HAL (Le Centre pour la Communication Scientifique Directe), 2003
Cahier n°2003-002 Soit G=(V,E) un graphe non-orienté et 2-arêtes connexe. Chaque arête et sommet ... more Cahier n°2003-002 Soit G=(V,E) un graphe non-orienté et 2-arêtes connexe. Chaque arête et sommet de G est muni d'un poids. Le problème du sous-graphe 2-arêtes connexe de poids minimum dans G (2ECSP), est de trouver un sous-graphe 2-arêtes connexe de G tel que la somme des poids sur ses sommets et ses arête soit minimum. Le 2ECSP généralise le problème, bien connu, du sousgraphe Steiner 2-arêtes connexe. Dans cet article, l'enveloppe convexe des vecteurs d'incidences des solutions de 2ECSP est étudiée. Une formulation naturelle du problème par un programme linéaire en nombres entiers est premièrement établie. Il est aussi montré que la relaxation ne suffit pas pour décrire l'enveloppe convexe associée au 2ECSP même dans une classe restreinte comme celle des graphes série-parallèles. Une nouvelle classe d'inégalités valides pour le 2ECSP est introduite. Il est monté qu'une sousclasse de ces inégalités peut être séparée en temps polynomial. Comme conséquence, ces nouvelles inégalités peuvent être séparées en temps polynomial dans la classe des graphes série-parallèles.
Computer Science and Information Systems (FedCSIS), 2019 Federated Conference on, Sep 26, 2021
We deal here with a fleet of autonomous vehicles which is required to perform internal logistics ... more We deal here with a fleet of autonomous vehicles which is required to perform internal logistics tasks inside some protected area. This fleet is supposed to be ruled by a hierarchical supervision architecture, which, at the top level distributes and schedules Pick up and Delivery tasks, and, at the lowest level, ensures safety at the crossroads and controls the trajectories. We focus here on the top level, while introducing a time dependent estimation of the risk induced by the traversal of any arc at a given time. We set a model, state some structural results, and design, in order to route and schedule the vehicles according to a well-fitted compromise between speed and risk, a bi-level algorithm and a A* algorithm which both relies on a reinforcement learning scheme.
HAL (Le Centre pour la Communication Scientifique Directe), May 22, 2023
The balanced traveling salesman problem (BTSP) is a variant of the traveling salesman problem, in... more The balanced traveling salesman problem (BTSP) is a variant of the traveling salesman problem, in which one seeks a tour that minimizes the difference between the largest and smallest edge costs in the tour. The BTSP, which is obviously NP-hard, was first investigated by Larusic and Punnen in 2011 [9]. They proposed several heuristics based on the double-threshold framework, which converge to good-quality solutions though not always optimal (e.g. 27 provably optimal solutions were found among 65 TSPLIB instances of at most 500 vertices). In this paper, we design a special-purpose branch-and-cut algorithm for solving exactly the BTSP. In contrast with the classical TSP, due to the BTSP's objective function, the efficiency of algorithms for solving the BTSP depends heavily on determining correctly the largest and smallest edge costs in the tour. In the proposed branch-and-cut algorithm, we develop several mechanisms based on local cutting planes, edge elimination, and variable fixing to locate more and more precisely those edge costs. Other important ingredients of our algorithm are heuristics for improving the lower and upper bounds of the branch-and-bound tree. Experiments on the same TSPLIB instances show that our algorithm was able to solve to optimality 63 out of 65 instances.
HAL (Le Centre pour la Communication Scientifique Directe), Jun 4, 2023
Branch-and-Cut is a widely-used method for solving integer programming problems exactly. In recen... more Branch-and-Cut is a widely-used method for solving integer programming problems exactly. In recent years, researchers have been exploring ways to use Machine Learning to improve the decision-making process of Branch-and-Cut algorithms. While much of this research focuses on selecting nodes, variables, and cuts [11,9,26], less attention has been paid to designing efficient cut generation strategies in Branch-and-Cut algorithms, despite its large impact on the algorithm performance. In this paper, we focus on improving the generation of subtour elimination constraints, a core and compulsory class of cuts in Branch-and-Cut algorithms devoted to solving the Traveling Salesman Problem, which is one of the most studied combinatorial optimization problems. Our approach takes advantage of Machine Learning to address two questions before launching the separation routine to find cuts at a node of the search tree: 1) Do violated subtour elimination constraints exist? 2) If yes, is it worth generating them? We consider the former as a binary classification problem and adopt a Graph Neural Network as a classifier. By formulating subtour elimination constraint generation as a Markov decision problem, the latter can be handled through an agent trained by reinforcement learning. Our method can leverage the underlying graph structure of fractional solutions in the search tree to enhance its decision-making. Furthermore, once trained, the proposed Machine Learning model can be applied to any graph of any size (in terms of the number of vertices and edges). Numerical results show that our approach can significantly accelerate the performance of subtour elimination constraints in Branchand-Cut algorithms for the Traveling Salesman Problem.
In this paper, we introduce a new cooperative game theory model that we call productiondistributi... more In this paper, we introduce a new cooperative game theory model that we call productiondistribution game to address a major open problem for operations research in forestry, raised by Rönnqvist et al. in 2015, namely, that of modelling and proposing efficient sharing principles for practical collaboration in transportation in this sector. The originality of our model lies in the fact that the value/strength of a player does not only depend on the individual cost or benefit of the objects she owns but also depends on her market shares (customers demand). We show however that the production-distribution game is an interesting special case of a market game introduced by Shapley and Shubik in 1969. As such it exhibits the nice property of having a non-empty core. We then prove that we can compute both the nucleolus and the Shapley value efficiently, in a nontrivial and interesting special case. We in particular provide two different algorithms to compute the nucleolus: a simple separation algorithm and a fast primal-dual algorithm. Our results can be used to tackle more general versions of the problem and we believe that our contribution paves the way towards solving the challenging open problem herein.
HAL (Le Centre pour la Communication Scientifique Directe), 2003
Mars 2003 Cahier n°2003-002 Résumé: Soit G=(V,E) un graphe non-orienté et 2-arêtes connexe. Chaqu... more Mars 2003 Cahier n°2003-002 Résumé: Soit G=(V,E) un graphe non-orienté et 2-arêtes connexe. Chaque arête et sommet de G est muni d'un poids. Le problème du sous-graphe 2-arêtes connexe de poids minimum dans G (2ECSP), est de trouver un sous-graphe 2-arêtes connexe de G tel que la somme des poids sur ses sommets et ses arête soit minimum. Le 2ECSP généralise le problème, bien connu, du sousgraphe Steiner 2-arêtes connexe. Dans cet article, l'enveloppe convexe des vecteurs d'incidences des solutions de 2ECSP est étudiée. Une formulation naturelle du problème par un programme linéaire en nombres entiers est premièrement établie. Il est aussi montré que la relaxation ne suffit pas pour décrire l'enveloppe convexe associée au 2ECSP même dans une classe restreinte comme celle des graphes série-parallèles. Une nouvelle classe d'inégalités valides pour le 2ECSP est introduite. Il est monté qu'une sousclasse de ces inégalités peut être séparée en temps polynomial. Comme conséquence, ces nouvelles inégalités peuvent être séparées en temps polynomial dans la classe des graphes série-parallèles.
HAL (Le Centre pour la Communication Scientifique Directe), Nov 29, 2021
We give a combinatorial algorithm to find a maximum packing of hypertrees in a capacitated hyperg... more We give a combinatorial algorithm to find a maximum packing of hypertrees in a capacitated hypergraph. Based on this we extend to hypergraphs several algorithms for the k-cut problem, that are based on packing spanning trees in a graph. In particular we give a γapproximation algorithm for hypergraphs of rank γ, extending the work of Ravi and Sinha [22] for graphs. We also extend the work of Chekuri, Quanrud and Xu [7] in graphs, to give an algorithm for the k-cut problem in hypergraphs that is polynomial if k and the rank of the hypergraph are fixed. We also give a combinatorial algorithm to solve a linear programming relaxation of this problem in hypergraphs.
In this article, we consider a particular case of bi‐objective optimization (BOO), called bi‐obje... more In this article, we consider a particular case of bi‐objective optimization (BOO), called bi‐objective minimization (BOM), where the two objective functions to be minimized take only positive values. As well as for BOO, most of the methods proposed in the literature for solving BOM focus on computing the Pareto‐optimal solutions representing different trade‐offs between two objectives. However, it may be difficult for a central decision‐maker to determine the preferred solutions due to the huge number of solutions in the Pareto set. We propose a novel criterion for selecting the preferred Pareto‐optimal solutions by introducing the concept of ‐Nash Fairness (‐) solutions inspired by the definition of proportional fairness. The ‐ solutions are the feasible solutions achieving some proportional nash equilibrium between the two objectives. The positive parameter is introduced to reflect the relative importance of the first objective to the second one. For this work, we will discuss exi...
In dial-a-ride systems involving autonomous vehicles circulating along a circuit, a fleet of vehi... more In dial-a-ride systems involving autonomous vehicles circulating along a circuit, a fleet of vehicles must serve clients who request rides between stations of the circuit such that the total number of pickup and drop-off operations is minimized. In this paper, we focus on a unitary variant where each client requests a single place in the vehicles and all the clients must be served within a single tour of the circuit. Such unitary variant induces a combinatorial optimization problem for which we introduce a nontrivial special case that is polynomially solvable when the capacity of each vehicle is at most 2 but it is NP-Hard otherwise. We also study the polytope associated with the solutions to this problem. We introduce new families of valid inequalities and give necessary and sufficient conditions under which they are facet-defining. Based on these inequalities, we devise an efficient branch-and-cut algorithm that outperforms the state-of-the-art commercial solvers.
We give a combinatorial algorithm to find a maximum packing of hypertrees in a capacitated hyperg... more We give a combinatorial algorithm to find a maximum packing of hypertrees in a capacitated hypergraph. Based on this we extend to hypergraphs several algorithms for the k-cut problem, that are based on packing spanning trees in a graph. In particular we give a γapproximation algorithm for hypergraphs of rank γ, extending the work of Ravi and Sinha [22] for graphs. We also extend the work of Chekuri, Quanrud and Xu [7] in graphs, to give an algorithm for the k-cut problem in hypergraphs that is polynomial if k and the rank of the hypergraph are fixed. We also give a combinatorial algorithm to solve a linear programming relaxation of this problem in hypergraphs.
2022 12th International Workshop on Resilient Networks Design and Modeling (RNDM)
We consider a fair version of combinatorial optimization, which aims for both Pareto-efficiency a... more We consider a fair version of combinatorial optimization, which aims for both Pareto-efficiency and fairness of a solution. A possible approach to achieve the objectives simultaneously is to use the Ordered Weighted Average (OWA) aggregating function, which can be formulated into mix-integer programming (MIP) formulations. In this paper, we study two MIP formulations proposed in the literature for the OWA in the context of fair combinatorial optimization. On the one hand, we prove that both MIP formulations are equivalent in terms of linear relaxations. On the other hand, we estimate the quality with regard to the OWA value of an optimal solution of original combinatorial optimization. An experimental evaluation of the MIP formulations in tackling OWA Traveling Salesman Problem is also presented.
Hypergraphics matroids were studied first by Lorea [20] and later by Frank et al [10]. They can b... more Hypergraphics matroids were studied first by Lorea [20] and later by Frank et al [10]. They can be seen as generalizations of graphic matroids. Here we show that several algorithms developed for the graphic case can be extended to hypergraphic matroids. We treat the following: the separation problem for the associated polytope, testing independence, separation of partition inequalities, computing the rank of a set, computing the strength, computing the arboricity and network reinforcement.
In this paper, we consider a variant of the Traveling Salesman Problem (TSP), called Balanced Tra... more In this paper, we consider a variant of the Traveling Salesman Problem (TSP), called Balanced Traveling Salesman Problem (BTSP) [7]. The BTSP seeks to find a tour which has the smallest maxmin distance : the difference between the maximum edge cost and the minimum one. We present a Mixed Integer Program (MIP) to find optimal solutions minimizing the max-min distance for BTSP. However, minimizing only the max-min distance may lead to a tour with an inefficient total cost in many situations. Hence, we propose a fair way based on Nash equilibrium [5], [11] to inject the total cost into the objective function of the BTSP. We consider a Nash equilibrium as it is defined in a context of fair competition based on proportional-fair scheduling. For BTSP, we are interested in solutions achieving a Nash equilibrium between two players: the first aims at minimizing the total cost and the second aims at minimizing the max-min distance. We call such solutions Nash Fairness (NF) solutions. We first show that NF solutions for BTSP exist and may be more than one. We show that NF solutions are Pareto-optimal [10] and can be found by optimizing a sequence of linear combinations of the two players objectives based on Weighted Sum Method [13]. We then focus on extreme NF solutions which are NF solutions having either the smallest value of total cost or the smallest max-min distance. Finally, we propose a Newton-based iterative algorithm which converges to extreme NF solutions in a polynomial number of iterations. Computational results on smallsize instances from TSPLIB will be presented and commented.
The Stop Number Problem arises in the management of a dial-a-ride system served by a fleet of aut... more The Stop Number Problem arises in the management of a dial-a-ride system served by a fleet of autonomous electric vehicles. In such a system, clients request for a ride from an origin station to a destination station, and a fleet of capacitated vehicles must satisfy all requests. The goal is to minimize the number of pick-up/drop-off operations. In this paper we focus on a special case of this problem that was recently conjectured to be NP-Hard. In this regard, we show how such special case relates to other problems known in the literature in order to derive some polynomial-time solvable variants. Moreover, we provide a positive answer to the conjecture by showing that the problem is NP-Hard for any fixed capacity greater than or equal to 2, even for the case where the graph of requests is restricted to the class of planar bipartite graphs. Our proof of NP-Hardness also improves the complexity results known in the literature for the related problems identified.
HAL (Le Centre pour la Communication Scientifique Directe), 2003
Cahier n°2003-002 Soit G=(V,E) un graphe non-orienté et 2-arêtes connexe. Chaque arête et sommet ... more Cahier n°2003-002 Soit G=(V,E) un graphe non-orienté et 2-arêtes connexe. Chaque arête et sommet de G est muni d'un poids. Le problème du sous-graphe 2-arêtes connexe de poids minimum dans G (2ECSP), est de trouver un sous-graphe 2-arêtes connexe de G tel que la somme des poids sur ses sommets et ses arête soit minimum. Le 2ECSP généralise le problème, bien connu, du sousgraphe Steiner 2-arêtes connexe. Dans cet article, l'enveloppe convexe des vecteurs d'incidences des solutions de 2ECSP est étudiée. Une formulation naturelle du problème par un programme linéaire en nombres entiers est premièrement établie. Il est aussi montré que la relaxation ne suffit pas pour décrire l'enveloppe convexe associée au 2ECSP même dans une classe restreinte comme celle des graphes série-parallèles. Une nouvelle classe d'inégalités valides pour le 2ECSP est introduite. Il est monté qu'une sousclasse de ces inégalités peut être séparée en temps polynomial. Comme conséquence, ces nouvelles inégalités peuvent être séparées en temps polynomial dans la classe des graphes série-parallèles.
Computer Science and Information Systems (FedCSIS), 2019 Federated Conference on, Sep 26, 2021
We deal here with a fleet of autonomous vehicles which is required to perform internal logistics ... more We deal here with a fleet of autonomous vehicles which is required to perform internal logistics tasks inside some protected area. This fleet is supposed to be ruled by a hierarchical supervision architecture, which, at the top level distributes and schedules Pick up and Delivery tasks, and, at the lowest level, ensures safety at the crossroads and controls the trajectories. We focus here on the top level, while introducing a time dependent estimation of the risk induced by the traversal of any arc at a given time. We set a model, state some structural results, and design, in order to route and schedule the vehicles according to a well-fitted compromise between speed and risk, a bi-level algorithm and a A* algorithm which both relies on a reinforcement learning scheme.
HAL (Le Centre pour la Communication Scientifique Directe), May 22, 2023
The balanced traveling salesman problem (BTSP) is a variant of the traveling salesman problem, in... more The balanced traveling salesman problem (BTSP) is a variant of the traveling salesman problem, in which one seeks a tour that minimizes the difference between the largest and smallest edge costs in the tour. The BTSP, which is obviously NP-hard, was first investigated by Larusic and Punnen in 2011 [9]. They proposed several heuristics based on the double-threshold framework, which converge to good-quality solutions though not always optimal (e.g. 27 provably optimal solutions were found among 65 TSPLIB instances of at most 500 vertices). In this paper, we design a special-purpose branch-and-cut algorithm for solving exactly the BTSP. In contrast with the classical TSP, due to the BTSP's objective function, the efficiency of algorithms for solving the BTSP depends heavily on determining correctly the largest and smallest edge costs in the tour. In the proposed branch-and-cut algorithm, we develop several mechanisms based on local cutting planes, edge elimination, and variable fixing to locate more and more precisely those edge costs. Other important ingredients of our algorithm are heuristics for improving the lower and upper bounds of the branch-and-bound tree. Experiments on the same TSPLIB instances show that our algorithm was able to solve to optimality 63 out of 65 instances.
HAL (Le Centre pour la Communication Scientifique Directe), Jun 4, 2023
Branch-and-Cut is a widely-used method for solving integer programming problems exactly. In recen... more Branch-and-Cut is a widely-used method for solving integer programming problems exactly. In recent years, researchers have been exploring ways to use Machine Learning to improve the decision-making process of Branch-and-Cut algorithms. While much of this research focuses on selecting nodes, variables, and cuts [11,9,26], less attention has been paid to designing efficient cut generation strategies in Branch-and-Cut algorithms, despite its large impact on the algorithm performance. In this paper, we focus on improving the generation of subtour elimination constraints, a core and compulsory class of cuts in Branch-and-Cut algorithms devoted to solving the Traveling Salesman Problem, which is one of the most studied combinatorial optimization problems. Our approach takes advantage of Machine Learning to address two questions before launching the separation routine to find cuts at a node of the search tree: 1) Do violated subtour elimination constraints exist? 2) If yes, is it worth generating them? We consider the former as a binary classification problem and adopt a Graph Neural Network as a classifier. By formulating subtour elimination constraint generation as a Markov decision problem, the latter can be handled through an agent trained by reinforcement learning. Our method can leverage the underlying graph structure of fractional solutions in the search tree to enhance its decision-making. Furthermore, once trained, the proposed Machine Learning model can be applied to any graph of any size (in terms of the number of vertices and edges). Numerical results show that our approach can significantly accelerate the performance of subtour elimination constraints in Branchand-Cut algorithms for the Traveling Salesman Problem.
In this paper, we introduce a new cooperative game theory model that we call productiondistributi... more In this paper, we introduce a new cooperative game theory model that we call productiondistribution game to address a major open problem for operations research in forestry, raised by Rönnqvist et al. in 2015, namely, that of modelling and proposing efficient sharing principles for practical collaboration in transportation in this sector. The originality of our model lies in the fact that the value/strength of a player does not only depend on the individual cost or benefit of the objects she owns but also depends on her market shares (customers demand). We show however that the production-distribution game is an interesting special case of a market game introduced by Shapley and Shubik in 1969. As such it exhibits the nice property of having a non-empty core. We then prove that we can compute both the nucleolus and the Shapley value efficiently, in a nontrivial and interesting special case. We in particular provide two different algorithms to compute the nucleolus: a simple separation algorithm and a fast primal-dual algorithm. Our results can be used to tackle more general versions of the problem and we believe that our contribution paves the way towards solving the challenging open problem herein.
HAL (Le Centre pour la Communication Scientifique Directe), 2003
Mars 2003 Cahier n°2003-002 Résumé: Soit G=(V,E) un graphe non-orienté et 2-arêtes connexe. Chaqu... more Mars 2003 Cahier n°2003-002 Résumé: Soit G=(V,E) un graphe non-orienté et 2-arêtes connexe. Chaque arête et sommet de G est muni d'un poids. Le problème du sous-graphe 2-arêtes connexe de poids minimum dans G (2ECSP), est de trouver un sous-graphe 2-arêtes connexe de G tel que la somme des poids sur ses sommets et ses arête soit minimum. Le 2ECSP généralise le problème, bien connu, du sousgraphe Steiner 2-arêtes connexe. Dans cet article, l'enveloppe convexe des vecteurs d'incidences des solutions de 2ECSP est étudiée. Une formulation naturelle du problème par un programme linéaire en nombres entiers est premièrement établie. Il est aussi montré que la relaxation ne suffit pas pour décrire l'enveloppe convexe associée au 2ECSP même dans une classe restreinte comme celle des graphes série-parallèles. Une nouvelle classe d'inégalités valides pour le 2ECSP est introduite. Il est monté qu'une sousclasse de ces inégalités peut être séparée en temps polynomial. Comme conséquence, ces nouvelles inégalités peuvent être séparées en temps polynomial dans la classe des graphes série-parallèles.
HAL (Le Centre pour la Communication Scientifique Directe), Nov 29, 2021
We give a combinatorial algorithm to find a maximum packing of hypertrees in a capacitated hyperg... more We give a combinatorial algorithm to find a maximum packing of hypertrees in a capacitated hypergraph. Based on this we extend to hypergraphs several algorithms for the k-cut problem, that are based on packing spanning trees in a graph. In particular we give a γapproximation algorithm for hypergraphs of rank γ, extending the work of Ravi and Sinha [22] for graphs. We also extend the work of Chekuri, Quanrud and Xu [7] in graphs, to give an algorithm for the k-cut problem in hypergraphs that is polynomial if k and the rank of the hypergraph are fixed. We also give a combinatorial algorithm to solve a linear programming relaxation of this problem in hypergraphs.
In this article, we consider a particular case of bi‐objective optimization (BOO), called bi‐obje... more In this article, we consider a particular case of bi‐objective optimization (BOO), called bi‐objective minimization (BOM), where the two objective functions to be minimized take only positive values. As well as for BOO, most of the methods proposed in the literature for solving BOM focus on computing the Pareto‐optimal solutions representing different trade‐offs between two objectives. However, it may be difficult for a central decision‐maker to determine the preferred solutions due to the huge number of solutions in the Pareto set. We propose a novel criterion for selecting the preferred Pareto‐optimal solutions by introducing the concept of ‐Nash Fairness (‐) solutions inspired by the definition of proportional fairness. The ‐ solutions are the feasible solutions achieving some proportional nash equilibrium between the two objectives. The positive parameter is introduced to reflect the relative importance of the first objective to the second one. For this work, we will discuss exi...
In dial-a-ride systems involving autonomous vehicles circulating along a circuit, a fleet of vehi... more In dial-a-ride systems involving autonomous vehicles circulating along a circuit, a fleet of vehicles must serve clients who request rides between stations of the circuit such that the total number of pickup and drop-off operations is minimized. In this paper, we focus on a unitary variant where each client requests a single place in the vehicles and all the clients must be served within a single tour of the circuit. Such unitary variant induces a combinatorial optimization problem for which we introduce a nontrivial special case that is polynomially solvable when the capacity of each vehicle is at most 2 but it is NP-Hard otherwise. We also study the polytope associated with the solutions to this problem. We introduce new families of valid inequalities and give necessary and sufficient conditions under which they are facet-defining. Based on these inequalities, we devise an efficient branch-and-cut algorithm that outperforms the state-of-the-art commercial solvers.
We give a combinatorial algorithm to find a maximum packing of hypertrees in a capacitated hyperg... more We give a combinatorial algorithm to find a maximum packing of hypertrees in a capacitated hypergraph. Based on this we extend to hypergraphs several algorithms for the k-cut problem, that are based on packing spanning trees in a graph. In particular we give a γapproximation algorithm for hypergraphs of rank γ, extending the work of Ravi and Sinha [22] for graphs. We also extend the work of Chekuri, Quanrud and Xu [7] in graphs, to give an algorithm for the k-cut problem in hypergraphs that is polynomial if k and the rank of the hypergraph are fixed. We also give a combinatorial algorithm to solve a linear programming relaxation of this problem in hypergraphs.
2022 12th International Workshop on Resilient Networks Design and Modeling (RNDM)
We consider a fair version of combinatorial optimization, which aims for both Pareto-efficiency a... more We consider a fair version of combinatorial optimization, which aims for both Pareto-efficiency and fairness of a solution. A possible approach to achieve the objectives simultaneously is to use the Ordered Weighted Average (OWA) aggregating function, which can be formulated into mix-integer programming (MIP) formulations. In this paper, we study two MIP formulations proposed in the literature for the OWA in the context of fair combinatorial optimization. On the one hand, we prove that both MIP formulations are equivalent in terms of linear relaxations. On the other hand, we estimate the quality with regard to the OWA value of an optimal solution of original combinatorial optimization. An experimental evaluation of the MIP formulations in tackling OWA Traveling Salesman Problem is also presented.
Hypergraphics matroids were studied first by Lorea [20] and later by Frank et al [10]. They can b... more Hypergraphics matroids were studied first by Lorea [20] and later by Frank et al [10]. They can be seen as generalizations of graphic matroids. Here we show that several algorithms developed for the graphic case can be extended to hypergraphic matroids. We treat the following: the separation problem for the associated polytope, testing independence, separation of partition inequalities, computing the rank of a set, computing the strength, computing the arboricity and network reinforcement.
In this paper, we consider a variant of the Traveling Salesman Problem (TSP), called Balanced Tra... more In this paper, we consider a variant of the Traveling Salesman Problem (TSP), called Balanced Traveling Salesman Problem (BTSP) [7]. The BTSP seeks to find a tour which has the smallest maxmin distance : the difference between the maximum edge cost and the minimum one. We present a Mixed Integer Program (MIP) to find optimal solutions minimizing the max-min distance for BTSP. However, minimizing only the max-min distance may lead to a tour with an inefficient total cost in many situations. Hence, we propose a fair way based on Nash equilibrium [5], [11] to inject the total cost into the objective function of the BTSP. We consider a Nash equilibrium as it is defined in a context of fair competition based on proportional-fair scheduling. For BTSP, we are interested in solutions achieving a Nash equilibrium between two players: the first aims at minimizing the total cost and the second aims at minimizing the max-min distance. We call such solutions Nash Fairness (NF) solutions. We first show that NF solutions for BTSP exist and may be more than one. We show that NF solutions are Pareto-optimal [10] and can be found by optimizing a sequence of linear combinations of the two players objectives based on Weighted Sum Method [13]. We then focus on extreme NF solutions which are NF solutions having either the smallest value of total cost or the smallest max-min distance. Finally, we propose a Newton-based iterative algorithm which converges to extreme NF solutions in a polynomial number of iterations. Computational results on smallsize instances from TSPLIB will be presented and commented.
The Stop Number Problem arises in the management of a dial-a-ride system served by a fleet of aut... more The Stop Number Problem arises in the management of a dial-a-ride system served by a fleet of autonomous electric vehicles. In such a system, clients request for a ride from an origin station to a destination station, and a fleet of capacitated vehicles must satisfy all requests. The goal is to minimize the number of pick-up/drop-off operations. In this paper we focus on a special case of this problem that was recently conjectured to be NP-Hard. In this regard, we show how such special case relates to other problems known in the literature in order to derive some polynomial-time solvable variants. Moreover, we provide a positive answer to the conjecture by showing that the problem is NP-Hard for any fixed capacity greater than or equal to 2, even for the case where the graph of requests is restricted to the class of planar bipartite graphs. Our proof of NP-Hardness also improves the complexity results known in the literature for the related problems identified.
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Papers by Mourad Baiou