Fixed-Parameter Algorithms for Rectilinear Steiner tree and Rectilinear Traveling Salesman Problem in the plane

Given a set $P$ of $n$ points with their pairwise distances, the traveling salesman problem (TSP) asks for a shortest tour that visits each point exactly once. A TSP instance is rectilinear when the points lie in the plane and the distance considered between two points is the $l_1$ distance. In this paper, a fixed-parameter algorithm for the Rectilinear TSP is presented and relies on techniques for solving TSP on bounded-treewidth graphs. It proves that the problem can be solved in $O\left(nh7^h\right)$ where $h \leq n$ denotes the number of horizontal lines containing the points of $P$. The same technique can be directly applied to the problem of finding a shortest rectilinear Steiner tree that interconnects the points of $P$ providing a $O\left(nh5^h\right)$ time complexity. Both bounds improve over the best time bounds known for these problems.Comment: 24 pages, 13 figures, 6 table

Exact algorithms for the order picking problem

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 mixed-integer 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

Des explications pour reconnaître et exploiter les structures cachées

http://www710.univ-lyon1.fr/~csolnonL'identification de structures propres à un problème est souvent une étape clef pour la conception d'heuristiques de recherche comme la compréhension de la complexité du problème. De nombreuses approches en Recherche Opérationnelle emploient des stratégies de relaxations ou décompositions dès lors que certaines structures idoines ont été identifiées. L'étape suivante est la conception d'algorithmes de résolution qui puisse intégrer à la volée, pendant la résolution, ce type d'information. Cet article propose d'utiliser un solveur de contraintes à base d'explications pour collecter de l'information pertinente sur les structures dynamiques et statiques inhérentes au problème. Par ailleurs, la reconnaissance de relations spécifiques entre les variables suggère l'adaptation d'algorithmes dédiés issus du monde de la Recherche Opérationnelle au contexte de la programmation par contraintes. Une telle adaptation est discutée dans le cadre de la décomposition de Benders

Des explications pour reconnaître et exploiter les structures cachées d'un problème combinatoire

L'identification de structures propres à un problème est souvent une étape clef pour la conception d'heuristiques de recherche comme pour la compréhension de la complexité du problème. De nombreuses approches en Recherche Opérationnelle emploient des stratégies de relaxation ou de décomposition dès lors que certaines structures idoines ont été identifiées. L'étape suivante est la conception d'algorithmes de résolution qui puissent intégrer à la volée, pendant la résolution, ce type d'information. Cet article propose d'utiliser un solveur de contraintes à base d'explications pour collecter une information pertinente sur les structures dynamiques et statiques inhérentes au problème. Identifying structure in a given combinatorial problem is often a key step for designing efficient search heuristics or for understanding the inherent complexity of the problem. Several Operations Research approaches apply decomposition or relaxation strategies upon such a structure identified within a given problem. The next step is to design algorithms that adaptively integrate that kind of information during search. We claim in this paper, inspired by previous work on impact-based search strategies for constraint programming, that using an explanation-based constraint solver may lead to collect invaluable information on the intimate dynamic and static structure of a problem instance

An Integer Programming Formulation Using Convex Polygons for the Convex Partition Problem

A convex partition of a point set P in the plane is a planar partition of the convex hull of P into 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 with 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

Roadef Challenge 2014: A Modeling Approach, Rolling stock unit management on railway sites

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

The Deployment of a Constraint-Based Dental School Timetabling System

International audienceWe describe a constraint-based timetabling system that was developed for the dental school based at Cork University Hospital in Ireland.This system has been deployed since 2010.Dental school timetabling differs from other university course scheduling in that certain clinic sessions can be used by multiple courses at the same time, provided a limit on room capacity is satisfied.Starting from a constraint programming solution using a web interface, we have moved to a mixed integer programming-based solver to deal with multiple objective functions, along with a dedicated Java application, which provides a rich user interface.Solutions for the years 2010, 2011 and 2012 have been used in the dental school, replacing a manual timetabling process, which could no longer cope with increasing student numbers and resulting resource bottlenecks.The use of the automated system allowed the dental school to increase student numbers to the maximum possible given the available resources.It also provides the school with a valuable "what-if" analysis tool

A shortest path-based approach to the multileaf collimator sequencing problem

AbstractThe multileaf collimator sequencing problem is an important component in effective cancer treatment delivery. The problem can be formulated as finding a decomposition of an integer matrix into a weighted sequence of binary matrices whose rows satisfy a consecutive ones property. Minimising the cardinality of the decomposition is an important objective and has been shown to be strongly NP-hard, even for a matrix restricted to a single column or row. We show that in this latter case it can be solved efficiently as a shortest path problem, giving a simple proof that the one-row problem is fixed-parameter tractable in the maximum intensity. We develop new linear and constraint programming models exploiting this result. Our approaches significantly improve the best known for the problem, bringing real-world sized problem instances within reach of exact algorithms

Techniques rétrospectives pour résoudre le Minimum Open Stacks Problem

Nous présentons une étude de cas sur le MOSP (minimum number of open stacks problem). Ce problème se rencontre dans des environnements de production où la réalisation d'un ensemble de tâches n´ecessite une uti- lisation simultanée de diff´erentes ressources (les piles – stacks). `A travers ce problème, nous cherchons com- ment des raisonnements rétrospectifs assez classiques mais bas´es sur les explications peuvent ˆetre utilis´es pour ´elaguer l'espace de recherche. Les explications ont, en effet, souvent ´et´e utilis´ees pour mettre en place des m´ecanismes sophistiqu´es de retour arri`ere dans le cadre de la programmation par contraintes mais tr`es peu dans des sch´emas de type nogood recording. Ce n'est pas le cas dans la communaut´e SAT o`u la notion de clause apprise joue un grand rˆole aussi bien pour l'´elagage de l'espace de recherche que pour l'exploration elle-mˆeme. Dans cet article, nous introduisons un nogood g´en´eralis´e (bas´e sur les explications) pour le MOSP qui nous per- met d'introduire une nouvelle technique de r´esolution. Des r´esultats exp´erimentaux montrent l'int´erˆet d'une telle approche pour le MOSP et la capacit´e des explica- tions `a identifier et `a exploiter dynamiquement la struc- ture de ces probl`emes