In the gas station problem we want to find the cheapest path between two vertices of an n-vertex graph. Our car has a specific fuel capacity and at each vertex we can fill our car with gas, with the fuel cost depending on the vertex. Furthermore, we are allowed at most ∆ stops for refuelling.In this short paper we provide an algorithm solving the problem in O(∆n 2 + n 2 log n) steps improving an earlier result by Khuller, Malekian and Mestre.Date: June 2, 2017. 2010 Mathematics Subject Classification. 90B06,90C35,68Q25.

We study a problem where k autonomous mobile agents are initially located on distinct nodes of a weighted graph (with n nodes and m edges). Each autonomous mobile agent has a predefined velocity and is only allowed to move along the edges of the graph. We are interested in delivering a package, initially positioned in a source node s, to a destination node y. The delivery is achieved by the collective effort of the autonomous mobile agents, which can carry and exchange the package among them. The objective is to compute a delivery schedule that minimizes the delivery time of the package. In this paper, we propose an O(kn log n + km) time algorithm for this problem. This improves the previous state-of-the-art O(k 2 m + kn 2 + APSP) time algorithm for this problem, where APSP stands for the running-time of an algorithm for the All-Pairs Shortest Paths problem.

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Given a 2-edge connected, unweighted, and undirected graph G with n vertices and m edges, a σ-tree spanner is a spanning tree T of G in which the ratio between the distance in T of any pair of vertices and the corresponding distance in G is upper bounded by σ. The minimum value of σ for which T is a σ-tree spanner of G is also called the stretch factor of T . We address the fault-tolerant scenario in which each edge e of a given tree spanner may temporarily fail and has to be replaced by a best swap edge, i.e. an edge that reconnects T − e at a minimum stretch factor. More precisely, we design an O(n 2 ) time and space algorithm that computes a best swap edge of every tree edge. Previously, an O(n 2 log 4 n) time and O(n 2 + m log 2 n) space algorithm was known for edge-weighted graphs [Bilò et al., ISAAC 2017]. Even if our improvements on both the time and space complexities are of a polylogarithmic factor, we stress the fact that the design of a o(n 2 ) time and space algorithm would be considered a breakthrough. ACM Subject Classification Theory of computation → Graph algorithms analysisKeywords and phrases Transient edge failure, best swap edges, tree spanner.

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