Frédéric Meunier

Bike sharing systems: Solving the static rebalancing problem

Abstract This paper deals with a new problem that is a generalization of the many to many pickup and delivery problem and which is motivated by operating self-service bike sharing systems. There is only one commodity, initially distributed among the vertices of a graph, and a capacitated single vehicle aims to redistribute the commodity in order to reach a target distribution. Each vertex can be visited several times and also can be used as a buffer in which the commodity is stored for a later visit. This problem is NP-hard, since it contains several NP-hard problems as special cases (the TSP being maybe the most obvious one). Even finding a tractable exact formulation remains problematic. This paper presents efficient algorithms for solving instances of reasonable size, and contains several theoretical results related to these algorithms. A branch-and-cut algorithm is proposed for solving a relaxation of the problem. An upper bound of the optimal solution of the problem is obtained by a tabu search, which is based on some theoretical properties of the solution, once fixed the sequence of the visited vertices. The possibility of using the information provided by the relaxation receives a special attention, both from a theoretical and a practical point of view. It is proven that to build a feasible solution of the problem by using the one obtained by the relaxation is an NP-hard problem. Nevertheless, a tabu search initialized with the optimal solution of the relaxation often shows that it is the optimal one. The algorithms have been tested on a set of instances coming from the literature, proving their effectiveness.

BALANCING THE STATIONS OF A SELF SERVICE \BIKE HIRE" SYSTEM

This paper is motivated by operating self service transport systems that ourish nowa- days. In cities where such systems have been set up with bikes, trucks travel to maintain a suitable number of bikes per station. It is natural to study a version of the C-delivery TSP dened by Chalasani and Motwani in which, unlike their denition, C is part of the input: each vertex v of a graph G = (V;E) has a certain amount xv of a commodity and wishes to have an amount equal to yv (we assume that P v2V xv = P v2V yv and all quantities are assumed to be integers); given a vehicle of capacity C, nd a minimal route that balances all vertices, that is, that allows to have an amount yv of the commodity on each vertex v. This paper presents among other things complexity results, lower bounds, approximation algo- rithms, and a polynomial algorithm when G is a tree. Mathematics Subject Classication. ???, ???

Carathéodory, Helly and the Others in the Max-Plus World

Carathéodory’s, Helly’s and Radon’s theorems are three basic results in discrete geometry. Their max-plus or tropical analogues have been proved by various authors. We show that more advanced results in discrete geometry also have max-plus analogues, namely, the colorful Carathéodory theorem and the Tverberg theorem. A conjecture connected to the Tverberg theorem—Sierksma’s conjecture—although still open for the usual convexity, is shown to be true in the max-plus setting.

Shared mobility systems

Shared mobility systems for bicycles and cars have grown in popularity in recent years and have attracted the attention of the operational research community. Researchers have investigated several problems arising at the strategic, tactical and operational levels. This survey paper classifies the relevant literature under five main headings: station location, fleet dimensioning, station inventory, rebalancing incentives, and vehicle repositioning. It closes with some open research questions.

A Further Generalization of the Colourful Carathéodory Theorem

Given d+1 sets, or colours, \(\mathbf{S}_{1},\mathbf{S}_{2},\ldots,\mathbf{S}_{d+1}\) of points in \({\mathbb{R}}^{d}\), a colourful set is a set \(S \subseteq \bigcup _{i}\mathbf{S}_{i}\) such that \(\vert S \cap \mathbf{S}_{i}\vert \leq 1\) for \(i = 1,\ldots,d + 1\). The convex hull of a colourful set S is called a colourful simplex. Barany’s colourful Caratheodory theorem asserts that if the origin 0 is contained in the convex hull of S i for \(i = 1,\ldots,d + 1\), then there exists a colourful simplex containing 0. The sufficient condition for the existence of a colourful simplex containing 0 was generalized to 0 being contained in the convex hull of \(\mathbf{S}_{i} \cup \mathbf{S}_{j}\) for 1≤i<j≤d+1 by Arocha etal. and by Holmsen etal. We further generalize the sufficient condition and obtain new colourful Caratheodory theorems. We also give an algorithm to find a colourful simplex containing 0 under the generalized condition. In the plane an alternative, and more general, proof using graphs is given. In addition, we observe that any condition implying the existence of a colourful simplex containing 0 actually implies the existence of min i |S i |such simplices.

Discrete Splittings of the Necklace

This paper deals with direct proofs and combinatorial proofs of the famous necklace theorem of Alon, Goldberg, and West. The new results are a direct proof for the case of two thieves and three types of beads, and an efficient constructive proof for the general case with two thieves. This last proof uses a theorem of Ky Fan which is a version of Tuckers lemma concerning cubical complexes instead of simplicial complexes.

Large Momentum Bounds from Flow Equations

Abstract. We analyze the large momentum behaviour of 4-dimensional massive euclidean

Equilibrium Results for Dynamic Congestion Games

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Sperner labellings: a combinatorial approach

theory using the flow equations of Wilson‚s renormalization group.The flow equations give access to a simple inductive proof of perturbative renormalizability. By sharpening the induction hypothesis we prove new and, as it seems, close to optimal bounds on the large momentum behaviour of the correlation functions. The bounds are related to what is generally called Weinberg‚s theorem.

Colorful linear programming, Nash equilibrium, and pivots

Consider the following game. Given a network with a continuum of users at some origins, suppose users wish to reach specific destinations but they are not indifferent to the cost to reach them. They may have multiple possible routes but their choices modify the travel costs on the network. Hence, each user faces the following problem: Given a pattern of travel costs for the different possible routes that reach the destination, find a path of minimal cost. This kind of game belongs to the class of congestion games. In the traditional static approach, travel times are assumed constant during the period of the game. In this paper, we consider the so-called dynamic case where the time-varying nature of traffic conditions is explicitly taken into account. In transportation science, the question of whether there is an equilibrium and how to compute it for such a model is referred to as the dynamic user equilibrium problem. Until now, there was no general model for this problem. Our paper attempts to resolve this issue. We define a new class of games, dynamic congestion games, which capture this time-dependency aspect. Moreover, we prove that under some natural assumptions there is a Nash equilibrium. When we apply this result to the dynamic user equilibrium problem, we get most of the previous known results.