The Quickest Transshipment Problem is to route flow as quickly as possible from sources with supplies to sinks with demands in a network with capacities and transit times on the arcs. It is of fundamental importance for numerous applications in areas such as logistics, production, traffic, evacuation, and finance. More than 25 years ago, Hoppe and Tardos presented the first (strongly) polynomial-time algorithm for this problem. Their approach, as well as subsequently derived algorithms with strongly polynomial running time, are hardly practical as they rely on parametric submodular function minimization via Megiddo's method of parametric search. The main contribution of this paper is a considerably faster algorithm for the Quickest Transshipment Problem that instead employs a subtle extension of the Discrete Newton Method. This improves the previously best known running time of Õ(m4k14) to O(m2k5 + m3k3 + m3n), where n is the number of nodes, m the number of arcs, and k the number of sources and sinks.

We consider the online traveling salesperson problem (TSP), where requests appear online over time on the real line and need to be visited by a server initially located at the origin. We distinguish between closed and open online TSP, depending on whether the server eventually needs to return to the origin or not. While online TSP on the line is a very natural online problem that was introduced more than two decades ago, no tight competitive analysis was known to date. We settle this problem by providing tight bounds on the competitive ratios for both the closed and the open variant of the problem. In particular, for closed online TSP, we provide a 1.64-competitive algorithm, thus matching a known lower bound. For open online TSP, we give a new upper bound as well as a matching lower bound that establish the remarkable competitive ratio of 2.04. Additionally, we consider the online D IAL -A-R IDE problem on the line, where each request needs to be transported to a specified destination. We provide an improved non-preemptive lower bound of 1.75 for this setting, as well as an improved preemptive algorithm with competitive ratio 2.41. Finally, we generalize known and give new complexity results for the underlying offline problems. In particular, we give an algorithm with running time O ( n 2 ) for closed offline TSP on the line with release dates and show that both variants of offline D IAL -A-R IDE on the line are NP-hard for any capacity c ≥ 2 of the server.

We consider the online traveling salesperson problem (TSP), where requests appear online over time on the real line and need to be visited by a server initially located at the origin. We distinguish between closed and open online TSP, depending on whether the server eventually needs to return to the origin or not. While online TSP on the line is a very natural online problem that was introduced more than two decades ago, no tight competitive analysis was known to date. We settle this problem by providing tight bounds on the competitive ratios for both the closed and the open variant of the problem. In particular, for closed online TSP, we provide a 1.64-competitive algorithm, thus matching a known lower bound. For open online TSP, we give a new upper bound as well as a matching lower bound that establish the remarkable competitive ratio of 2.04. Additionally, we consider the online Dial-A-Ride problem on the line, where each request needs to be transported to a specified destination. We provide an improved non-preemptive lower bound of 1.75 for this setting, as well as an improved preemptive algorithm with competitive ratio 2.41. Finally, we generalize known and give new complexity results for the underlying offline problems. In particular, we give an algorithm with running time O(n2) for closed offline TSP on the line with release dates and show that both variants of offline Dial-A-Ride on the line are NP-hard for any capacity c ≥ 2 of the server.

We study two classical flow over time problems that capture the essence of evacuation planning. Given a network with capacities and transit times on the arcs and sources/sinks with supplies/demands, a quickest transshipment sends the supplies from the sources to meet the demands at the sinks as quickly as possible. In a 1995 landmark paper, Hoppe and Tardos describe the first strongly polynomial time algorithm solving the quickest transshipment problem. Their algorithm relies on repeatedly calling an oracle for parametric submodular function minimization. We present a somewhat simpler and more efficient algorithm for the quickest transshipment problem. Our algorithm (i) relies on only one parametric submodular function minimization and, as a consequence, has considerably improved running time, (ii) uses not only the solution of a submodular function minimization but actually exploits the underlying algorithmic approach to determine a quickest transshipment as a convex combination of simple lex-max flows over time, and (iii) in this way determines a structurally easier solution in the form of a generalized temporally repeated flow. Our second main result is an entirely novel algorithm for computing earliest arrival transshipments, which feature a particularly desirable property in the context of evacuation planning. An earliest arrival transshipment - which in general only exists in networks with a single sink - is a quickest transshipment maximizing the amount of flow which has reached the sink for every point in time simultaneously. In contrast to previous approaches, our algorithm solely works on the given network and, as a consequence, requires only polynomial space.