Search

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., arXiv admin note: substantial text overlap with arXiv:1704.07284, arXiv:2103.06536

International audience; We consider a non-monotone activation process $(X_t)_{t\in\{ 0,1,2,\ldots\}}$ on a graph $G$, where $X_0\subseteq V(G)$, $X_t=\{ u\in V(G):|N_G(u)\cap X_{t-1}|\geq \tau(u)\}$ for every positive integer $t$, and $\tau:V(G)\to \mathbb{Z}$ is a threshold function. The set $X_0$ is a so-called non-monotone target set for $(G,\tau)$ 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 $\tau$ satisfying $\tau(u)\in \{ 0,1,d_G(u)\}$ 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.

International audience

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 $\Delta$, and no isolated vertex, has an acyclic matching of size at least $(1-o(1))\frac{6n}{\Delta^2},$ and we explain how to find such an acyclic matching in polynomial time.

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., 17 pages, 10 figures

Given two integers $\ell$ and $p$ as well as $\ell$ graph classes $\mathcal{H}_1,\ldots,\mathcal{H}_\ell$, the problems $\mathsf{GraphPart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell,p)$, $\mathsf{VertPart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell)$, 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 show that, in some known cases, the obtained running times are comparable to those of the best know algorithms.

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: $f_H(tw)=2^{\Theta(tw)}$ if $H$ is a contraction of the chair or the banner, and $f_H(tw)=2^{\Theta(tw\cdot \log tw)}$ otherwise., Comment: 51 pages, 17 figures

A matching in a graph is induced if no two of its edges are joined by an edge, and finding a large induced matching is a very hard problem. Lin et al. (Approximating weighted induced matchings, Discrete Applied Mathematics 243 (2018) 304-310) provide an approximation algorithm with ratio $\Delta$ for the weighted version of the induced matching problem on graphs of maximum degree $\Delta$. Their approach is based on an integer linear programming formulation whose integrality gap is at least $\Delta-1$, that is, their approach offers only little room for improvement in the weighted case. For the unweighted case though, we conjecture that the integrality gap is at most $\frac{5}{8}\Delta+O(1)$, and that also the approximation ratio can be improved at least to this value. We provide primal-dual approximation algorithms with ratios $(1-\epsilon) \Delta + \frac{1}{2}$ for general $\Delta$ with $\epsilon \approx 0.02005$, and $\frac{7}{3}$ for $\Delta=3$. Furthermore, we prove a best-possible bound on the fractional induced matching number in terms of the order and the maximum degree.

International audience; For a fixed graph H, we are interested in the parameterized complexity of the following problem, called {H}-M-Deletion, parameterized by the treewidth tw of the input graph: given an n-vertex graph G and an integer k, decide whether there exists S ⊆ V (G) with |S| ≤ k such that G\S does not contain H as a minor. In previous work [IPEC, 2017] we proved that if H is planar and connected, then the problem cannot be solved in time 2 o(tw) · n O(1) under the ETH, and can be solved in time 2 O(tw·log tw) · n O(1). In this article we manage to classify the optimal asymptotic complexity of {H}-M-Deletion when H is a connected planar graph on at most 5 vertices. Out of the 29 possibilities (discarding the trivial case H = K 1), we prove that 9 of them are solvable in time 2 Θ(tw) · n O(1) , and that the other 20 ones are solvable in time 2 Θ(tw·log tw) · n O(1). Namely, we prove that K 4 and the diamond are the only graphs on at most 4 vertices for which the problem is solvable in time 2 Θ(tw·log tw) · n O(1) , and that the chair and the banner are the only graphs on 5 vertices for which the problem is solvable in time 2 Θ(tw) · n O(1). For the version of the problem where H is forbidden as a topological minor, the case H = K 1,4 can be solved in time 2 Θ(tw) · n O(1). This exhibits, to the best of our knowledge, the first difference between the computational complexity of both problems. 2012 ACM Subject Classification Mathematics of computing → Graph algorithms, Theory of computation → Parameterized complexity and exact algorithms

A matching in a graph is induced if no two of its edges are joined by an edge, and finding a large induced matching is a very hard problem. Lin et al. (Approximating weighted induced matchings, Discrete Applied Mathematics 243 (2018) 304-310) provide an approximation algorithm with ratio $��$ for the weighted version of the induced matching problem on graphs of maximum degree $��$. Their approach is based on an integer linear programming formulation whose integrality gap is at least $��-1$, that is, their approach offers only little room for improvement in the weighted case. For the unweighted case though, we conjecture that the integrality gap is at most $\frac{5}{8}��+O(1)$, and that also the approximation ratio can be improved at least to this value. We provide primal-dual approximation algorithms with ratios $(1-��) ��+ \frac{1}{2}$ for general $��$ with $��\approx 0.02005$, and $\frac{7}{3}$ for $��=3$. Furthermore, we prove a best-possible bound on the fractional induced matching number in terms of the order and the maximum degree.

International audience; Given a graph G, we define bcg(G) as the minimum k for which G can be contracted to the uniformly triangulated grid Γ k. A graph class G has the SQGC property if every graph G ∈ G has treewidth O(bcg(G) c) for some 1 ≤ c < 2. The SQGC 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 general family of graph classes that satisfy the SQGC property and includes bounded-degree string graphs. This considerably extends the applicability of bidimensionality theory for several intersection graph classes of 2-dimensional geometrical objects.

International audience; For a fixed collection of graphs F, the F-M-DELETION problem consists in, given a graph G and an integer k, decide whether there exists a subset S of V(G) of size at most k such that G-S does not contain any of the graphs in F as a minor. We are interested in the parameterized complexity of F-M-DELETION when the parameter is the treewidth of G, denoted by tw. Our objective is to determine, for a fixed F}, the smallest function f_F such that F-M-DELETION can be solved in time f_F(tw)n^{O(1)} on n-vertex graphs. Using and enhancing the machinery of boundaried graphs and small sets of representatives introduced by Bodlaender et al. [J ACM, 2016], we prove that when all the graphs in F are connected and at least one of them is planar, then f_F(w) = 2^{O(wlog w)}. When F is a singleton containing a clique, a cycle, or a path on i vertices, we prove the following asymptotically tight bounds: - f_{K_4}(w) = 2^{Theta(wlog w)}. - f_{C_i}(w) = 2^{Theta(w)} for every i4. - f_{P_i}(w) = 2^{Theta(w)} for every i5. The lower bounds hold unless the Exponential Time Hypothesis fails, and the superexponential ones are inspired by a reduction of Marcin Pilipczuk [Discrete Appl Math, 2016]. The single-exponential algorithms use, in particular, the rank-based approach introduced by Bodlaender et al. [Inform Comput, 2015]. We also consider the version of the problem where the graphs in F are forbidden as topological minors, and prove essentially the same set of results holds.

For two integers $r, \ell \geq 0$, a graph $G = (V, E)$ is an $(r,\ell)$-graph if $V$ can be partitioned into $r$ independent sets and $\ell$ cliques. In the parameterized $(r,\ell)$-Vertex Deletion problem, given a graph $G$ and an integer $k$, one has to decide whether at most $k$ vertices can be removed from $G$ to obtain an $(r,\ell)$-graph. This problem is NP-hard if $r+\ell \geq 1$ and encompasses several relevant problems such as Vertex Cover and Odd Cycle Transversal. The parameterized complexity of $(r,\ell)$-Vertex Deletion was known for all values of $(r,\ell)$ except for $(2,1)$, $(1,2)$, and $(2,2)$. We prove that each of these three cases is FPT and, furthermore, solvable in single-exponential time, which is asymptotically optimal in terms of $k$. We consider as well the version of $(r,\ell)$-Vertex Deletion where the set of vertices to be removed has to induce an independent set, and provide also a parameterized complexity dichotomy for this problem., Comment: After the first version of this article appeared in arXiv, we learnt that Kolay and Panolan [abs/1504.08120] obtained simultaneously and independently some of the results of this article

For some years, it was believed that for "connectivity" problems such as Hamiltonian Cycle, algorithms running in time 2O(tw) nO(1) -called singleexponential- existed only on planar and other sparse graph classes, where tw stands for the treewidth of the n-vertex input graph. This was recently disproved by Cygan et al. [FOCS 2011] and Bodlaender et al. [ICALP 2013], who provided single-exponential algorithms on general graphs for essentially all connectivity problems that were known to be solvable in single-exponential time on sparse graphs. During my internship, we further investigate the role of planarity in connectivity problems parameterized by treewidth, and convey that several problems can indeed be distinguished according to their behavior on planar graphs. In particular, we show that there exist problems that cannot be solved in time 2o(twlog tw) nO(1) on general graphs but that can be solved in time 2O(tw) nO(1) when restricted to planar graphs, and problems that can be solved in time 2O(twlog tw) nO(1) on general graphs but that cannot be solved in time 2o(twlog tw) nO(1) even when restricted to planar graphs, the negative results holding unless the ETH fails. We feel that our results constitute a first step in a subject that can be much exploited.