Computing a (short) path between two vertices is one of the most fundamental primitives in graph algorithmics. In recent years, the study of paths in temporal graphs, that is, graphs where the vertex set is fixed but the edge set changes over time, gained more and more attention. A path is time-respecting, or temporal, if it uses edges with non-decreasing time stamps. We investigate a basic constraint for temporal paths, where the time spent at each vertex must not exceed a given duration $$\varDelta $$ Δ , referred to as $$\varDelta $$ Δ -restless temporal paths. This constraint arises naturally in the modeling of real-world processes like packet routing in communication networks and infection transmission routes of diseases where recovery confers lasting resistance. While finding temporal paths without waiting time restrictions is known to be doable in polynomial time, we show that the “restless variant” of this problem becomes computationally hard even in very restrictive settings. For example, it is W[1]-hard when parameterized by the distance to disjoint path of the underlying graph, which implies W[1]-hardness for many other parameters like feedback vertex number and pathwidth. A natural question is thus whether the problem becomes tractable in some natural settings. We explore several natural parameterizations, presenting FPT algorithms for three kinds of parameters: (1) output-related parameters (here, the maximum length of the path), (2) classical parameters applied to the underlying graph (e.g., feedback edge number), and (3) a new parameter called timed feedback vertex number, which captures finer-grained temporal features of the input temporal graph, and which may be of interest beyond this work.

Temporal graphs are graphs with time-stamped edges. We study the problem of finding a small vertex set (the separator) with respect to two designated terminal vertices such that the removal of the set eliminates all temporal paths connecting one terminal to the other. Herein, we consider two models of temporal paths: paths that pass through arbitrarily many edges per time step (non-strict) and paths 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 (NPhardness 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. We further show that, on temporal graphs with planar underlying graph, if additionally the number of time steps is constant, then the problem variant for strict paths is solvable in quasi-linear time. Finally, we introduce and motivate the notion of a temporal core (vertices whose incident edges change over time). We prove that the non-strict variant is fixed-parameter tractable when parameterized by the size of the temporal core, while the strict variant remains NP-complete, even for constant-size temporal cores. * Supported by the Stiftung Begabtenförderung berufliche Bildung (SBB). † Supported by the DFG, project DAMM (NI 369/13) and project TORE (NI 369/18). ‡ Supported by DFG, project MATE (NI 369/17). 2 We also refer to Himmel [23] for a thorough discussion and comparison of temporal path concepts. 3 Temporal (s, z)-Separation with τ = 1 is equivalent to (s, z)-Separation on static graphs.

We investigate the computational complexity of separating two distinct vertices s and z by vertex deletion in a temporal graph. In a temporal graph, the vertex set is fixed but the edges have (discrete) time labels. Since the corresponding Temporal (s, z)-Separation problem is NP-hard, it is natural to investigate whether relevant special cases exist that are computationally tractable. To this end, we study restrictions of the underlying (static) graph-there we observe polynomial-time solvability in the case of bounded treewidth-as well as restrictions concerning the "temporal evolution" along the time steps. Systematically studying partially novel concepts in this direction, we identify sharp borders between tractable and intractable cases. * Supported by the DFG, project DAMM (NI 369/13).

No abstract

We consider a variant of the matroid median problem introduced by Krishnaswamy et al. [SODA 2011]: an uncapacitated discrete facility location problem where the task is to decide which facilities to open and which clients to serve for maximum profit so that the facilities form an independent set in given facility-side matroids and the clients form an independent set in given client-side matroids. We show that the problem is fixed-parameter tractable parameterized by the number of matroids and the minimum rank among the client-side matroids. To this end, we derive fixed-parameter algorithms for computing representative families for matroid intersections and maximum-weight set packings with multiple matroid constraints. To illustrate the modeling capabilities of the new problem, we use it to obtain algorithms for a problem in social network analysis. We complement our tractability results by lower bounds.Keywords: matroid set packing · matroid parity · matroid median · representative families · social network analysis · strong triadic closure 4 The State Register of Waste Disposal Facilities of the Russian Federation lists 18 dumps for municipal solid waste in Moscow region-the largest city of Europe.