@misc{Bley, author = {Bley, Andreas}, title = {Inapproximability Results for the Inverse Shortest Paths Problem with Integer Length and Unique Shortest Paths}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-8388}, number = {05-04}, abstract = {We study the complexity of two Inverse Shortest Paths (ISP) problems with integer arc lengths and the requirement for uniquely determined shortest paths. Given a collection of paths in a directed graph, the task is to find positive integer arc lengths such that the given paths are uniquely determined shortest paths between their respective terminals. The first problem seeks for arc lengths that minimize the length of the longest of the prescribed paths. In the second problem, the length of the longest arc is to be minimized. We show that it is \$np-hard\$ to approximate the minimal longest path length within a factor less than \$8/7\$ or the minimal longest arc length within a factor less than \$9/8\$. This answers the (previously) open question whether these problems are \$np-hard\$ or not. We also present a simple algorithm that achieves an \$\mathcal{O}(|V|)\$-approximation guarantee for both variants. Both ISP problems arise in the planning of telecommunication networks with shortest path routing protocols. Our results imply that it is \$\mathcal{NP}\$-hard to decide whether a given path set can be realized with a real shortest path routing protocol such as OSPF, IS-IS, or RIP.}, language = {en} } @misc{Hiller, author = {Hiller, Benjamin}, title = {Probabilistic Competitive Analysis of a Dial-a-Ride Problem on Trees Under High Load}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-8896}, number = {05-56}, abstract = {In this paper we consider a simple variant of the Online Dial-a-Ride Problem from a probabilistic point of view. To this end, we look at a probabilistic version of this online Dial-a-Ride problem and introduce a probabilistic notion of the competitive ratio which states that an algorithm performs well on the vast majority of the instances. Our main result is that under the assumption of high load a certain online algorithm is probabilistically \$(1+o(1))\$-competitive if the underlying graph is a tree. This result can be extended to general graphs by using well-known approximation techniques at the expense of a distortion factor~\$O(\log\|V\|)\$.}, language = {en} }