Till Fluschnik’s research while affiliated with Technische Universität Berlin and other places

Kernelization – a mathematical key concept for provably effective polynomial-time preprocessing of NP-hard problems – plays a central role in parameterized complexity and has triggered an extensive line of research. This is in part due to a lower bounds framework that allows to exclude polynomial-size kernels under the assumption of [Formula: see text]. In this paper we consider a restricted yet natural variant of kernelization, namely strict kernelization, where one is not allowed to increase the parameter of the reduced instance (the kernel) by more than an additive constant. Building on earlier work of Chen, Flum, and Müller [CiE 2009, Theory Comput. Syst. 2011], we underline the applicability of their framework by showing that a variety of fixed-parameter tractable problems, including graph problems and Turing machine computation problems, does not admit strict polynomial kernels under the assumption of [Formula: see text], an assumption being weaker than the assumption of [Formula: see text]. Finally, we study an adaption of the framework to a relaxation of the notion of strict kernels, where in the latter one is not allowed to increase the parameter of the reduced instance by more than a constant times the input parameter.

Given a graph G = (V, E), two vertices s, t ∈ V, and two integers k, ℓ, the Short Secluded Path problem is to find a simple s‐t‐path with at most k vertices and ℓ neighbors. We study the parameterized complexity of the problem with respect to four structural graph parameters: the vertex cover number, treewidth, feedback vertex number, and feedback edge number. In particular, we completely settle the question of the existence of problem kernels with size polynomial in these parameters and their combinations with k and ℓ. We also obtain a 2O(tw) · ℓ2 · n‐time algorithm for n‐vertex graphs of treewidth tw, which yields subexponential‐time algorithms in several graph classes.

Temporal graphs have time-stamped edges. Building on previous work, we study the problem of finding a small vertex set (the separator) whose removal destroys all temporal paths between two designated terminal vertices. Herein, we consider two models of temporal paths: those that pass through arbitrarily many edges per time step (non-strict) and those that pass through at most one edge per time step (strict). Regarding the number of time steps of a temporal graph, we show a complexity dichotomy (NP-completeness versus polynomial-time solvability) for both problem variants. Moreover, we prove both problem variants to be NP-complete even on temporal graphs whose underlying graph is planar. Finally, we introduce the notion of a temporal core (vertices whose incident edges change over time) and prove that the non-strict variant is fixed-parameter tractable when parameterized by the temporal core size, while the strict variant remains NP-complete, even for constant-size temporal cores.

Given a graph G=(V,E) with edge weights and a subset $R\subseteq E$ of required edges, the NP-hard Rural Postman Problem (RPP) is to find a closed walk of minimum total weight containing all edges of R. The number b of vertices incident to an odd number of edges of R and the number c of connected components formed by the edges in R are both bounded from above by the number of edges that has to be traversed additionally to the required ones. We show how to reduce any RPP instance I to an RPP instance $I'$ with $2b+O(c/\varepsilon )$ vertices in $O(n^3)$ time so that any $\alpha$-approximate solution for $I'$ gives an $\alpha (1+\varepsilon )$-approximate solution for I, for any $\alpha \ge 1$ and $\varepsilon >0$. That is, we provide a polynomial-size approximate kernelization scheme (PSAKS). We make first steps towards a PSAKS with respect to the parameter c.

Given a graph G=(V,E), two vertices s,t∈V, and two integers k,ℓ, the Short Secluded Path problem is to find a simple s-t-path with at most k vertices and ℓ neighbors. We study the parameterized complexity of the problem with respect to four structural graph parameters: the vertex cover number, treewidth, feedback vertex number, and feedback edge number. In particular, we completely settle the question of the existence of problem kernels with size polynomial in these parameters and their combinations with k and ℓ. We also obtain a 2^{O(w)} ℓ^2 n-time algorithm for graphs of treewidth w, which yields subexponential-time algorithms in several graph classes.

Covering all edges of a graph by a minimum number of vertices, this is the NP-hard Vertex Cover problem, is among the most fundamental algorithmic tasks. Following a recent trend in studying dynamic and temporal graphs, we initiate the study of Multistage Vertex Cover. Herein, having a series of graphs with same vertex set but over time changing edge sets (known as temporal graph consisting of various layers), the goal is to find for each layer of the temporal graph a small vertex cover and to guarantee that the two vertex cover sets between two subsequent layers differ not too much (specified by a given parameter). We show that, different from classic Vertex Cover and some other dynamic or temporal variants of it, Multistage Vertex Cover is computationally hard even in fairly restricted settings. On the positive side, however, we also spot several fixed-parameter tractability results based on some of the most natural parameterizations.

We study the computational complexity of routing multiple objects through a network while avoiding collision: Given a graph G with two distinct terminals and two positive integers p,k, the question is whether one can connect the terminals by at least p routes (walks, trails, paths; the latter two without repeated edges or vertices, respectively) such that in at most k edges it happens that we traverse them in more than one route at the same time. We prove that for paths and trails the problem is NP-hard on undirected and directed planar graphs even if the maximum vertex degree or k≥0 is constant. For walks we prove polynomial-time solvability on undirected graphs for unbounded k and on directed graphs if k≥0 is constant. We additionally study variants where the maximum length of a route is restricted. For walks this variant becomes NP-hard on undirected graphs.

Hyperbolicity is a distance-based measure of how close a given graph is to being a tree. Due to its relevance in modeling real-world networks, hyperbolicity has seen intensive research over the last years. Unfortunately, the best known algorithms used in practice for computing the hyperbolicity number of an n-vertex graph have running time $O(n^4)$. Exploiting the framework of parameterized complexity analysis, we explore possibilities for “linear-time FPT” algorithms to compute hyperbolicity. For example, we show that hyperbolicity can be computed in $2^{O(k)} + O(n +m)$ time (where m and k denote the number of edges and the size of a vertex cover in the input graph, respectively) while at the same time, unless the Strong Exponential Time Hypothesis (SETH) fails, there is no $2^{o(k)}\cdot n^{2-\varepsilon }$-time algorithm for every $\varepsilon >0$.