In this work we consider temporal networks, i.e. networks defined by a labeling λ assigning to each edge of an underlying graph G a set of discrete time-labels. The labels of an edge, which are natural numbers, indicate the discrete time moments at which the edge is available. We focus on path problems of temporal networks. In particular, we consider time-respecting paths, i.e. paths whose edges are assigned by λ a strictly increasing sequence of labels. We begin by giving two efficient algorithms for computing shortest time-respecting paths on a temporal network. We then prove that there is a natural analogue of Menger's theorem holding for arbitrary temporal networks. Finally, we propose two cost minimization parameters for temporal network design. One is the temporality of G, in which the goal is to minimize the maximum number of labels of an edge, and the other is the temporal cost of G, in which the goal is to minimize the total number of labels used. Optimization of these parameters is performed subject to some connectivity constraint. We prove several lower and upper bounds for the temporality and the temporal cost of some very basic graph families such as rings, directed acyclic graphs, and trees.

Citation for published item:h¡ % zD tF nd qold ergD vFeF nd wertziosD qFfF nd i her yD hF nd ern D wF nd pir kisD FqF @PHIRA 9epproxim ting (x tion pro ilities in the gener lized wor n pro essF9D elgorithmi FD TW @IAF ppF UVEWIFFurther information on publisher's website:The nal publication is available at Springer via http://dx.doi.org/10.1007/s00453-012-9722-7.Additional information:A preliminary version of this work appeared in Proceedings of the ACM SIAM Symposium on Discrete Algorithms (SODA), pp. 954 960, 2012. Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-pro t purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO• the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. AbstractWe consider the Moran process, as generalized by Lieberman, Hauert and Nowak (Nature, 433:312-316, 2005). A population resides on the vertices of a finite, connected, undirected graph and, at each time step, an individual is chosen at random with probability proportional to its assigned "fitness" value. It reproduces, placing a copy of itself on a neighbouring vertex chosen uniformly at random, replacing the individual that was there. The initial population consists of a single mutant of fitness r > 0 placed uniformly at random, with every other vertex occupied by an individual of fitness 1. The main quantities of interest are the probabilities that the descendants of the initial mutant come to occupy the whole graph (fixation) and that they die out (extinction); almost surely, these are the only possibilities. In general, exact computation of these quantities by standard Markov chain techniques requires solving a system of linear equations of size exponential in the order of the graph so is not feasible. We show that, with high probability, the number of steps needed to reach fixation or extinction is bounded by a polynomial in the number of vertices in the graph. This bound allows us to construct fully polynomial randomized approximation schemes (FPRAS) for the probability of fixation (when r 1) and of extinction (for all r > 0). * A preliminary version of this work appeared in

Modern, inherently dynamic systems are usually characterized by a network structure, i.e. an underlying graph topology, which is subject to discrete changes over time. Given a static underlying graph G, a temporal graph can be represented via an assignment of a set of integer time-labels to every edge of G, indicating the discrete time steps when this edge is active. While most of the recent theoretical research on temporal graphs has focused on the notion of a temporal path and other "path-related" temporal notions, only few attempts have been made to investigate "non-path" temporal graph problems. In this paper, motivated by applications in sensor and in transportation networks, we introduce and study two natural temporal extensions of the classical problem Vertex Cover. In both cases we wish to minimize the total number of "vertex appearances" that are needed to "cover" the whole temporal graph. In our first problem, Temporal Vertex Cover, the aim is to cover every edge at least once during the lifetime of the temporal graph, where an edge can be covered by one of its endpoints, only at a time step when it is active. In our second, more pragmatic variation Sliding Window Temporal Vertex Cover, we are also given a natural number ∆, and our aim is to cover every edge at least once at every ∆ consecutive time steps. We present a thorough investigation of the computational complexity and approximability of these two temporal covering problems. In particular, we provide strong hardness results, complemented by various approximation and exact algorithms. Some of our algorithms are polynomial-time, while others are asymptotically almost optimal under the Exponential Time Hypothesis (ETH) and other plausible complexity assumptions.

We study the design of small cost temporally connected graphs, under various constraints. We mainly consider undirected graphs of n vertices, where each edge has an associated set of discrete availability instances (labels). A journey from vertex u to vertex v is a path from u to v where successive path edges have strictly increasing labels. A graph is temporally connected iff there is a (u, v)-journey for any pair of vertices u, v, u = v. We first give a simple polynomial-time algorithm to check whether a given temporal graph is temporally connected. We then consider the case in which a designer of temporal graphs can freely choose availability instances for all edges and aims for temporal connectivity with very small cost; the cost is the total number of availability instances used. We achieve this via a simple polynomialtime procedure which derives designs of cost linear in n. We also show that the above procedure is (almost) optimal when the underlying graph is a tree, by proving a lower bound on the cost for any tree. However, there are pragmatic cases where one is not free to design a temporally connected graph anew, but is instead given a temporal graph design with the claim that it is temporally connected, and wishes to make it more cost-efficient by removing labels without destroying temporal connectivity (redundant labels). Our main technical result is that computing the maximum number of redundant labels is APX-hard, i.e., there is no PTAS unless P = NP . On the positive side, we show that in dense graphs with random edge availabilities, there is asymptotically almost surely a very large number of redundant labels. A temporal design may, however, be minimal, i.e., no redundant labels exist. We show the existence of minimal temporal designs with at least n log n labels.

We consider the problems of finding optimal identifying codes, (open) locating-dominating sets and resolving sets of an interval or a permutation graph. In these problems, one asks to find a subset of vertices, normally called a solution set, using which all vertices of the graph are distinguished. The identification can be done by considering the neighborhood within the solution set, or by employing the distances to the solution vertices. Normally the goal is to minimize the size of the solution set then. Here we study the case of interval graphs, unit interval graphs, (bipartite) permutation graphs and cographs. For these classes of graphs we give tight lower bounds for the size of such solution sets depending on the order of the input graph. While such lower bounds for the general class of graphs are in logarithmic order, the improved bounds in these special classes are of the order of either quadratic root or linear in terms of number of vertices. Moreover, the results for cographs lead to linear-time algorithms to solve the considered problems on inputs that are cographs. * u, v if it (totally) dominates exactly one of them. A set S (totally) separates the vertices of a set X if all pairs of X are (totally) separated by a vertex of S. Whenever it is clear from the context, we will only say "separate" and omit the word "totally". We have the three key definitions, that merge the concepts of (total) domination and (total) separation:Definition 1 (Slater [33,34]). A set S of vertices of a graph G is a locating-dominating set if it is a dominating set and it separates the vertices of V (G) \ S. The smallest size of a locating-dominating set of G is the location-domination number of G, denoted γ LD (G). Without the domination constraint, this concept has also been used under the name distinguishing set in [2] and sieve in [28].Definition 2 (Karpovsky, Chakrabarty and Levitin [27]). A set S of vertices of a graph G is an identifying code if it is a dominating set and it separates all vertices of V (G). The smallest size of an identifying code of G is the identifying code number of G, denoted γ ID (G). Definition 3 (Seo and Slater [31]). A set S of vertices of a graph G is an open locating-dominating set if it is a total dominating set and it totally separates all vertices of V (G). The smallest size of an open locating-dominating set of G is the open location-domination number of G, denoted γ OLD (G). This concept has also been called identifying open code in [25]. Separation could also be done using distances from the members of the solution set. Let d(x, u) denote the distance between two vertices x and u. Definition 4 (Harary and Melter [24], Slater [32]). A set B of vertices of a graph G is a resolving set if for each pair u, v of distinct vertices, there is a vertex x of B with d(x, u) = d(x, v). 1 The smallest size of a resolving set of G is the metric dimension of G, denoted dim(G).