Alain Faye

A new upper bound for the 0-1 quadratic knapsack problem

The 0-1 quadratic knapsack problem (QKP) consists in maximizing a positive quadratic pseudo-Boolean function subject to a linear capacity constraint. We present in this paper a new method, based on Lagrangian decomposition, for computing an upper bound of QKP. We report computational experiments which demonstrate the sharpness of the bound (relative error very often less than 1%) for large size instances (up to 500 variables).

Optimizing splitter and fiber location in a multilevel optical FTTH network

Due to the emergence of bandwidth-requiring services, telecommunication operators are brought to renew their fixed access network, most of them favoring the Fiber To The Home (FTTH) technology. This paper focuses on the optimization of FTTH deployment, which is of prime importance due to the economic stakes. The key design issue here is locating splitters and routing fibers in an existing network infrastructure to which is associated a graph with given capacities on the edges. No assumption is made on the structure of the graph. First we propose a mixed integer formulation for this decision problem and we prove it is NP-hard. Then, valid inequalities and problem size reduction schemes are presented. Finally, the efficiency of solution approaches is assessed through extensive numerical tests performed on France Telecom-Orange real-life data.

Partial Lagrangian relaxation for general quadratic programming

We give a complete characterization of constant quadratic functions over an affine variety. This result is used to convexify the objective function of a general quadratic programming problem (Pb) which contains linear equality constraints. Thanks to this convexification, we show that one can express as a semidefinite program the dual of the partial Lagrangian relaxation of (Pb) where the linear constraints are not relaxed. We apply these results by comparing two semidefinite relaxations made from two sets of null quadratic functions over an affine variety.

Solving the Aircraft Landing Problem with time discretization approach

This paper studies the multiple runway Aircraft Landing Problem. The aim is to schedule arriving aircraft to available runways at the airport. Landing times lie within predefined time windows and safety separation constraints between two successive landings must be satisfied. We propose a new approach for solving the problem. The method is based on an approximation of the separation time matrix and on time discretization. The separation matrix is approximated by a rank two matrix. This provides lower bounds or upper bounds depending on the choice of the approximating matrix. These bounds are used in a constraint generation algorithm to, exactly or heuristically, solve the problem. Computational tests, performed on publicly available problems involving up to 500 aircraft, show the efficiency of the approach.

Decomposition and Linearization for 0-1 Quadratic Programming

This paper presents a general decomposition method to compute bounds for constrained 0-1 quadratic programming. The best decomposition is found by using a Lagrangian decomposition of the problem. Moreover, in its simplest version this method is proved to give at least the bound obtained by the LP-relaxation of a non-trivial linearization. To illustrate this point, some computational results are given for the 0-1 quadratic knapsack problem.

Solving the Two-Stage Robust FTTH network design Problem under Demand Uncertainty

For the past few years, the increase in high bandwidth requiring services forced telecommunication operators like France Telecom - Orange to engage the deployment of optical networks, the Fiber To The Home Gigabit Passive Optical Network (FTTH GPON) technology, leading to new design problems. Such problems have already been studied. However, to the best of our knowledge, without taking into account the future demand uncertainty. In this paper, we propose a model for a two-stage robust optimization FTTH network design problem tackling the demand uncertainty. We propose an exact algorithm, based on column and constraint generation algorithms, and we show some preliminary results.

A lower bound for a constrained quadratic 0-1 minimization problem

Abstract Given a quadratic pseudo-Boolean functionf(x1, …, xn) written as a multilinear polynomial in its variables, Hammer et al. [7]have studied, in their paper “Roof duality, complementation and persistency in quadratic 0–1 optimization”, the greatest constant c such that there exists a quadratic posiform φ satisfyingf = c + φ for all xϵ {0, 1}n. Obviously c is a lower bound to the minimum of f. In this paper we consider the problem of minimizing a quadratic pseudo- Boolean function subject to the cardinality constraint ∑i = 1, n xi = k and we propose a linear programming method to compute the greatest constant c such that there exists a quadratic posiform φ satisfying f = c + φ for all x ϵ {0, 1}n with ∑i = 1, n xi = k. As in the unconstrained case c is a lower bound to the optimum. Some computational tests showing how sharp this bound is in practice are reported.

A cutting planes algorithm based upon a semidefinite relaxation for the quadratic assignment problem

We present a cutting planes algorithm for the Quadratic Assignment Problem based upon a semidefinite relaxation, and we report experiments for classical instances. Our lower bound is compared with the ones obtained by linear and semidefinite approaches. Our tests show that the cuts we use (originally proposed for a linear approach) allow to improve significantly on the bounds obtained by the other approaches. Moreover, this is achieved within a moderate additional computing effort, and even in a shorter total time sometimes. Indeed, thanks to the strong tailing off effect of the SDP solver we have used (SB), we obtain in a reasonable time an approximate solution which is suitable to generate efficient cutting planes which speed up the convergence of SB.

Un Algorithme De Génération De Coupes Pour Le Problème De L’Affectation Quadratique

Abstract We address the Quadratic Assignment Problem following a polyhedral method. We consider the Quadratic Assignment Polytopc defined as the convex hull of the solutions of the linearized problem. Its dimension and a minimal description of its affine hull have been given by Padberg and Rijal (1996). Here we propose a large family of valid inequalities inducing facets. We show that the separation problem of this family is NP-complete. We propose a heuristic for the separation problem and a cutting plane algorithm based on this heuristic. Numerical results show the practical interest of this family of inequalities

Taxi-sharing: Parameterized complexity and approximability of the dial-a-ride problem with money as an incentive

Abstract We study, in this paper, a taxi-sharing problem, called Dial-a-Ride problem with money as an incentive (DARP-M ). This problem consists in defining a set of taxis that will be shared by different clients in order to reduce their bill by a given factor α 1 . To achieve this, each client shares the cost of the ride with other passengers. More precisely, the fragments of the ride in which the client is alone is fully paid by this client and, for each fragment in which the client shares the taxi with other passengers, the cost is equally divided between the passengers. In addition to this cost constraint, the taxi must satisfy a time window constraint for each passenger and a capacity constraint. We define three versions of the problem: max-DARP-M where the objective is to drive the maximum number of clients with an arbitrarily large number of taxis; max-1-DARP-M in which we want to drive the maximum number of clients with one taxi; and 1-DARP-M which consists in deciding whether it is possible to drive at least one client while satisfying the constraints. We study the parameterized complexity and approximability of those problems with respect to four parameters: the factor α, the capacity c a p a of the taxis, the maximum size TW of the time windows of the clients, and the value S of an optimal solution. Among other results, we prove that 1-DARP-M is NP-Complete and max-DARP-M and max-1-DARP-M cannot be approximated in polynomial time to within any variable ratio even if α, c a p a and TW are fixed and if the road network is a planar graph. We also give a polynomial algorithm for max-1-DARP-M for the case where c a p a and TW are fixed and where the network does not contain a circuit. This algorithm implies a 1 n -polynomial approximation for max-DARP-M.