Fabio Furini | Université Paris Dauphine - PSL (original) (raw)

Papers by Fabio Furini

We consider the Weighted Vertex Coloring Problem (WVCP), in which a positive weight is associated... more We consider the Weighted Vertex Coloring Problem (WVCP), in which a positive weight is associated to each vertex of a graph. In WVCP, one is required to assign a color to each vertex in such a way that colors on adjacent vertices are different, and the objective is to minimize the sum of the costs of the colors used, where the cost of each color is given by the maximum weight of the vertices assigned to that color. This NP-hard problem arises in practical scheduling applications, where it is also known as Scheduling on a Batch Machine with Job Compatibilities. We propose the first exact algorithm for the problem, which is based on column generation and branch-and-price. Computational results on a large set of instances from the literature are reported, showing excellent performance when compared with the best heuristic algorithms from the literature.

The integration of drones into civil airspace is one of the most challenging problems for the aut... more The integration of drones into civil airspace is one of the most challenging problems for the automation of the controlled airspace, and the optimization of the drone route is a key step for this process. In this paper, we optimize the route planning of a drone mission that consists of departing from an airport, flying over a set of mission way points and coming back to the initial airport. We assume that during the mission a set of piloted aircraft flies in the same airspace and thus the cost of the drone route depends on the air traffic and on the avoidance maneuvers used to prevent possible conflicts. Two Air Traffic Management techniques, i.e., routing and holding, are modeled in order to maintain a minimum separation between the drone and the piloted aircraft. The considered problem, called the Time Dependent Traveling Salesman Planning Problem in Controlled Airspace (TDTSPPCA), relates to the drone route planning phase and aims to minimize the total operational cost. Two heuristic algorithms are proposed for the solution of the problem. A mathematical formulation based on a particular version of the Time Dependent Traveling Salesman Problem, which allows holdings at mission way points, and a Branch and Cut algorithm are proposed for solving the TDTSPPCA to optimality. An additional formulation, based on a Travelling Salesman Problem variant that uses specific penalties to model the holding times, is proposed and a Cutting Plane algorithm is designed. Finally, computational experiments on real-world air traffic data from Milano Linate Terminal Maneuvering Area are reported to evaluate the performance of the proposed formulations and of the heuristic algorithms.

We study a natural generalization of the knapsack problem, in which each item exists only for a g... more We study a natural generalization of the knapsack problem, in which each item exists only for a given time interval. One has to select a subset of the items (as in the classical case), guaranteeing that for each time instant the set of existing selected items has total weight not larger than the knapsack capacity. We focus on the exact solution of the problem, noting that prior to our work the best method was the straightforward application of a general-purpose solver to the natural ILP formulation. Our results indicate that much better results can be obtained by using the same general-purpose solver to tackle a nonstandard Dantzig-Wolfe reformulation in which subproblems are associated with groups of constraints. This is also interesting since the more natural Dantzig-Wolfe reformulation of single constraints performs extremely poorly in practice.

—A Look-Up-Table-based method is proposed to generate random instances of an antipodal n-dimensio... more —A Look-Up-Table-based method is proposed to generate random instances of an antipodal n-dimensional vector so that its 2-nd order statistics are as close as possible to a given specification. The method is based on linear optimization and exploits column-generation techniques to cope with the exponential complexity of the task. It yields a LUT whose storage requirements are only O(n 3) and thus are compatible with hardware implementation for non-negligible n. Applications are shown in the fields of Compressive Sensing and of Ultra Wide Band systems based on Direct Sequence-Code Division Multiple Acces.

The Temporal Knapsack Problem (TKP) is a generalization of the standard Knapsack Problem where a ... more The Temporal Knapsack Problem (TKP) is a generalization of the standard Knapsack Problem where a time horizon is considered, and each item consumes the knapsack capacity during a limited time interval only. In this paper we solve the TKP using what we call a Recursive Dantzig-Wolfe Reformulation (DWR) method. The generic idea of Recursive DWR is to solve a Mixed Integer Program (MIP) by recursively applying DWR, i.e., by using DWR not only for solving the original MIP but also for recursively solving the pricing sub-problems. In a binary case (like the TKP), the Recursive DWR method can be performed in such a way that the only two components needed during the optimization are a Linear Programming solver and an algorithm for solving Knapsack Problems. The Recursive DWR allows us to solve Temporal Knapsack Problem instances through computation of strong dual bounds, which could not be obtained by exploiting the best-known previous approach based on DWR.

We consider the Train Timetabling Problem (TTP) in a railway node (i.e. a set of stations in an u... more We consider the Train Timetabling Problem (TTP) in a railway node (i.e. a set of stations in an urban area interconnected by tracks), which calls for determining the best schedule for a given set of trains during a given time horizon, while satisfying several track operational constraints. In particular, we consider the context of a highly congested railway node in which different Train Operators wish to run trains according to timetables that they propose, called ideal timetables. The ideal timetables altogether may be (and usually are) conflicting, i.e. they do not respect one or more of the track operational constraints. The goal is to determine conflict-free timetables that differ as little as possible from the ideal ones. The problem was studied for a research project funded by Rete Ferroviaria Italiana (RFI), the main Italian railway Infrastructure Manager, who also provided us with real-world instances. We present an Integer Linear Programming (ILP) model for the problem, which adapts previous ILP models from the literature to deal with the case of a railway node. The Linear Programming (LP) relaxation of the model is used to derive a dual bound. In addition, we propose an iterative heuristic algorithm that is able to obtain good solutions to real-world instances with up to 1500 trains in short computing times. The proposed algorithm is also used to evaluate the capacity saturation of the railway nodes.

We consider a generalization of the 0-1 knapsack problem in which the profit of each item can tak... more We consider a generalization of the 0-1 knapsack problem in which the profit of each item can take any value in a range characterized by a minimum and a maximum possible profit. A set of specific profits is called a scenario. Each feasible solution associated with a scenario has a regret, given by the difference between the optimal solution value for such scenario and the value of the considered solution. The interval min-max regret knapsack problem (MRKP) is then to find a feasible solution such that the maximum regret over all scenarios is minimized. The problem is extremely challenging both from a theoretical and a practical point of view. Its decision version is complete for the complexity class Σ p 2 hence it is most probably not in N P. In addition, even computing the regret of a solution with respect to a scenario requires the solution of an N P-hard problem. We examine the behavior of classical combinatorial optimization approaches when adapted to the solution of the MRKP. We introduce an iterated local search approach and a Lagrangian-based branch-and-cut algorithm, and evaluate their performance through extensive computational experiments.

In Vertex Coloring Problems, one is required to assign a color to each vertex of an undirected gr... more In Vertex Coloring Problems, one is required to assign a color to each vertex of an undirected graph in such a way that adjacent vertices receive different colors, and the objective is to minimize the cost of the used colors. In this work we solve four different coloring problems formulated as Maximum Weight Stable Set Problems on an associated graph. We exploit the transformation proposed by Cornaz and Jost [6], where given a graph G, an auxiliary graphĜ is constructed, such that the family of all stable sets ofĜ is in one-to-one correspondence with the family of all feasible colorings of G. The transformation in [6] was originally proposed for the classical Vertex Coloring and the Max-Coloring problems; we extend it to the Equitable Coloring Problem and the Bin Packing Problem with Conflicts. We discuss the relation between the Maximum Weight Stable formulation and a polynomial-size formulation for the VCP, proposed by Campelo, Correa and Campos [4] and called the Representative formulation. We report extensive computational experiments on benchmark instances of the four problems, and compare the solution method with the state-of-the-art algorithms. By exploiting the proposed method, we largely outperform the stateof-the-art algorithm for the Max-coloring Problem, and we are able to solve, for the first time to proven optimality, 14 Max-coloring and 2 Equitable Coloring instances.

In a scenario characterized by a continuous growth of air transportation demand, the runways of l... more In a scenario characterized by a continuous growth of air transportation demand, the runways of large airports serve hundreds of aircraft every day. Aircraft sequencing is a challenging problem that aims to increase runway capacity in order to reduce delays as well as the workload of air traffic controllers.

The Perspective Reformulation (PR) of a Mixed-Integer NonLinear Program with semicontinuous varia... more The Perspective Reformulation (PR) of a Mixed-Integer NonLinear Program with semicontinuous variables is obtained by replacing each term in the (separable) objective function with its convex envelope. Solving the corresponding continuous relaxation requires appropriate techniques. Under some rather restrictive assumptions, the Projected PR (P 2 R) can be defined where the integer variables are eliminated by projecting the solution set onto the space of the continuous variables only. This approach produces a simple piecewise-convex problem with the same structure as the original one; however, this prevents the use of general-purpose solvers, in that some variables are then only implicitly represented in the formulation. We show how to construct an Approximated Projected PR (AP 2 R) whereby the projected formulation is "lifted" back to the original variable space, with each integer variable expressing one piece of the obtained piecewise-convex function. In some cases, this produces a reformulation of the original problem with exactly the same size and structure as the standard continuous relaxation, but providing substantially improved bounds. In the process we also substantially extend the approach beyond the original P 2 R development by relaxing the requirement that the objective function be quadratic and the left endpoint of the domain of the variables be non-negative. While the AP 2 R bound can be weaker than that of the PR, this approach can be applied in many more cases and allows direct use of off-the-shelf MINLP software; this is shown to be competitive with previously proposed approaches in some applications.

Dantzig-Wolfe decomposition (or reformulation) is well-known to provide strong dual bounds for sp... more Dantzig-Wolfe decomposition (or reformulation) is well-known to provide strong dual bounds for specially structured mixed integer programs (MIPs). However, the method is not implemented in any state-of-the-art MIP solver as it is considered to require structural problem knowledge and tailoring to this structure. We provide a computational proof-of-concept that the reformulation can be automated. That is, we perform a rigorous experimental study, which results in identifying a score to estimate the quality of a decomposition: after building a set of potentially good candidates, we exploit such a score to detect which decomposition might be useful for Dantzig-Wolfe reformulation of a MIP. We experiment with general instances from MIPLIB2003 and MIPLIB2010 for which a decomposition method would not be the first choice, and demonstrate that strong dual bounds can be obtained from the automatically reformulated model using column generation. Our findings support the idea that Dantzig-Wolfe reformulation may

We propose a framework to model general guillotine restrictions in two-dimensional cutting proble... more We propose a framework to model general guillotine restrictions in two-dimensional cutting problems formulated as Mixed Integer Linear Programs (MIP). The modeling framework requires a pseudo-polynomial number of variables and constraints, which can be effectively enumerated for medium-size instances. Our modeling of general guillotine cuts is the first one that, once it is implemented within a state-of-the-art MIP solver, can tackle instances of challenging size. We mainly concentrate our analysis on the Guillotine Two Dimensional Knapsack Problem (G2KP), for which a model, and an exact procedure able to significantly improve the computational performance, are given. We also show how the modeling of general guillotine cuts can be extended to other relevant problems such as the Guillotine Two Dimensional Cutting Stock Problem (G2CSP) and the Guillotine Strip Packing Problem (GSPP). Finally, we conclude the paper discussing an extensive set of computational experiments on G2KP and GSPP benchmark instances from the literature.

We consider a Two-Dimensional Cutting Stock Problem where stock of different sizes is available, ... more We consider a Two-Dimensional Cutting Stock Problem where stock of different sizes is available, and a set of rectangular items has to be obtained through two-staged guillotine cuts. We propose a heuristic algorithm, based on column generation, which requires as subproblem the solution of a Two-Dimensional Knapsack Problem with two-staged guillotines cuts. A further contribution of the paper consists in the definition of a Mixed Integer Linear Programming Model for the solution of this Knapsack Problem, as well as a heuristic procedure based on dynamic programming. Computational experiments show the effectiveness of the proposed approach, which obtains very small optimality gaps and outperforms the heuristic algorithm proposed by Cintra et al. .

Lecture Notes in Computer Science, 2011

Dantzig-Wolfe decomposition is well-known to provide strong dual bounds for specially structured ... more Dantzig-Wolfe decomposition is well-known to provide strong dual bounds for specially structured mixed integer programs (MIPs) in practice. However, the method is not implemented in any state-of-the-art MIP solver: it needs tailoring to the particular problem; the decomposition must be determined from the typical bordered block-diagonal matrix structure; the resulting column generation subproblems must be solved efficiently; etc. We provide a computational proof-of-concept that the process can be automated in principle, and that strong dual bounds can be obtained on general MIPs for which a solution by a decomposition has not been the first choice. We perform an extensive computational study on the 0-1 dynamic knapsack problem (without block-diagonal structure) and on general MIPLIB2003 instances. Our results support that Dantzig-Wolfe reformulation may hold more promise as a general-purpose tool than previously acknowledged by the research community.

Lecture Notes in Computer Science, 2011