arXiv subjects
Publications and source records attributed to Guillaume Chèze.
The problem of the Nile has been stated by R. Fisher in 1936. This problem concerns the fair division of an agricultural land and is the eponymous problem of a general class of problems on measures and similar regions. In 1938, W. Feller has shown that the general problem has no solution. However, we will show that under a natural hypothesis, the original problem of the Nile is solvable with connected plots.
This article deals with the cake cutting problem. In this setting, there exists two notions of fair division: proportional division (when there are n players, each player thinks to get at least 1/n of the cake) and envy-free division (each player wants to keep his or her share because he or she does not envy the portion given to another player). Some results are valid for proportional division but not for envy-free division. Here, we introduce and study a scale between the proportional division and the envy-free division. The goal is to understand where is the gap between statements about proportional division and envy-free division. This scale comes from the notion introduced in this article: k-proportionality. When k = n this notion corresponds to the proportional division and when k = 2 it corresponds to envy-free division. With k-proportionality we can understand where some difficulties in fair division lie. First, we show that there are situations in which there is no k-proportional and equitable division of a pie with connected pieces when k $\le$ n -1. This result was known only for envy-free division, ie k = 2. Next, we prove that there are situations in which there is no Pareto-optimal k-proportional division of a cake with connected pieces when k $\le$ n -1. This result was known only for k = 2. These theorems say that we can get an impossibility result even if we do not consider an envy-free division but a weaker notion. Finally, k-proportionality allows to give a generalization with a uniform statement of theorems about strong envy-free and strong proportional divisions.
In this article we propose a probabilistic framework in order to study the fair division of a divisible good, e.g., a cake, between n players. Our framework follows the same idea than the ''Full independence model'' used in the study of fair division of indivisible goods. We show that, in this framework, there exists an envy-free division algorithm satisfying the following probability estimate:$$\mathbb{P}\big( C(μ_1, \ldots,μ_n) \geq n^{7+b}\big) = \mathcal{O}\Big(n^{-\frac{b-1}{3}+1+o(1)}\Big),$$where $μ_1,\ldots, μ_n$ correspond to the preferences of the $n$ players,$C(μ_1, \ldots,μ_n)$ is the number of queries used by the algorithm and $b>4$. In particular, this gives$$\lim_{n \rightarrow + \infty}\mathbb{P}\big( C(μ_1, \ldots,μ_n) \geq n^{12}\big) = 0.$$ It must be noticed that nowadays few things are known about the complexity of envy-free division algorithms. Indeed, Procaccia has given a lower bound in $Ω(n^2)$ and Aziz and Mackenzie have given an upper bound in $n^{n^{n^{n^{n^{n}}}}}$. As our estimate means that we have $C(μ_1, \ldots, μ_n)<n^{12}$ with a high probability, this gives a new insight on the complexity of envy-free cake cutting algorithms. Our result follows from a study of Webb's algorithm and a theorem of Tao and Vu about the smallest singular value of a random matrix.
In this article we study a cake cutting problem. More precisely, we study symmetric fair division algorithms, that is to say we study algorithms where the order of the players do not influence the value obtained by each player. In the first part of the article, we give a symmetric and envy-free fair division algorithm. More precisely, we show how to get a symmetric and envy-free fair division algorithm from an envy-free division algorithm. In the second part, we give a proportional and symmetric fair division algorithm with a complexity in O(n 3) in the Robertson-Webb model of complexity. This algorithm is based on Kuhn's algorithm. Furthermore, our study has led us to introduce a new notion: aristotelian fair division. This notion is an interpretation of Aristotle's principle: give equal shares to equal people. We conclude this article with a discussion and some questions about the Robertson-Webb model of computation.
In this article we suggest a model of computation for the cake cutting problem. In this model the mediator can ask the same queries as in the Robertson-Webb model but he or she can only perform algebraic operations as in the Blum-Shub-Smale model. All existing algorithms described in the Robertson-Webb model can be described in this new model.We show that in this model there exist explicit couples of measures for which no algorithm outputs an equitable fair division with connected parts.We also show that there exist explicit set of measures for which no algorithm in this model outputs a fair division which maximizes the utilitarian social welfare function.The main tool of our approach is Galois theory.
In this article we show how to generalize to the Darbouxian, Liouvillian and Riccati case the extactic curve introduced by J. Pereira. With this approach, we get new algorithms for computing, if it exists, a rational, Darbouxian, Liouvillian or Riccati first integral with bounded degree of a polynomial planar vector field. We give probabilistic and deterministic algorithms. The arithmetic complexity of our probabilistic algorithm is in $\tilde{\mathcal{O}}(N^{ω+1})$, where $N$ is the bound on the degree of a representation of the first integral and $ω\in [2;3]$ is the exponent of linear algebra. This result improves previous algorithms. Our algorithms have been implemented in Maple and are available on authors' websites. In the last section, we give some examples showing the efficiency of these algorithms.
In this note we study a problem of fair division in the absence of full information. We give an algorithm which solves the following problem: n $\ge$ 2 persons want to cut a cake into n shares so that each person will get at least 1/n of the cake for his or her own measure, furthermore the preferences of one person are secret. How can we construct such shares? Our algorithm is a slight modification of the Even-Paz algorithm and allows to give a connected part to each agent. Moreover, the number of cuts used during the algorithm is optimal: O (n log(n)) .
In this article we study the problem of fair division. In particular we study a notion introduced by J. Barbanel that generalizes super envy-free fair division. We give a new proof of his result. Our approach allows us to give an explicit bound for this kind of fair division. Furthermore, we also give a theoretical answer to an open problem posed by Barbanel in 1996. Roughly speaking, this question is: how can we decide if there exists a fair division satisfying some inequalities constraints? Furthermore, when all the measures are given with piecewise constant density functions then we show how to construct effectively such a fair division.
In this note we study how to share a good between n players in a simple and equitable way. We give a short proof for the existence of such fair divisions.
Consider the framework of topological aggregation introduced by Chichilnisky (1980). We prove that in this framework the Twin Paradox and the No Show Paradox cannot be avoided. Anonymity and unanimity are not needed to obtain these results.
In this article we bound the number of remarkable values of a polynomial vector field. The proof is short and based on Jouanolou's theorem about rational first integrals of planar polynomial derivations. Our bound is given in term of the size of a Newton polygon associated to the vector field. We prove that this bound is almost reached.
We present fast algorithms for computing rational first integrals with bounded degree of a planar polynomial vector field. Our approach is inspired by an idea of Ferragut and Giacomini. We improve upon their work by proving that rational first integrals can be computed via systems of linear equations instead of systems of quadratic equations. This leads to a probabilistic algorithm with arithmetic complexity $\bigOsoft(N^{2 ω})$ and to a deterministic algorithm solving the problem in $\bigOsoft(d^2N^{2 ω+1})$ arithmetic operations, where $N$ denotes the given bound for the degree of the rational first integral, and where $d \leq N$ is the degree of the vector field, and $ω$ the exponent of linear algebra. We also provide a fast heuristic variant which computes a rational first integral, or fails, in $\bigOsoft(N^{ω+2})$ arithmetic operations. By comparison, the best previous algorithm uses at least $d^{ω+1}\, N^{4ω+4}$ arithmetic operations. We then show how to apply a similar method to the computation of Darboux polynomials. The algorithms are implemented in a Maple package which is available to interested readers with examples showing its efficiency.
Darboux's theorem and Jouanolou's theorem deal with the existence of first integrals and rational first integrals of a polynomial vector field. These results are given in terms of the degree of the polynomial vector field. Here we show that we can get the same kind of results if we consider the size of a Newton polytope associated to the vector field. Furthermore, we show that in this context the bound is optimal.
In this paper, the problem of bounding the number of reducible curves in a pencil of algebraic plane curves is addressed. Unlike most of the previous related works, each reducible curve of the pencil is here counted with its appropriate multiplicity. It is proved that this number of reducible curves, counted with multiplicity, is bounded by d^2-1 where d is the degree of the pencil. Then, a sharper bound is given by taking into account the Newton's polygon of the pencil.
In this paper we show how we can compute in a deterministic way the decomposition of a multivariate rational function with a recombination strategy. The key point of our recombination strategy is the used of Darboux polynomials. We study the complexity of this strategy and we show that this method improves the previous ones. In appendix, we explain how the strategy proposed recently by J. Berthomieu and G. Lecerf for the sparse factorization can be used in the decomposition setting. Then we deduce a decomposition algorithm in the sparse bivariate case and we give its complexity
In this paper we study planar polynomial differential systems of this form: dX/dt=A(X, Y), dY/dt= B(X, Y), where A,B belongs to Z[X, Y], degA \leq d, degB \leq d, and the height of A and B is smaller than H. A lot of properties of planar polynomial differential systems are related to irreducible Darboux polynomials of the corresponding derivation: D =A(X, Y)dX + B(X, Y)dY . Darboux polynomials are usually computed with the method of undetermined coefficients. With this method we have to solve a polynomial system. We show that this approach can give rise to the computation of an exponential number of reducible Darboux polynomials. Here we show that the Lagutinskii-Pereira's algorithm computes irreducible Darboux polynomials with degree smaller than N, with a polynomial number, relatively to d, log(H) and N, binary operations. We also give a polynomial-time method to compute, if it exists, a rational first integral with bounded degree.
The extended Lüroth's Theorem says that if the transcendence degree of $\KK(\mathsf{f}_1,\dots,\mathsf{f}_m)/\KK$ is 1 then there exists $f \in \KK(\underline{X})$ such that $\KK(\mathsf{f}_1,\dots,\mathsf{f}_m)$ is equal to $\KK(f)$. In this paper we show how to compute $f$ with a probabilistic algorithm. We also describe a probabilistic and a deterministic algorithm for the decomposition of multivariate rational functions. The probabilistic algorithms proposed in this paper are softly optimal when $n$ is fixed and $d$ tends to infinity. We also give an indecomposability test based on gcd computations and Newton's polytope. In the last section, we show that we get a polynomial time algorithm, with a minor modification in the exponential time decomposition algorithm proposed by Gutierez-Rubio-Sevilla in 2001.
Let $f(X,Y) \in \ZZ[X,Y]$ be an irreducible polynomial over $\QQ$. We give a Las Vegas absolute irreducibility test based on a property of the Newton polytope of $f$, or more precisely, of $f$ modulo some prime integer $p$. The same idea of choosing a $p$ satisfying some prescribed properties together with $LLL$ is used to provide a new strategy for absolute factorization of $f(X,Y)$. We present our approach in the bivariate case but the techniques extend to the multivariate case. Maple computations show that it is efficient and promising as we are able to factorize some polynomials of degree up to 400.