For a fixed connected graph H, the {H}-M-Deletion problem asks, given a graph G, for the minimum number of vertices that intersect all minor models of H in G. It is known that this problem can be solved in time f (tw) · n O(1) , where tw is the treewidth of G. We determine the asymptotically optimal function f (tw), for each possible choice of H. Namely, we prove that, under the ETH, f (tw) = 2 Θ(tw) if H is a contraction of the chair or the banner, and f (tw) = 2 Θ(tw · log tw) otherwise. Prior to this work, such a complete characterization was only known when H is a planar graph with at most five vertices. For the upper bounds, we present an algorithm in time 2 Θ(tw · log tw) · n O(1) for the more general problem where all minor models of connected graphs in a finite family F need to be hit. 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. In particular, this algorithm vastly generalizes a result of Jansen et al. [SODA 2014] for the particular case F = {K 5 , K 3,3 }. For the lower bounds, our reductions are based on a generic construction building on the one given by the authors in [IPEC 2018], which uses the framework introduced by Lokshtanov et al. [SODA 2011] to obtain superexponential lower bounds.

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. We consider an intuitive notion of diversity of a collection of solutions which suits a large variety of combinatorial problems of practical interest. Our main contribution is 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.

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.

For a finite collection of graphs F , the F-M-Deletion problem consists in, given a graph G and an integer k, deciding whether there exists S ⊆ V (G) with |S| ≤ 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. We prove that f F (tw) = 2 2 O(tw•log tw) for every collection F , that f F (tw) = 2 O(tw•log tw) if F contains a planar graph, and that f 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 F are forbidden as topological minors, called F-TM-Deletion. We prove similar results for this problem, except that in the last two algorithms, instead of requiring 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.

A matching M in a graph G is r-degenerate if the subgraph of G induced by the set of vertices incident with an edge in M is r-degenerate. Goddard, Hedetniemi, Hedetniemi, and Laskar (Generalized subgraph-restricted matchings in graphs, Discrete Mathematics 293 (2005) 129-138) introduced the notion of acyclic matchings, which coincide with 1-degenerate matchings. Solving a problem they posed, we describe an efficient algorithm to determine the maximum size of an r-degenerate matching in a given chordal graph. Furthermore, we study the r-chromatic index of a graph defined as the minimum number of r-degenerate matchings into which its edge set can be partitioned, obtaining upper bounds and discussing extremal graphs.

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A link stream is a sequence of pairs of the form (t, {u, v}), where t ∈ N represents a time instant and u = 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 ∈ 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.

For a finite collection of graphs F, the F-M-Deletion problem consists in, given a graph G and an integer k, decide whether there exists S ⊆ V (G) with |S| ≤ 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. We prove that f F (tw) = 2 2 O(tw·log tw) for every collection F, that f F (tw) = 2 O(tw·log tw) if all the graphs in F are connected and at least one of them is planar, and that f F (tw) = 2 O(tw) if in addition the input graph G is planar or embedded in a surface. When F contains a single connected planar graph H, we obtain a tight dichotomy about the asymptotic complexity of {H}-M-Deletion. Namely, we prove that f {H} (tw) = 2 Θ(tw) if H is a minor of the banner (that is, the graph consisting of a C 4 plus a pendent edge) that is different from P 5 , and that f {H} (tw) = 2 Θ(tw·log tw) otherwise. All the lower bounds hold under the ETH. We also consider the version of the problem where the graphs in F are forbidden as topological minors, and prove similar results, except that, in the algorithms, instead of requiring F to contain a planar graph, we need it to contain a subcubic planar graph. We also prove that, for this problem, f {K1,i} (tw) = 2 Θ(tw) for every i ≥ 1, while for the minor version it holds that f {K1,i} (tw) = 2 Θ(tw·log tw) for every i ≥ 4. Extended abstracts containing some of the results of this article appeared in the Proc. of the 12th International Symposium on Parameterized and Exact Computation (IPEC 2017) [3] and in the Proc. of the 13th International Symposium on Parameterized and Exact Computation (IPEC 2018) [4]. Work supported by French projects DEMOGRAPH (ANR-16-CE40-0028) and ESIGMA (ANR-17-CE23-0010).