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 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.

We introduce the integrality number of an integer program (IP) in inequality form. Roughly speaking, the integrality number is the smallest number of integer constraints needed to solve an IP via a mixed integer (MIP) relaxation. One notable property of this number is its invariance under unimodular transformations of the constraint matrix. Considering the largest minor ∆ of the constraint matrix, our analysis allows us to make statements of the following form: there exist numbers τ (∆) and κ(∆) such that an IP with n ≥ τ (∆) many variables and n + κ(∆) · √ n many inequality constraints can be solved via a MIP relaxation with fewer than n integer constraints. A special instance of our results shows that IPs defined by only n constraints can be solved via a MIP relaxation with O( √ ∆) many integer constraints.1 Introduction.Let A ∈ Z m×n satisfy rank(A) = n and c ∈ Z n . We denote the integer linear program parameterized by right hand side b ∈ Z m by IP A,c (b) := max{c ⊺ x : Ax ≤ b and x ∈ Z n }.See [8] for more on parametric integer programs. We are interested in solving IP A,c (b) by relaxing it to have fewer integer constraints. In special cases we can solve IP A,c (b) with zero integer constraints by only solving its linear relaxation LP A,c (b) := max{c ⊺ x : Ax ≤ b}.However, these special cases require the underlying polyhedron to have optimal integral vertices, which occurs for instance when A is totally unimodular. Our target is to consider general matrices A and relaxations in the form of a mixed integer program:where k ∈ Z ≥0 and W ∈ Z k×n satisfies rank(W ) = k.One sufficient condition for solving IP A,c (b) using a mixed integer relaxation is that the vertices of W-MIP A,c (b) are integral. The vertices of W-MIP A,c (b) are the vertices of the polyhedron conv {x ∈ R n : Ax ≤ b, W x ∈ Z k } .

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