arXiv subjects

Nicolas Bousquet

Publications and source records attributed to Nicolas Bousquet.

At least 73 records · Page 4Linked to original sources

arXiv2021

(Sub)linear kernels for edge modification problems towards structured graph classes

In a (parameterized) graph edge modification problem, we are given a graph $G$, an integer $k$ and a (usually well-structured) class of graphs $\mathcal{G}$, and ask whether it is possible to transform $G$ into a graph $G' \in \mathcal{G}$ by adding and/or removing at most $k$ edges. Parameterized graph edge modification problems received considerable attention in the last decades. In this paper, we focus on finding small kernels for edge modification problems. One of the most studied problems is the Cluster Editing problem, in which the goal is to partition the vertex set into a disjoint union of cliques. Even if this problem admits a $2k$ kernel [Cao, 2012], this kernel does not reduce the size of most instances. Therefore, we explore the question of whether linear kernels are a theoretical limit in edge modification problems, in particular when the target graphs are very structured (such as a partition into cliques for instance). We prove, as far as we know, the first sublinear kernel for an edge modification problem. Namely, we show that Clique + Independent Set Deletion, which is a restriction of Cluster Deletion, admits a kernel of size $O(k/\log k)$. We also obtain small kernels for several other edge modification problems. We prove that Split Addition (and the equivalent Split Deletion) admits a linear kernel, improving the existing quadratic kernel of Ghosh et al. [Ghosh et al., 2015]. We complement this result by proving that Trivially Perfect Addition admits a quadratic kernel (improving the cubic kernel of Guo [Guo, 2007]), and finally prove that its triangle-free version (Starforest Deletion) admits a linear kernel, which is optimal under ETH.

cs.DS↗

arXiv2021

Local certification of graph decompositions and applications to minor-free classes

Local certification consists in assigning labels to the nodes of a network to certify that some given property is satisfied, in such a way that the labels can be checked locally. In the last few years, certification of graph classes received a considerable attention. The goal is to certify that a graph $G$ belongs to a given graph class~$\mathcal{G}$. Such certifications with labels of size $O(\log n)$ (where $n$ is the size of the network) exist for trees, planar graphs and graphs embedded on surfaces. Feuilloley et al. ask if this can be extended to any class of graphs defined by a finite set of forbidden minors. In this work, we develop new decomposition tools for graph certification, and apply them to show that for every small enough minor $H$, $H$-minor-free graphs can indeed be certified with labels of size $O(\log n)$. We also show matching lower bounds with a simple new proof technique.

cs.DC↗

arXiv2021

Distributed recoloring of interval and chordal graphs

One of the fundamental and most-studied algorithmic problems in distributed computing on networks is graph coloring, both in bounded-degree and in general graphs. Recently, the study of this problem has been extended in two directions. First, the problem of recoloring, that is computing an efficient transformation between two given colorings (instead of computing a new coloring), has been considered, both to model radio network updates, and as a useful subroutine for coloring. Second, as it appears that general graphs and bounded-degree graphs do not model real networks very well (with, respectively, pathological worst-case topologies and too strong assumptions), coloring has been studied in more specific graph classes. In this paper, we study the intersection of these two directions: distributed recoloring in two relevant graph classes, interval and chordal graphs.

cs.DC↗

arXiv2021

Asymptotic Dimension of Minor-Closed Families and Assouad-Nagata Dimension of Surfaces

The asymptotic dimension is an invariant of metric spaces introduced by Gromov in the context of geometric group theory. In this paper, we study the asymptotic dimension of metric spaces generated by graphs and their shortest path metric and show their applications to some continuous spaces. The asymptotic dimension of such graph metrics can be seen as a large scale generalisation of weak diameter network decomposition which has been extensively studied in computer science. We prove that every proper minor-closed family of graphs has asymptotic dimension at most 2, which gives optimal answers to a question of Fujiwara and Papasoglu and (in a strong form) to a problem raised by Ostrovskii and Rosenthal on minor excluded groups. For some special minor-closed families, such as the class of graphs embeddable in a surface of bounded Euler genus, we prove a stronger result and apply this to show that complete Riemannian surfaces have Assouad-Nagata dimension at most 2. Furthermore, our techniques allow us to prove optimal results for the asymptotic dimension of graphs of bounded layered treewidth and graphs of polynomial growth, which are graph classes that are defined by purely combinatorial notions and properly contain graph classes with some natural topological and geometric flavours.

math.CO↗

arXiv2021

Distributed algorithms for fractional coloring

In this paper we study fractional coloring from the angle of distributed computing. Fractional coloring is the linear relaxation of the classical notion of coloring, and has many applications, in particular in scheduling. It was proved by Hasemann, Hirvonen, Rybicki and Suomela (2016) that for every real $α>1$ and integer $Δ$, a fractional coloring of total weight at most $α(Δ+1)$ can be obtained deterministically in a single round in graphs of maximum degree $Δ$, in the LOCAL model of computation. However, a major issue of this result is that the output of each vertex has unbounded size. Here we prove that even if we impose the more realistic assumption that the output of each vertex has constant size, we can find fractional colorings of total weight arbitrarily close to known tight bounds for the fractional chromatic number in several cases of interest. More precisely, we show that for any fixed $ε> 0$ and $Δ$, a fractional coloring of total weight at most $Δ+ε$ can be found in $O(\log^*n)$ rounds in graphs of maximum degree $Δ$ with no $K_{Δ+1}$, while finding a fractional coloring of total weight at most $Δ$ in this case requires $Ω(\log \log n)$ rounds for randomized algorithms and $Ω( \log n)$ rounds for deterministic algorithms. We also show how to obtain fractional colorings of total weight at most $2+ε$ in grids of any fixed dimension, for any $ε>0$, in $O(\log^*n)$ rounds. Finally, we prove that in sparse graphs of large girth from any proper minor-closed family we can find a fractional coloring of total weight at most $2+ε$, for any $ε>0$, in $O(\log n)$ rounds.

cs.DC↗

arXiv2021

TS-Reconfiguration of Dominating Sets in circle and circular-arc graphs

We study the dominating set reconfiguration problem with the token sliding rule. It consists, given a graph G=(V,E) and two dominating sets D_s and D_t of G, in determining if there exists a sequence S= of dominating sets of G such that for any two consecutive dominating sets D_r and D_{r+1} with r<t, D_{r+1}=(D_r\ u) U v, where uv is an edge of G. In a recent paper, Bonamy et al studied this problem and raised the following questions: what is the complexity of this problem on circular arc graphs? On circle graphs? In this paper, we answer both questions by proving that the problem is polynomial on circular-arc graphs and PSPACE-complete on circle graphs.

math.CO↗

arXiv2020

Recoloring graphs of treewidth 2

Two (proper) colorings of a graph are adjacent if they differ on exactly one vertex. Jerrum proved that any $(d + 2)$-coloring of any d-degenerate graph can be transformed into any other via a sequence of adjacent colorings. A result of Bonamy et al. ensures that a shortest transformation can have a quadratic length even for $d = 1$. Bousquet and Perarnau proved that a linear transformation exists for between $(2d + 2)$-colorings. It is open to determine if this bound can be reduced. In this note, we prove that it can be reduced for graphs of treewidth 2, which are 2-degenerate. There exists a linear transformation between 5-colorings. It completes the picture for graphs of treewidth 2 since there exist graphs of treewidth 2 such a linear transformation between 4-colorings does not exist.

cs.DM↗

arXiv2020

Surfaces have (asymptotic) dimension 2

The asymptotic dimension is an invariant of metric spaces introduced by Gromov in the context of geometric group theory. When restricted to graphs and their shortest paths metric, the asymptotic dimension can be seen as a large scale version of weak diameter colorings (also known as weak diameter network decompositions), i.e. colorings in which each monochromatic component has small weak diameter. In this paper, we prove that for any $p$, the class of graphs excluding $K_{3,p}$ as a minor has asymptotic dimension at most 2. This implies that the class of all graphs embeddable on any fixed surface (and in particular the class of planar graphs) has asymptotic dimension 2, which gives a positive answer to a recent question of Fujiwara and Papasoglu. Our result extends from graphs to Riemannian surfaces. We also prove that graphs of bounded pathwidth have asymptotic dimension at most 1 and graphs of bounded layered pathwidth have asymptotic dimension at most 2. We give some applications of our techniques to graph classes defined in a topological or geometrical way, and to graph classes of polynomial growth. Finally we prove that the class of bounded degree graphs from any fixed proper minor-closed class has asymptotic dimension at most 2. This can be seen as a large scale generalization of the result that bounded degree graphs from any fixed proper minor-closed class are 3-colorable with monochromatic components of bounded size. This also implies that (infinite) Cayley graphs avoiding some minor have asymptotic dimension at most 2, which solves a problem raised by Ostrovskii and Rosenthal.

math.CO↗

arXiv2020

On girth and the parameterized complexity of token sliding and token jumping

In the Token Jumping problem we are given a graph $G = (V,E)$ and two independent sets $S$ and $T$ of $G$, each of size $k \geq 1$. The goal is to determine whether there exists a sequence of $k$-sized independent sets in $G$, $\langle S_0, S_1, \ldots, S_\ell \rangle$, such that for every $i$, $|S_i| = k$, $S_i$ is an independent set, $S = S_0$, $S_\ell = T$, and $|S_i ΔS_{i+1}| = 2$. In other words, if we view each independent set as a collection of tokens placed on a subset of the vertices of $G$, then the problem asks for a sequence of independent sets which transforms $S$ to $T$ by individual token jumps which maintain the independence of the sets. This problem is known to be PSPACE-complete on very restricted graph classes, e.g., planar bounded degree graphs and graphs of bounded bandwidth. A closely related problem is the Token Sliding problem, where instead of allowing a token to jump to any vertex of the graph we instead require that a token slides along an edge of the graph. Token Sliding is also known to be PSPACE-complete on the aforementioned graph classes. We investigate the parameterized complexity of both problems on several graph classes, focusing on the effect of excluding certain cycles from the input graph. In particular, we show that both Token Sliding and Token Jumping are fixed-parameter tractable on $C_4$-free bipartite graphs when parameterized by $k$. For Token Jumping, we in fact show that the problem admits a polynomial kernel on $\{C_3,C_4\}$-free graphs. In the case of Token Sliding, we also show that the problem admits a polynomial kernel on bipartite graphs of bounded degree. We believe both of these results to be of independent interest. We complement these positive results by showing that, for any constant $p \geq 4$, both problems are W[1]-hard on $\{C_4, \dots, C_p\}$-free graphs and Token Sliding remains W[1]-hard even on bipartite graphs.

cs.CC↗

arXiv2020

Packing and covering balls in graphs excluding a minor

We prove that for every integer $t\ge 1$ there exists a constant $c_t$ such that for every $K_t$-minor-free graph $G$, and every set $S$ of balls in $G$, the minimum size of a set of vertices of $G$ intersecting all the balls of $S$ is at most $c_t$ times the maximum number of vertex-disjoint balls in $S$. This was conjectured by Chepoi, Estellon, and Vaxès in 2007 in the special case of planar graphs and of balls having the same radius.

math.CO↗

arXiv2020

Linear transformations between dominating sets in the TAR-model

Given a graph $G$ and an integer $k$, a token addition and removal ({\sf TAR} for short) reconfiguration sequence between two dominating sets $D_{\sf s}$ and $D_{\sf t}$ of size at most $k$ is a sequence $S= \langle D_0 = D_{\sf s}, D_1 \ldots, D_\ell = D_{\sf t} \rangle$ of dominating sets of $G$ such that any two consecutive dominating sets differ by the addition or deletion of one vertex, and no dominating set has size bigger than $k$. We first improve a result of Haas and Seyffarth, by showing that if $k=Γ(G)+α(G)-1$ (where $Γ(G)$ is the maximum size of a minimal dominating set and $α(G)$ the maximum size of an independent set), then there exists a linear {\sf TAR} reconfiguration sequence between any pair of dominating sets. We then improve these results on several graph classes by showing that the same holds for $K_{\ell}$-minor free graph as long as $k \ge Γ(G)+O(\ell \sqrt{\log \ell})$ and for planar graphs whenever $k \ge Γ(G)+3$. Finally, we show that if $k=Γ(G)+tw(G)+1$, then there also exists a linear transformation between any pair of dominating sets.

cs.DM↗

arXiv2020

Reconfiguration of Spanning Trees with Many or Few Leaves

Let $G$ be a graph and $T_1,T_2$ be two spanning trees of $G$. We say that $T_1$ can be transformed into $T_2$ via an edge flip if there exist two edges $e \in T_1$ and $f$ in $T_2$ such that $T_2= (T_1 \setminus e) \cup f$. Since spanning trees form a matroid, one can indeed transform a spanning tree into any other via a sequence of edge flips, as observed by Ito et al. We investigate the problem of determining, given two spanning trees $T_1,T_2$ with an additional property $Π$, if there exists an edge flip transformation from $T_1$ to $T_2$ keeping property $Π$ all along. First we show that determining if there exists a transformation from $T_1$ to $T_2$ such that all the trees of the sequence have at most $k$ (for any fixed $k \ge 3$) leaves is PSPACE-complete. We then prove that determining if there exists a transformation from $T_1$ to $T_2$ such that all the trees of the sequence have at least $k$ leaves (where $k$ is part of the input) is PSPACE-complete even restricted to split, bipartite or planar graphs. We complete this result by showing that the problem becomes polynomial for cographs, interval graphs and when $k=n-2$.

cs.DS↗

arXiv2020

A note on the simultaneous edge coloring

Let $G=(V,E)$ be a graph. A (proper) $k$-edge-coloring is a coloring of the edges of $G$ such that any pair of edges sharing an endpoint receive distinct colors. A classical result of Vizing ensures that any simple graph $G$ admits a $(Δ(G)+1)$-edge coloring where $Δ(G)$ denotes the maximum degreee of $G$. Recently, Cabello raised the following question: given two graphs $G_1,G_2$ of maximum degree $Δ$ on the same set of vertices $V$, is it possible to edge-color their (edge) union with $Δ+2$ colors in such a way the restriction of $G$ to respectively the edges of $G_1$ and the edges of $G_2$ are edge-colorings? More generally, given $\ell$ graphs, how many colors do we need to color their union in such a way the restriction of the coloring to each graph is proper? In this short note, we prove that we can always color the union of the graphs $G_1,\ldots,G_\ell$ of maximum degree $Δ$ with $Ω(\sqrt{\ell} \cdot Δ)$ colors and that there exist graphs for which this bound is tight up to a constant multiplicative factor. Moreover, for two graphs, we prove that at most $\frac 32 Δ+4$ colors are enough which is, as far as we know, the best known upper bound.

math.CO↗

arXiv2019

When Maximum Stable Set can be solved in FPT time

Maximum Independent Set (MIS for short) is in general graphs the paradigmatic $W[1]$-hard problem. In stark contrast, polynomial-time algorithms are known when the inputs are restricted to structured graph classes such as, for instance, perfect graphs (which includes bipartite graphs, chordal graphs, co-graphs, etc.) or claw-free graphs. In this paper, we introduce some variants of co-graphs with parameterized noise, that is, graphs that can be made into disjoint unions or complete sums by the removal of a certain number of vertices and the addition/deletion of a certain number of edges per incident vertex, both controlled by the parameter. We give a series of FPT Turing-reductions on these classes and use them to make some progress on the parameterized complexity of MIS in $H$-free graphs. We show that for every fixed $t \geqslant 1$, MIS is FPT in $P(1,t,t,t)$-free graphs, where $P(1,t,t,t)$ is the graph obtained by substituting all the vertices of a four-vertex path but one end of the path by cliques of size $t$. We also provide randomized FPT algorithms in dart-free graphs and in cricket-free graphs. This settles the FPT/W[1]-hard dichotomy for five-vertex graphs $H$.

cs.DS↗

arXiv2019

Graph Isomorphism for $(H_1,H_2)$-free Graphs: An Almost Complete Dichotomy

We resolve the computational complexity of Graph Isomorphism for classes of graphs characterized by two forbidden induced subgraphs $H_1$ and $H_2$ for all but six pairs $(H_1,H_2)$. Schweitzer had previously shown that the number of open cases was finite, but without specifying the open cases. Grohe and Schweitzer proved that Graph Isomorphism is polynomial-time solvable on graph classes of bounded clique-width. Our work combines known results such as these with new results. By exploiting a relationship between Graph Isomorphism and clique-width, we simultaneously reduce the number of open cases for boundedness of clique-width for $(H_1,H_2)$-free graphs to five.

cs.DM↗

arXiv2019

Linear transformations between colorings in chordal graphs

Let $k$ and $d$ be such that $k \ge d+2$. Consider two $k$-colorings of a $d$-degenerate graph $G$. Can we transform one into the other by recoloring one vertex at each step while maintaining a proper coloring at any step? Cereceda et al. answered that question in the affirmative, and exhibited a recolouring sequence of exponential length. If $k=d+2$, we know that there exists graphs for which a quadratic number of recolorings is needed. And when $k=2d+2$, there always exists a linear transformation. In this paper, we prove that, as long as $k \ge d+4$, there exists a transformation of length at most $f(Δ) \cdot n$ between any pair of $k$-colorings of chordal graphs (where $Δ$ denotes the maximum degree of the graph). The proof is constructive and provides a linear time algorithm that, given two $k$-colorings $c_1,c_2$ computes a linear transformation between $c_1$ and $c_2$.

cs.DM↗

arXiv2019

The Perfect Matching Reconfiguration Problem

We study the perfect matching reconfiguration problem: Given two perfect matchings of a graph, is there a sequence of flip operations that transforms one into the other? Here, a flip operation exchanges the edges in an alternating cycle of length four. We are interested in the complexity of this decision problem from the viewpoint of graph classes. We first prove that the problem is PSPACE-complete even for split graphs and for bipartite graphs of bounded bandwidth with maximum degree five. We then investigate polynomial-time solvable cases. Specifically, we prove that the problem is solvable in polynomial time for strongly orderable graphs (that include interval graphs and strongly chordal graphs), for outerplanar graphs, and for cographs (also known as $P_4$-free graphs). Furthermore, for each yes-instance from these graph classes, we show that a linear number of flip operations is sufficient and we can exhibit a corresponding sequence of flip operations in polynomial time.

cs.DS↗

arXiv2019

A polynomial version of Cereceda's conjecture

Let $k$ and $d$ be such that $k \ge d+2$. Consider two $k$-colourings of a $d$-degenerate graph $G$. Can we transform one into the other by recolouring one vertex at each step while maintaining a proper coloring at any step? Cereceda et al. answered that question in the affirmative, and exhibited a recolouring sequence of exponential length. However, Cereceda conjectured that there should exist one of quadratic length. The $k$-reconfiguration graph of $G$ is the graph whose vertices are the proper $k$-colourings of $G$, with an edge between two colourings if they differ on exactly one vertex. Cereceda's conjecture can be reformulated as follows: the diameter of the $(d+2)$-reconfiguration graph of any $d$-degenerate graph on $n$ vertices is $O(n^2)$. So far, the existence of a polynomial diameter is open even for $d=2$. In this paper, we prove that the diameter of the $k$-reconfiguration graph of a $d$-degenerate graph is $O(n^{d+1})$ for $k \ge d+2$. Moreover, we prove that if $k \ge \frac 32 (d+1)$ then the diameter of the $k$-reconfiguration graph is quadratic, improving the previous bound of $k \ge 2d+1$. We also show that the $5$-reconfiguration graph of planar bipartite graphs has quadratic diameter, confirming Cereceda's conjecture for this class of graphs.

cs.DM↗