Concept# Problème d'affectation

Résumé

En informatique, plus précisément en recherche opérationnelle et d'optimisation combinatoire, le problème d'affectation consiste à attribuer au mieux des tâches à des agents. Chaque agent peut réaliser une unique tâche pour un coût donné et chaque tâche doit être réalisée par un unique agent. Les affectations (c'est-à-dire les couples agent-tâche) ont toutes un coût défini. Le but est de minimiser le coût total des affectations afin de réaliser toutes les tâches.
thumb|L'affectation optimale entre le groupe d'agent et de tâche est représenté ici par les arcs rouges.
Plus formellement, l'objectif est de déterminer un couplage d'une taille égale au nombre de tâches, de poids minimum dans un graphe biparti valué. Si il y a autant d'agents que de tâches, il s'agit de déterminer un couplage parfait de poids minimum dans un graphe biparti valué. Le problème d'affectation peut être résolu en temps polynomial par l'algorithme hongrois, il appartient par conséquent à la classe de complexité P

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Couplage (théorie des graphes)

En théorie des graphes, un couplage ou appariement (en anglais matching) d'un graphe est un ensemble d'arêtes de ce graphe qui n'ont pas de sommets en commun.
Définitions
Soit un graphe s

Optimisation combinatoire

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Optimisation linéaire

thumb|upright=0.5|Optimisation linéaire dans un espace à deux dimensions (x1, x2). La fonction-coût fc est représentée par les lignes de niveau bleues à gauche et par le plan bleu à droite. L'ensemble

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ME-454: Modelling and optimization of energy systems

The goal of the lecture is to present and apply techniques for the modelling and the thermo-economic optimisation of industrial process and energy systems. The lecture covers the problem statement, the solving methods for the simulation and the single and multi-objective optimisation problems.

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The student will learn state-of-the-art algorithms for solving differential equations. The analysis and implementation of these algorithms will be discussed in some detail.

EE-556: Mathematics of data: from theory to computation

This course provides an overview of key advances in continuous optimization and statistical analysis for machine learning. We review recent learning formulations and models as well as their guarantees, describe scalable solution techniques and algorithms, and illustrate the trade-offs involved.

Constraint Satisfaction Problems (CSPs) are ubiquitous in computer science. Many problems, ranging from resource allocation and scheduling to fault diagnosis and design, involve constraint satisfaction as an essential component. A CSP is given by a set of variables and constraints on small subsets of these variables. It is solved by finding assignments of values to the variables such that all constraints are satisfied. In its most general form, a CSP is combinatorial and complex. In this thesis, we consider constraint satisfaction problems with variables in continuous, numerical domains. Contrary to most existing techniques, which focus on computing a single optimal solution, we address the problem of computing a compact representation of the space of all solutions that satisfy the constraints. This has the advantage that no optimization criterion has to be formulated beforehand, and that the space of possibilities can be explored systematically. In certain applications, such as diagnosis and design, these advantages are crucial. In consistency techniques, the solution space is represented by labels assigned to individual variables or combinations of variables. When the labeling is globally consistent, each label contains only those values or combinations of values which appear in at least one solution. This kind of labeling is a compact, sound and complete representation of the solution space, and can be combined with other reasoning methods. In practice, computing a globally consistent labeling is too complex. This is usually tackled in two ways. One way is to enforce consistencies locally, using propagation algorithms. This prunes the search space and hence reduces the subsequent search effort. The other way is to identify simplifying properties which guarantee that global consistency can be enforced tractably using local propagation algorithms. When constraints are represented by mathematical expressions, implementing local consistency algorithms is difficult because it requires tools for solving arbitrary systems of equations. In this thesis, we propose to approximate feasible solution regions by 2k-trees, thus providing a means of combining constraints logically rather than numerically. This representation, commonly used in computer vision and image processing, avoids using complex mathematical tools. We propose simple and stable algorithms for computing labels of arbitrary degrees of consistency using this representation. For binary constraints, it is known that simplifying convexity properties reduces the complexity of solving a CSP. These properties guarantee that local degrees of consistency are sufficient to ensure global consistency. We show how, in continuous domains, these results can be generalized to ternary and in fact arbitrary n-ary constraints. This leads to polynomial-time algorithms for computing globally consistent labels for a large class of constraint satisfaction problems with continuous variables. We describe and justify our representation of constraints and our consistency algorithms. We also give a complete analysis of the theoretical results we present. Finally, the developed techniques are illustrated using practical examples.

As robots start pervading human environments, the need for new interfaces that would simplify human-robot interaction has become more pressing. Robot Programming by Demonstration (RbD) develops intuitive ways of programming robots, taking inspiration in strategies used by humans to transmit knowledge to apprentices. The user-friendliness of RbD is meant to allow lay users with no prior knowledge in computer science, electronics or mechanics to train robots to accomplish tasks the same way as they would with a co-worker. When a trainer teaches a task to a robot, he/she shows a particular way of fulfilling the task. For a robot to be able to learn from observing the trainer, it must be able to learn what the task entails (i.e. answer the so-called "What-to-imitate?" question), by inferring the user's intentions. But most importantly, the robot must be able to adapt its own controller to fit at best the demonstration (the so-called "How-to-imitate?" question) despite different setups and embodiments. The latter is the question that interested us in this thesis. It relates to the problem of optimizing the reproduction of the task under environmental constraints. The "How-to-imitate?" question is subdivided into two problems. The first problem, also known as the "correspondence problem", relates to resolving the discrepancy between the human demonstrator and robot's body that prevent the robot from doing an identical reproduction of the task. Even though we helped ourselves by considering solely humanoid platforms, that is platforms that have a joint configuration similar to that of the human, discrepancies in the number of degrees of freedom and range of motion remained. We resolved these by exploiting the redundant information conveyed through the demonstrations by collecting data through different frames of reference. By exploiting these redundancies in an algorithm comparable to the damped least square algorithm, we are able to reproduce a trajectory that minimizes the error between the desired trajectory and the reproduced trajectory across each frame of reference. The second problem consists in reproducing a trajectory in an unknown setup while respecting the task constraints learned during training. When the information learned from the demonstration no longer suffice to generalize the task constraints to a new set-up, the robot must re-learn the task; this time through trial-and-error. Here we considered the combination of trial-and-error learning to complement RbD. By adding a trial-and-error module to the original Imitation Learning algorithm, the robot can find a solution that is more adapted to the context and to its embodiment than the solution found using RbD. Specifically, we compared Reinforcement Learning (RL) – to other classical optimization techniques. We show that the system is advantageous in that: a) learning is more robust to unexpected events that have not been encountered during the demonstrations and b) the robot is able to optimize its own model of the task according to its own embodiment.

Submodular functions are a widely studied topic in theoretical computer science. They have found several applications both theoretical and practical in the fields of economics, combinatorial optimization and machine learning. More recently, there have also been numerous works that study combinatorial problems with submodular objective functions. This is motivated by their natural diminishing returns property which is useful in real-world applications. The thesis at hand is concerned with the study of streaming and matching problems with submodular functions.Firstly, motivated by developing robust algorithms, we propose a new adversarial injections model, in which the input is ordered randomly, but an adversary may inject misleading elements at arbitrary positions. We study the maximum matching problem and cardinality constrained monotone submodular maximization. We show that even under this seemingly powerful adversary, it is possible to break the barrier of 1/2 for both these problems in the streaming setting. Our main result is a novel streaming algorithm that computes a 0.55-approximation for cardinality constrained monotone submodular maximization.In the second part of the thesis, we study the problem of matroid intersection in the semi-streaming setting. Our main result is a (2 + e)-approximate semi-streaming algorithm for weighted matroid inter- section improving upon the previous best guarantee of 4 + e. While our algorithm is based on the local ratio technique, its analysis differs from the related problem of weighted maximum matching and uses the concept of matroid kernels. We are also able to generalize our results to work for submodular functions by adapting ideas from a recent result by Levin and Wajc (SODA'21) on submodular maximization subject to matching constraints.Finally, we study the submodular Santa Claus problem in the restricted assignment case. The submodular Santa Claus problem was introduced in a seminal work by Goemans, Harvey, Iwata, and Mirrokni (SODA'09) as an application of their structural result. In the mentioned problem n unsplittable resources have to be assigned to m players, each with a monotone submodular utility function fi. The goal is to maximize mini fi(Si) where S1, . . . , Sm is a partition of the resources. The result by Goemans et al. implies a polynomial time O(n1/2+e)-approximation algorithm. In the restricted assignment case, each player is given a set of desired resources Gi and the individual valuation functions are defined as fi(S)= f(SnGi). OurmainresultisaO(loglog(n))-approximation algorithm for the problem. Our proof is inspired by the approach of Bansal and Srividenko (STOC'06) to the Santa Claus problem. Com- pared to the more basic linear setting, the introduction of submodularity requires a much more involved analysis and several new ideas.