Search

International audience

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

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

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.