Fair Allocation of Two Types of Chores

We consider the problem of fair allocation of indivisible chores under additive valuations. We assume that the chores are divided into two types and under this scenario, we present several results. Our first result is a new characterization of Pareto optimal allocations in our setting, and a polynomial-time algorithm to compute an envy-free up to one item (EF1) and Pareto optimal allocation. We then turn to the question of whether we can achieve a stronger fairness concept called envy-free up any item (EFX). We present a polynomial-time algorithm that returns an EFX allocation. Finally, we show that for our setting, it can be checked in polynomial time whether an envy-free allocation exists or not

Manifold learning and convergence of Laplacian eigenmaps

In this thesis, we investigate the problem of obtaining meaningful low dimensional representation of high dimensional data, often referred to as manifold learning. We examine classical methods for manifold learning such as PCA and cMDS as well as some modern techniques of manifold learning namely Isomap, Locally Linear Embedding and Laplacian Eigenmaps. The algorithms for these individual methods are presented in mathematically consistent, concise and easy to understand fashion so that people with no computer science background can use the methods presented in their own research. Motivations and justifications of these manifold learning methods are provided. Finally we prove the convergence of Laplacian Eigenmaps method in a self contained and compact fashion following the work of Mikhail Belkin and Partha Niyogi.Dans cette thèse, on aborde le problème de l'apprentissage de variété afin de réduire la dimensionalité d'ensembles de données pour obtenir une représentation significative de basse dimension. On examine les méthodes classiques d'apprentissage de variété telles que l'analyse en composantes principales (PCA) et le positionnement multidimensionnel classique (cMDS). On présente, de plus, trois techniques modernes pour résoudre ce problème: les méthodes «Isomap», «Locally Linear Embedding» et «Laplacian Eigenmaps». On expose les détails mathématiques de ces algorithmes en termes concis et simples tout en préservant leur cohérence mathématique. Ces développements aideront sans doute d'autres chercheurs sans expérience en informatique à utiliser ces méthodes. Par la suite, on justifie l'applicabilité de ces techniques pour résoudre le problème de réduction dimensionelle. Finalement, on montre la convergence de la méthode «Laplacian Eigenmaps» d'une façon compacte en suivant l'approche de Mikhail Belkin et de Partha Nyogi

Random Rank: The One and Only Strategyproof and Proportionally Fair Randomized Facility Location Mechanism

Proportionality is an attractive fairness concept that has been applied to a range of problems including the facility location problem, a classic problem in social choice. In our work, we propose a concept called Strong Proportionality, which ensures that when there are two groups of agents at different locations, both groups incur the same total cost. We show that although Strong Proportionality is a well-motivated and basic axiom, there is no deterministic strategyproof mechanism satisfying the property. We then identify a randomized mechanism called Random Rank (which uniformly selects a number $k$ between $1$ to $n$ and locates the facility at the $k$'th highest agent location) which satisfies Strong Proportionality in expectation. Our main theorem characterizes Random Rank as the unique mechanism that achieves universal truthfulness, universal anonymity, and Strong Proportionality in expectation among all randomized mechanisms. Finally, we show via the AverageOrRandomRank mechanism that even stronger ex-post fairness guarantees can be achieved by weakening universal truthfulness to strategyproofness in expectation

Coordinating Monetary Contributions in Participatory Budgeting

We formalize a framework for coordinating the funding of projects and sharing the costs among agents with quasi-linear utility functions and individual budgets. Our model contains the classical discrete participatory budgeting model as a special case, while capturing other well-motivated problems. We propose several important axioms and objectives and study how well they can be simultaneously satisfied. One of our main results is that whereas welfare maximization admits an FPTAS, welfare maximization subject to a well-motivated and very weak participation requirement leads to a strong inapproximability result. We show that this result is bypassed if we consider some natural restricted valuations or when we take an average-case heuristic approach

Best-of-Both-Worlds Fairness in Committee Voting

The best-of-both-worlds paradigm advocates an approach that achieves desirable properties both ex-ante and ex-post. We launch a best-of-both-worlds fairness perspective for the important social choice setting of approval-based committee voting. To this end, we initiate work on ex-ante proportional representation properties in this domain and formalize a hierarchy of properties including Individual Fair Share (IFS), Unanimous Fair Share (UFS), Group Fair Share (GFS), and their stronger variants. We establish their compatibility with well-studied ex-post concepts such as extended justified representation (EJR) and fully justified representation (FJR). Our first main result is a polynomial-time algorithm that simultaneously satisfies ex-post EJR, ex-ante GFS and ex-ante Strong UFS. Subsequently, we strengthen our ex-post guarantee to FJR and present an algorithm that outputs a lottery which is ex-post FJR and ex-ante Strong UFS, but does not run in polynomial time.Comment: This version contains a new section on Fully Justified Representation (FJR) as well as additional results on stronger ex-ante fair share notion