminus S$ does not contain any of the graphs in ${\cal F}$ as a minor. We are interested in the parameterized complexity of ${\cal F}$-M-DELETION when the parameter is the treewidth of $G$, denoted by $tw$. Our objective is to determine, for a fixed ${\cal F}$, the smallest function $f_{\cal F}$ such that ${\cal F}$-M-DELETION can be solved in time $f_{\cal F}(tw) \cdot n^{O(1)}$ on $n$-vertex graphs. We provide lower bounds under the ETH on $f_{\cal F}$ for several collections ${\cal F}$. We first prove that for any ${\cal F}$ containing connected graphs of size at least two, $f_{\cal F}(tw)= 2^{Ω(tw)}$, even if the input graph $G$ is planar. Our main contribution consists of superexponential lower bounds for a number of collections ${\cal F}$, inspired by a reduction of Bonnet et al.~[IPEC, 2017]. In particular, we prove that when ${\cal F}$ contains a single connected graph $H$ that is either $P_5$ or is not a minor of the banner (that is, the graph consisting of a $C_4$ plus a pendent edge), then $f_{\cal F}(tw)= 2^{Ω(tw \cdot \log tw)}$. This is the third of a series of articles on this topic, and the results given here together with other ones allow us, in particular, to provide a tight dichotomy on the complexity of $\{H\}$-M-DELETION, in terms of $H$, when $H$ is connected.

cs.DS↗

arXiv2021

A Neighborhood-preserving Graph Summarization

We introduce in this paper a new summarization method for large graphs. Our summarization approach retains only a user-specified proportion of the neighbors of each node in the graph. Our main aim is to simplify large graphs so that they can be analyzed and processed effectively while preserving as many of the node neighborhood properties as possible. Since many graph algorithms are based on the neighborhood information available for each node, the idea is to produce a smaller graph which can be used to allow these algorithms to handle large graphs and run faster while providing good approximations. Moreover, our compression allows users to control the size of the compressed graph by adjusting the amount of information loss that can be tolerated. The experiments conducted on various real and synthetic graphs show that our compression reduces considerably the size of the graphs. Moreover, we conducted several experiments on the obtained summaries using various graph algorithms and applications, such as node embedding, graph classification and shortest path approximations. The obtained results show interesting trade-offs between the algorithms runtime speed-up and the precision loss.

cs.DS↗

arXiv2020

Acyclic matchings in graphs of bounded maximum degree

A matching $M$ in a graph $G$ is acyclic if the subgraph of $G$ induced by the set of vertices that are incident to an edge in $M$ is a forest. We prove that every graph with $n$ vertices, maximum degree at most $Δ$, and no isolated vertex, has an acyclic matching of size at least $(1-o(1))\frac{6n}{Δ^2},$ and we explain how to find such an acyclic matching in polynomial time.

math.CO↗

arXiv2019

FPT Algorithms for Diverse Collections of Hitting Sets

In this work, we study the $d$-Hitting Set and Feedback Vertex Set problems through the paradigm of finding diverse collections of $r$ solutions of size at most $k$ each, which has recently been introduced to the field of parameterized complexity [Baste et al., 2019]. This paradigm is aimed at addressing the loss of important side information which typically occurs during the abstraction process which models real-world problems as computational problems. We use two measures for the diversity of such a collection: the sum of all pairwise Hamming distances, and the minimum pairwise Hamming distance. We show that both problems are FPT in $k + r$ for both diversity measures. A key ingredient in our algorithms is a (problem independent) network flow formulation that, given a set of `base' solutions, computes a maximally diverse collection of solutions. We believe that this could be of independent interest.

cs.DS↗

arXiv2019

Domination versus edge domination

We propose the conjecture that the domination number $γ(G)$ of a $Δ$-regular graph $G$ with $Δ\geq 1$ is always at most its edge domination number $γ_e(G)$, which coincides with the domination number of its line graph. We prove that $γ(G)\leq \left(1+\frac{2(Δ-1)}{Δ2^Δ}\right)γ_e(G)$ for general $Δ\geq 1$, and $γ(G)\leq \left(\frac{7}{6}-\frac{1}{204}\right)γ_e(G)$ for $Δ=3$. Furthermore, we verify our conjecture for cubic claw-free graphs.

math.CO↗

arXiv2019

Bounding and approximating minimum maximal matchings in regular graphs

The edge domination number $γ_e(G)$ of a graph $G$ is the minimum size of a maximal matching in $G$. It is well known that this parameter is computationally very hard, and several approximation algorithms and heuristics have been studied. In the present paper, we provide best possible upper bounds on $γ_e(G)$ for regular and non-regular graphs $G$ in terms of their order and maximum degree. Furthermore, we discuss algorithmic consequences of our results and their constructive proofs.

math.CO↗

arXiv2019

Composing dynamic programming tree-decomposition-based algorithms

Given two integers $\ell$ and $p$ as well as $\ell$ graph classes $\mathcal{H}_1,\ldots,\mathcal{H}_\ell$, the problems $\mathsf{GraphPart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell,p)$, \break $\mathsf{VertPart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell)$, and $\mathsf{EdgePart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell)$ ask, given graph $G$ as input, whether $V(G)$, $V(G)$, $E(G)$ respectively can be partitioned into $\ell$ sets $S_1, \ldots, S_\ell$ such that, for each $i$ between $1$ and $\ell$, $G[S_i] \in \mathcal{H}_i$, $G[S_i] \in \mathcal{H}_i$, $(V(G),S_i) \in \mathcal{H}_i$ respectively. Moreover in $\mathsf{GraphPart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell,p)$, we request that the number of edges with endpoints in different sets of the partition is bounded by $p$. We show that if there exist dynamic programming tree-decomposition-based algorithms for recognizing the graph classes $\mathcal{H}_i$, for each $i$, then we can constructively create a dynamic programming tree-decomposition-based algorithms for $\mathsf{GraphPart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell,p)$, $\mathsf{VertPart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell)$, and $\mathsf{EdgePart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell)$. We apply this approach to known problems. For well-studied problems, like VERTEX COVER and GRAPH $q$-COLORING, we obtain running times that are comparable to those of the best known problem-specific algorithms. For an exotic problem from bioinformatics, called DISPLAYGRAPH, this approach improves the known algorithm parameterized by treewidth.

cs.DS↗

arXiv2019

Diversity of Solutions: An Exploration Through the Lens of Fixed-Parameter Tractability Theory

When modeling an application of practical relevance as an instance of a combinatorial problem X, we are often interested not merely in finding one optimal solution for that instance, but in finding a sufficiently diverse collection of good solutions. In this work we initiate a systematic study of diversity from the point of view of fixed-parameter tractability theory. First, we consider an intuitive notion of diversity of a collection of solutions which suits a large variety of combinatorial problems of practical interest. We then present an algorithmic framework which --automatically-- converts a tree-decomposition-based dynamic programming algorithm for a given combinatorial problem X into a dynamic programming algorithm for the diverse version of X. Surprisingly, our algorithm has a polynomial dependence on the diversity parameter.

cs.DS↗

arXiv2019

Temporal Matching

A link stream is a sequence of pairs of the form $(t,\{u,v\})$, where $t\in\mathbb N$ represents a time instant and $u\neq v$. Given an integer $γ$, the $γ$-edge between vertices $u$ and $v$, starting at time $t$, is the set of temporally consecutive edges defined by $\{(t',\{u,v\}) | t' \in [t,t+γ-1]\}$. We introduce the notion of temporal matching of a link stream to be an independent $γ$-edge set belonging to the link stream. We show that the problem of computing a temporal matching of maximum size is NP-hard as soon as $γ>1$. We depict a kernelization algorithm parameterized by the solution size for the problem. As a byproduct we also give a $2$-approximation algorithm. Both our $2$-approximation and kernelization algorithms are implemented and confronted to link streams collected from real world graph data. We observe that finding temporal matchings is a sensitive question when mining our data from such a perspective as: managing peer-working when any pair of peers $X$ and $Y$ are to collaborate over a period of one month, at an average rate of at least two email exchanges every week. We furthermore design a link stream generating process by mimicking the behaviour of a random moving group of particles under natural simulation, and confront our algorithms to these generated instances of link streams. All the implementations are open source.

cs.DS↗