Nicolas Catusse - Academia.edu (original) (raw)

Papers by Nicolas Catusse

arXiv (Cornell University), Mar 2, 2017

Order picking is the problem of collecting a set of products in a warehouse in a minimum amount o... more Order picking is the problem of collecting a set of products in a warehouse in a minimum amount of time. It is currently a major bottleneck in supply-chain because of its cost in time and labor force. This article presents two exact and effective algorithms for this problem. Firstly, a sparse formulation in mixedinteger programming is strengthened by preprocessing and valid inequalities. Secondly, a dynamic programming approach generalizing known algorithms for two or three cross-aisles is proposed and evaluated experimentally. Performances of these algorithms are reported and compared with the Traveling Salesman Problem (TSP) solver Concorde.

arXiv (Cornell University), Dec 14, 2020

A convex partition of a point set P in the plane is a planar partition of the convex hull of P wi... more A convex partition of a point set P in the plane is a planar partition of the convex hull of P with empty convex polygons or internal faces whose extreme points belong to P. In a convex partition, the union of the internal faces give the convex hull of P and the interiors of the polygons are pairwise disjoint. Moreover, no polygon is allowed to contain a point of P in its interior. The problem is to find a convex partition based on the minimum number of internal faces. The problem has been shown to be NP-Hard and was recently used in the CG:SHOP Challenge 2020. We propose a new integer linear programming (IP) formulation that considerably improves over the existing one. It relies on the representation of faces as opposed to segments and points. A number of geometric properties are used to strengthen it. Data sets of 100 points are easily solved to optimality and the lower bounds provided by the model can be computed up to 300 points.

arXiv (Cornell University), Jul 11, 2022

Computing lower and upper bounds on the competitive ratio of online algorithms is a challenging q... more Computing lower and upper bounds on the competitive ratio of online algorithms is a challenging question: For a minimization combinatorial problem, proving a competitive ratio for a given algorithm leads to an upper bound. However computing lower bounds requires a proof on all algorithms. This can be modeled as a 2-player game where a strategy for one of the players is a proof for the lower bound. The tree representing the proof can can be found computationally. This method has been used with success on the online bin stretching problem where a set of items must be packed online in m bins. The items are guaranteed to fit into the m bins. However, the online procedure might require to stretch the bins to a larger capacity in order to be able to pack all the items. This stretching factor is the objective to be minimized. We propose original ideas to strongly improve the speed of computer searches for lower bound: propagate the game states that can be pruned from the search and improve the speed and memory usage in the dynamic program which is used in the search. These improvements allowed to increase significantly the speed of the search and hence to prove new lower bounds for the bin stretching problem for 6, 7 and 8 bins.

Theoretical Computer Science, May 1, 2011

In this paper, we present an optimal O(n 2) time algorithm for deciding if a metric space (X, d) ... more In this paper, we present an optimal O(n 2) time algorithm for deciding if a metric space (X, d) on n points can be isometrically embedded into the plane endowed with the l1-metric. It improves the O(n 2 log 2 n) time algorithm of J. Edmonds (2008). Together with some ingredients introduced by Edmonds, our algorithm uses the concept of tight span and the injectivity of the l1-plane. A different O(n 2) time algorithm was recently proposed by D. Eppstein (2009).

Algorithmica, Aug 31, 2011

Let B be a centrally symmetric convex polygon of R 2 and ||p − q|| be the distance between two po... more Let B be a centrally symmetric convex polygon of R 2 and ||p − q|| be the distance between two points p, q ∈ R 2 in the normed plane whose unit ball is B. For a set T of n points (terminals) in R 2 , a B-Manhattan network on T is a network N (T) = (V, E) with the property that its edges are parallel to the directions of B and for every pair of terminals t i and t j , the network N (T) contains a shortest B-path between them, i.e., a path of length ||t i − t j ||. A minimum B-Manhattan network on T is a B-Manhattan network of minimum possible length. The problem of finding minimum B-Manhattan networks has been introduced by Gudmundsson, Levcopoulos, and Narasimhan (APPROX'99) in the case when the unit ball B is a square (and hence the distance ||p − q|| is the l 1 or the l ∞-distance between p and q) and it has been shown recently by Chin, Guo, and Sun [6] to be strongly NP-complete. Several approximation algorithms (with factors 8,4,3, and 2) for the minimum Manhattan problem are known. In this paper, we propose a factor 2.5 approximation algorithm for the minimum B-Manhattan network problem. The algorithm employs a simplified version of the strip-staircase decomposition proposed in our paper [5] and subsequently used in other factor 2 approximation algorithms for the minimum Manhattan problem.

HAL (Le Centre pour la Communication Scientifique Directe), 2011

In this paper, we present an optimal O(n 2) time algorithm for deciding if a metric space (X, d) ... more In this paper, we present an optimal O(n 2) time algorithm for deciding if a metric space (X, d) on n points can be isometrically embedded into the plane endowed with the l1-metric. It improves the O(n 2 log 2 n) time algorithm of J. Edmonds (2008). Together with some ingredients introduced by Edmonds, our algorithm uses the concept of tight span and the injectivity of the l1-plane. A different O(n 2) time algorithm was recently proposed by D. Eppstein (2009).

HAL (Le Centre pour la Communication Scientifique Directe), Jul 3, 2022

arXiv (Cornell University), Mar 31, 2023

Warehouses are the scene of complex logistic problems integrating different decision layers. This... more Warehouses are the scene of complex logistic problems integrating different decision layers. This paper addresses the Joint Order Batching, Picker Routing and Sequencing Problem with Deadlines (JOBPRSP-D) in rectangular warehouses. To tackle the problem an exponential linear programming formulation is proposed. It is solved with a column generation heuristic able to provide valid lower and upper bounds on the optimal value. We start by showing that the JOBPRSP-D is related to the bin packing problem rather than the scheduling problem. We take advantage of this aspect to derive a number of valid inequalities that enhance the resolution of the master problem. The proposed algorithm is evaluated on publicly available data-sets. It is able to optimally solve instances with up to 18 orders in few minutes. It is also able to prove optimality or to provide high-quality lower bounds on larger instances with 100 orders. To the best of our knowledge this is the first paper that provides optimality guarantee on large size instances for the JOBPRSP-D, the results can therefore be used to assert the quality of heuristics proposed for the same problem.

HAL (Le Centre pour la Communication Scientifique Directe), 2016

Order picking is the problem of collecting a set of products in a warehouse in a minimum amount o... more Order picking is the problem of collecting a set of products in a warehouse in a minimum amount of time. It is currently a major bottleneck in supply-chain because of its cost in time and labor force. This article presents two exact and effective algorithms for this problem. Firstly, a sparse formulation in mixedinteger programming is strengthened by preprocessing and valid inequalities. Secondly, a dynamic programming approach generalizing known algorithms for two or three cross-aisles is proposed and evaluated experimentally. Performances of these algorithms are reported and compared with the Traveling Salesman Problem (TSP) solver Concorde.

HAL (Le Centre pour la Communication Scientifique Directe), Feb 10, 2016

HAL (Le Centre pour la Communication Scientifique Directe), Sep 9, 2022

Teaching graph theory with the most adequate tools requires time and ideas. We present how an ope... more Teaching graph theory with the most adequate tools requires time and ideas. We present how an open community of teachers shares contents and ideas on an innovative platform. The objective is to get the students autonomous in their training with activities that give them immediate feedback on their understanding. Beyond learning, the very large collection of exercises of various levels can also be used to evaluate the student's level. The proposed activities can be algorithm's code in classical programming languages (e.g. Java, Python) that the student can test with predefined tests proposed by the teacher or collections of generated questions.

We study an elementary path problem which appears in the pricing step of a column generation sche... more We study an elementary path problem which appears in the pricing step of a column generation scheme solving the kidney exchange problem. The latter aims at finding exchanges of donations in a pool of patients and donors of kidney transplantations. Informally, the problem is to determine a set of cycles and chains of limited length maximizing a medical benefit in a directed graph. The cycle formulation, a large-scale model of the problem restricted to cycles of donation, is efficiently solved via branch-and-price. When including chains of donation however, the pricing subproblem becomes NP-hard. This article proposes a new complete column generation scheme that takes into account these chains initiated by altruistic donors. The development of non-exact dynamic approaches for the pricing problem, the NG-route relaxation and the color coding heuristic, leads to an efficient column generation process.

VeRoLog 2019 - Workshop of the EURO Working Group on Vehicle Routing and Logistics optimization, Jun 2, 2019

International audiencePicking is the process of retrieving products from inventory.It is mostly d... more International audiencePicking is the process of retrieving products from inventory.It is mostly done manually by dedicated employees called pickers and is considered the most expensive of warehouse operations.To reduce the picking cost, customer orders can be grouped into batches that are then collected by traveling the shortest possible distance.This work presents an exponential linear programming formulation to tackle the joint order batching and picker routing problem.Variables, or columns, are related to the picking routes in the warehouse.Computing such routess is generally an intractable routing problem and relates to the well known traveling salesman problem (TSP).Nonetheless, the rectangular warehouse's layouts can be used to efficiently solve the corresponding TSP and take into account in the development of an efficient subroutine, called oracle.We therefore investigate whether such an oracle allows for an effective exponential formulation.In order to tackle the exponential number of variables, we develop a column generation heuristic with performance guarantee.The pricing problem approximates the distances, and when finding a solution the oracle is used to get the optimal distance of this solution.The performance of the proposed column generation is strengthen using stabilization techniques and a rich column set.Experimented on a publicly available benchmark, the algorithm proves to be very effective.It improves many of the best known solutions and provides very strong lower bounds.Finally, this approach is applied to another industrial case to demonstrate its interest for this field of application

Let B be a centrally symmetric convex polygon of R 2 and ||p − q|| be the distance between two po... more Let B be a centrally symmetric convex polygon of R 2 and ||p − q|| be the distance between two points p, q ∈ R 2 in the normed plane whose unit ball is B. For a set T of n points (terminals) in R 2 , a B-Manhattan network on T is a network N (T) = (V, E) with the property that its edges are parallel to the directions of B and for every pair of terminals t i and t j , the network N (T) contains a shortest B-path between them, i.e., a path of length ||t i − t j ||. A minimum B-Manhattan network on T is a B-Manhattan network of minimum possible length. The problem of finding minimum B-Manhattan networks has been introduced by Gudmundsson, Levcopoulos, and Narasimhan (APPROX'99) in the case when the unit ball B is a square (and hence the distance ||p − q|| is the l 1 or the l ∞-distance between p and q) and it has been shown recently by Chin, Guo, and Sun [6] to be strongly NP-complete. Several approximation algorithms (with factors 8,4,3, and 2) for the minimum Manhattan problem are known. In this paper, we propose a factor 2.5 approximation algorithm for the minimum B-Manhattan network problem. The algorithm employs a simplified version of the strip-staircase decomposition proposed in our paper [5] and subsequently used in other factor 2 approximation algorithms for the minimum Manhattan problem.

Nous presentons un modele d’aide au dimensionnement et a la planification pluri-annuels des inves... more Nous presentons un modele d’aide au dimensionnement et a la planification pluri-annuels des investissements materiels et humains pour maximiser le revenu agricole d’une ferme maraichere en reponse a des demandes connues de clients. Ce modele en programmation lineaire mixte integre les couts et les temps lies a l’installation des infrastructures productives, telles que les serres, les tunnels ou les reseaux d’irrigation. Nous fondons notre formulation du modele sur les problemes de dimensionnement de lot et d'ordonnancement avec couts et temps fixes. Des resultats numeriques sur des instances realistes sont presentes.

We report here our analysis of the challenge Roadef/EURO 2014 problem and propose a methodology s... more We report here our analysis of the challenge Roadef/EURO 2014 problem and propose a methodology strongly based on modeling with MIP and CP technologies. Due to the complexity of the problem formulation, we believe that a robust engineering is easier to achieve by relying on models than dedicated code. Two core modeling ideas are presented for relating the daily maintenances limit and the linked departures to an assignment based MIP model. Additionnaly, a variant of the maximum matching problem lying at the heart of the problem is shown to be NP-Complete. Intermediate experimental results are given along the way to support the ideas reported.

arXiv (Cornell University), Mar 2, 2017

Order picking is the problem of collecting a set of products in a warehouse in a minimum amount o... more Order picking is the problem of collecting a set of products in a warehouse in a minimum amount of time. It is currently a major bottleneck in supply-chain because of its cost in time and labor force. This article presents two exact and effective algorithms for this problem. Firstly, a sparse formulation in mixedinteger programming is strengthened by preprocessing and valid inequalities. Secondly, a dynamic programming approach generalizing known algorithms for two or three cross-aisles is proposed and evaluated experimentally. Performances of these algorithms are reported and compared with the Traveling Salesman Problem (TSP) solver Concorde.

arXiv (Cornell University), Dec 14, 2020

A convex partition of a point set P in the plane is a planar partition of the convex hull of P wi... more A convex partition of a point set P in the plane is a planar partition of the convex hull of P with empty convex polygons or internal faces whose extreme points belong to P. In a convex partition, the union of the internal faces give the convex hull of P and the interiors of the polygons are pairwise disjoint. Moreover, no polygon is allowed to contain a point of P in its interior. The problem is to find a convex partition based on the minimum number of internal faces. The problem has been shown to be NP-Hard and was recently used in the CG:SHOP Challenge 2020. We propose a new integer linear programming (IP) formulation that considerably improves over the existing one. It relies on the representation of faces as opposed to segments and points. A number of geometric properties are used to strengthen it. Data sets of 100 points are easily solved to optimality and the lower bounds provided by the model can be computed up to 300 points.

arXiv (Cornell University), Jul 11, 2022

Computing lower and upper bounds on the competitive ratio of online algorithms is a challenging q... more Computing lower and upper bounds on the competitive ratio of online algorithms is a challenging question: For a minimization combinatorial problem, proving a competitive ratio for a given algorithm leads to an upper bound. However computing lower bounds requires a proof on all algorithms. This can be modeled as a 2-player game where a strategy for one of the players is a proof for the lower bound. The tree representing the proof can can be found computationally. This method has been used with success on the online bin stretching problem where a set of items must be packed online in m bins. The items are guaranteed to fit into the m bins. However, the online procedure might require to stretch the bins to a larger capacity in order to be able to pack all the items. This stretching factor is the objective to be minimized. We propose original ideas to strongly improve the speed of computer searches for lower bound: propagate the game states that can be pruned from the search and improve the speed and memory usage in the dynamic program which is used in the search. These improvements allowed to increase significantly the speed of the search and hence to prove new lower bounds for the bin stretching problem for 6, 7 and 8 bins.

Theoretical Computer Science, May 1, 2011

In this paper, we present an optimal O(n 2) time algorithm for deciding if a metric space (X, d) ... more In this paper, we present an optimal O(n 2) time algorithm for deciding if a metric space (X, d) on n points can be isometrically embedded into the plane endowed with the l1-metric. It improves the O(n 2 log 2 n) time algorithm of J. Edmonds (2008). Together with some ingredients introduced by Edmonds, our algorithm uses the concept of tight span and the injectivity of the l1-plane. A different O(n 2) time algorithm was recently proposed by D. Eppstein (2009).

Algorithmica, Aug 31, 2011

Let B be a centrally symmetric convex polygon of R 2 and ||p − q|| be the distance between two po... more Let B be a centrally symmetric convex polygon of R 2 and ||p − q|| be the distance between two points p, q ∈ R 2 in the normed plane whose unit ball is B. For a set T of n points (terminals) in R 2 , a B-Manhattan network on T is a network N (T) = (V, E) with the property that its edges are parallel to the directions of B and for every pair of terminals t i and t j , the network N (T) contains a shortest B-path between them, i.e., a path of length ||t i − t j ||. A minimum B-Manhattan network on T is a B-Manhattan network of minimum possible length. The problem of finding minimum B-Manhattan networks has been introduced by Gudmundsson, Levcopoulos, and Narasimhan (APPROX'99) in the case when the unit ball B is a square (and hence the distance ||p − q|| is the l 1 or the l ∞-distance between p and q) and it has been shown recently by Chin, Guo, and Sun [6] to be strongly NP-complete. Several approximation algorithms (with factors 8,4,3, and 2) for the minimum Manhattan problem are known. In this paper, we propose a factor 2.5 approximation algorithm for the minimum B-Manhattan network problem. The algorithm employs a simplified version of the strip-staircase decomposition proposed in our paper [5] and subsequently used in other factor 2 approximation algorithms for the minimum Manhattan problem.

HAL (Le Centre pour la Communication Scientifique Directe), 2011

In this paper, we present an optimal O(n 2) time algorithm for deciding if a metric space (X, d) ... more In this paper, we present an optimal O(n 2) time algorithm for deciding if a metric space (X, d) on n points can be isometrically embedded into the plane endowed with the l1-metric. It improves the O(n 2 log 2 n) time algorithm of J. Edmonds (2008). Together with some ingredients introduced by Edmonds, our algorithm uses the concept of tight span and the injectivity of the l1-plane. A different O(n 2) time algorithm was recently proposed by D. Eppstein (2009).

HAL (Le Centre pour la Communication Scientifique Directe), Jul 3, 2022

arXiv (Cornell University), Mar 31, 2023

Warehouses are the scene of complex logistic problems integrating different decision layers. This... more Warehouses are the scene of complex logistic problems integrating different decision layers. This paper addresses the Joint Order Batching, Picker Routing and Sequencing Problem with Deadlines (JOBPRSP-D) in rectangular warehouses. To tackle the problem an exponential linear programming formulation is proposed. It is solved with a column generation heuristic able to provide valid lower and upper bounds on the optimal value. We start by showing that the JOBPRSP-D is related to the bin packing problem rather than the scheduling problem. We take advantage of this aspect to derive a number of valid inequalities that enhance the resolution of the master problem. The proposed algorithm is evaluated on publicly available data-sets. It is able to optimally solve instances with up to 18 orders in few minutes. It is also able to prove optimality or to provide high-quality lower bounds on larger instances with 100 orders. To the best of our knowledge this is the first paper that provides optimality guarantee on large size instances for the JOBPRSP-D, the results can therefore be used to assert the quality of heuristics proposed for the same problem.

HAL (Le Centre pour la Communication Scientifique Directe), 2016

Order picking is the problem of collecting a set of products in a warehouse in a minimum amount o... more Order picking is the problem of collecting a set of products in a warehouse in a minimum amount of time. It is currently a major bottleneck in supply-chain because of its cost in time and labor force. This article presents two exact and effective algorithms for this problem. Firstly, a sparse formulation in mixedinteger programming is strengthened by preprocessing and valid inequalities. Secondly, a dynamic programming approach generalizing known algorithms for two or three cross-aisles is proposed and evaluated experimentally. Performances of these algorithms are reported and compared with the Traveling Salesman Problem (TSP) solver Concorde.

HAL (Le Centre pour la Communication Scientifique Directe), Feb 10, 2016

HAL (Le Centre pour la Communication Scientifique Directe), Sep 9, 2022

Teaching graph theory with the most adequate tools requires time and ideas. We present how an ope... more Teaching graph theory with the most adequate tools requires time and ideas. We present how an open community of teachers shares contents and ideas on an innovative platform. The objective is to get the students autonomous in their training with activities that give them immediate feedback on their understanding. Beyond learning, the very large collection of exercises of various levels can also be used to evaluate the student's level. The proposed activities can be algorithm's code in classical programming languages (e.g. Java, Python) that the student can test with predefined tests proposed by the teacher or collections of generated questions.

We study an elementary path problem which appears in the pricing step of a column generation sche... more We study an elementary path problem which appears in the pricing step of a column generation scheme solving the kidney exchange problem. The latter aims at finding exchanges of donations in a pool of patients and donors of kidney transplantations. Informally, the problem is to determine a set of cycles and chains of limited length maximizing a medical benefit in a directed graph. The cycle formulation, a large-scale model of the problem restricted to cycles of donation, is efficiently solved via branch-and-price. When including chains of donation however, the pricing subproblem becomes NP-hard. This article proposes a new complete column generation scheme that takes into account these chains initiated by altruistic donors. The development of non-exact dynamic approaches for the pricing problem, the NG-route relaxation and the color coding heuristic, leads to an efficient column generation process.

VeRoLog 2019 - Workshop of the EURO Working Group on Vehicle Routing and Logistics optimization, Jun 2, 2019

International audiencePicking is the process of retrieving products from inventory.It is mostly d... more International audiencePicking is the process of retrieving products from inventory.It is mostly done manually by dedicated employees called pickers and is considered the most expensive of warehouse operations.To reduce the picking cost, customer orders can be grouped into batches that are then collected by traveling the shortest possible distance.This work presents an exponential linear programming formulation to tackle the joint order batching and picker routing problem.Variables, or columns, are related to the picking routes in the warehouse.Computing such routess is generally an intractable routing problem and relates to the well known traveling salesman problem (TSP).Nonetheless, the rectangular warehouse's layouts can be used to efficiently solve the corresponding TSP and take into account in the development of an efficient subroutine, called oracle.We therefore investigate whether such an oracle allows for an effective exponential formulation.In order to tackle the exponential number of variables, we develop a column generation heuristic with performance guarantee.The pricing problem approximates the distances, and when finding a solution the oracle is used to get the optimal distance of this solution.The performance of the proposed column generation is strengthen using stabilization techniques and a rich column set.Experimented on a publicly available benchmark, the algorithm proves to be very effective.It improves many of the best known solutions and provides very strong lower bounds.Finally, this approach is applied to another industrial case to demonstrate its interest for this field of application

Let B be a centrally symmetric convex polygon of R 2 and ||p − q|| be the distance between two po... more Let B be a centrally symmetric convex polygon of R 2 and ||p − q|| be the distance between two points p, q ∈ R 2 in the normed plane whose unit ball is B. For a set T of n points (terminals) in R 2 , a B-Manhattan network on T is a network N (T) = (V, E) with the property that its edges are parallel to the directions of B and for every pair of terminals t i and t j , the network N (T) contains a shortest B-path between them, i.e., a path of length ||t i − t j ||. A minimum B-Manhattan network on T is a B-Manhattan network of minimum possible length. The problem of finding minimum B-Manhattan networks has been introduced by Gudmundsson, Levcopoulos, and Narasimhan (APPROX'99) in the case when the unit ball B is a square (and hence the distance ||p − q|| is the l 1 or the l ∞-distance between p and q) and it has been shown recently by Chin, Guo, and Sun [6] to be strongly NP-complete. Several approximation algorithms (with factors 8,4,3, and 2) for the minimum Manhattan problem are known. In this paper, we propose a factor 2.5 approximation algorithm for the minimum B-Manhattan network problem. The algorithm employs a simplified version of the strip-staircase decomposition proposed in our paper [5] and subsequently used in other factor 2 approximation algorithms for the minimum Manhattan problem.

Nous presentons un modele d’aide au dimensionnement et a la planification pluri-annuels des inves... more Nous presentons un modele d’aide au dimensionnement et a la planification pluri-annuels des investissements materiels et humains pour maximiser le revenu agricole d’une ferme maraichere en reponse a des demandes connues de clients. Ce modele en programmation lineaire mixte integre les couts et les temps lies a l’installation des infrastructures productives, telles que les serres, les tunnels ou les reseaux d’irrigation. Nous fondons notre formulation du modele sur les problemes de dimensionnement de lot et d'ordonnancement avec couts et temps fixes. Des resultats numeriques sur des instances realistes sont presentes.

We report here our analysis of the challenge Roadef/EURO 2014 problem and propose a methodology s... more We report here our analysis of the challenge Roadef/EURO 2014 problem and propose a methodology strongly based on modeling with MIP and CP technologies. Due to the complexity of the problem formulation, we believe that a robust engineering is easier to achieve by relying on models than dedicated code. Two core modeling ideas are presented for relating the daily maintenances limit and the linked departures to an assignment based MIP model. Additionnaly, a variant of the maximum matching problem lying at the heart of the problem is shown to be NP-Complete. Intermediate experimental results are given along the way to support the ideas reported.