Relocation in Carsharing Systems using Flows in Time-Expanded Networks

A manuscript on this topic can be found at: http://hal.archives-ouvertes.fr/hal-00908242International audienceIn a carsharing system, a fleet of cars is distributed at stations in an urban area, customers can take and return cars at any time and station, provided that there is a car available at the start station and a free place at the final station. To ensure the latter, customers have to book their demands in advance; hereby, customer requests can be accepted or rejected by the operator. The stations have to keep a good ratio between available cars and free places in each station, in order to serve already accepted customer requests and to refuse as few new customer requests as possible. This leads to the problem of relocating cars between stations, which can be modeled as Pickup and Delivery Problem in a metric space induced by the urban area or, alternatively, by means of flows of cars in convoys in a time-expanded network.Note that we consider an innovative carsharing system with partly autonomous cars which allows to build convoys of cars, each moved by only one driver. This leads to a similar situation as in bikesharing systems, where trucks can simultaneously move several bikes, but no requests are booked in advance. Hereby, two flows are coupled in the sense that the flow of cars is dependent from the flow of drivers (since cars can only be moved in convoys); the flow coupling constraints reflect the complexity of the studied problem.We present integer programming formulations for two variants of the relocation problem: a min-cost flow problem to serve a given set of customer requests at minimal costs (quality of service aspect), and a max-profit flow problem to additionally solve the decision problem of accepting or rejecting customer requests (economic aspect). Both models take advantage of users booking their demands in advance and can be applied to the offline as well as the online version of the relocation problem in order to fully capture the dynamic evolution of the system over time.</p

Online k-server routing problems

In an online k-server routing problem, a crew of k servers has to visit points in a metric space as they arrive in real time. Possible objective functions include minimizing the makespan (k-Traveling Salesman Problem) and minimizing the sum of completion times (k-Traveling Repairman Problem). We give competitive algorithms, resource augmentation results and lower bounds for k-server routing problems in a wide class of metric spaces. In some cases the competitive ratio is dramatically better than that of the corresponding single server problem. Namely, we give a 1+O((log¿k)/k)-competitive algorithm for the k-Traveling Salesman Problem and the k-Traveling Repairman Problem when the underlying metric space is the real line. We also prove that a similar result cannot hold for the Euclidean plane

Atomic routing in a deterministic queuing model

The issue of selfish routing through a network has received a lot of attention in recent years. We study an atomic dynamic routing scenario, where players allocate resources with load dependent costs only for some limited time. Our paper introduces a natural discrete version of the deterministic queuing model introduced by Koch and Skutella (2011). In this model the time a user needs to traverse an edge  e is given by a constant travel time and the waiting time in a queue at the end of  e. At each discrete time step the first ue  users of the queue proceed to the end vertex of  e, where ue denotes the capacity of the edge  e. An important aspect of this model is that it ensures the FIFO  property. We study the complexity of central algorithmic questions for this model such as determining an optimal flow in an empty network, an optimal path in a congested network or a maximum dynamic flow and the question whether a given flow is a Nash equilibrium. For the bottleneck case, where the cost of each user is the travel time of the slowest edge on her path, the main results here are mostly bad news. Computing social optima and Nash equilibria turns out to be NP-complete and the Price of Anarchy is given by the number of users. We also consider the makespan objective (arrival time of the last user) and show that optimal solutions and Nash equilibria in these games, where every user selfishly tries to minimize her travel time, can be found efficiently

On an online traveling repairman problem with flowtimes: Worst-case and average-case analysis

We consider an online problem where a server operates on an edge-weighted graph C and an adversarial sequence of requests to vertices is released over time. Each request requires one unit of servicetime. The server is free to choose the ordering of service and intends to minimize the total flowtime of the requests. A natural class of algorithms for this problem are IGNORE algorithms. From worst-case perspective we show that IGNORE algorithms are not competitive for flowtime minimization. From an average-case point of view, we obtain a more detailed picture. In our model, the adversary may still choose the vertices of the requests arbitrarily. But the arrival times are according to a stochastic process (with some rate lambda > 0), chosen by the adversary out of a natural class of processes. The class contains the Poisson-process and (some) deterministic arrivals as special cases. We then show that there is an IGNORE algorithm that is competitive if and only if lambda not equal 1. Specifically, for lambda not equal 1, the expected competitive ratio of the algorithm is within a constant of the length of a shortest cycle that visits all vertices of G. The reason for this is that if lambda not equal 1 the requests either arrive slow enough for our algorithm or too fast even for an offline optimal algorithm. For lambda = 1 the routing-mistakes of the online algorithm accumulate just as in the worst case. As an additional result, we show how IGNORE tours are constructed optimally in polynomial time, if the underlying graph G is a line

Approximating infeasible 2VPI-systems

It is a folklore result that testing whether a given system of equations with two variables per inequality (a 2VPI system) of the form x i -x j =c ij is solvable, can be done efficiently not only by Gaussian elimination but also by shortest-path computation on an associated constraint graph. However, when the system is infeasible and one wishes to delete a minimum weight set of inequalities to obtain feasibility (MinFs2 =), this task becomes NP-complete. Our main result is a 2-approximation for the problem MinFs2 = for the case when the constraint graph is planar using a primal-dual approach. We also give an -approximation for the related maximization problem MaxFs2 = where the goal is to maximize the weight of feasible inequalities. Here, denotes the arboricity of the constraint graph. Our results extend to obtain constant factor approximations for the case when the domains of the variables are further restricted