Differential approximation results for the Steiner tree problem

International audienceWe study the approximability of three versions of the Steiner tree problem. For the first one where the input graph is only supposed connected, we show that it is not approximable within better than |V \setminus N|^{-r} for any r in [0,1], where V and N are the vertex-set of the input graph and the set of terminal vertices, respectively. For the second of the Steiner tree versions considered, the one where the input graph is supposed complete and the edge distances are arbitrary, we prove that it can be differentially approximated within 1/2. For the third one defined on complete graphs with edge distances 1 or 2, we show that it is differentially approximable within 0.82. Also, we extend the result of (M. Bern and P. Plassmann, The Steiner problem with edge lengths 1 and 2, Inform. Process. Lett. 32, 1989), we show that the Steiner tree problem with edge lengths 1 and 2 is MaxSNP-complete even in the case where |V| 0. This allows us to finally show that Steiner tree problem with edge lengths 1 and 2 cannot by approximated by polynomial time differential approximation schemata

Maximizing the number of unused bins

We analyze the approximation behavior of some of the best-known polynomial-time approximation algorithms for bin-packing under an approximation criterion, called differential ratio, informally the ratio (n - apx(I))/(n - opt(I)), where n is the size of the input list, apx(I) is the size of the solution provided by an approximation algorithm and opt(I) is the size of the optimal one. This measure has originally been introduced by Ausiello, DÁtri and Protasi and more recently revisited, in a more systematic way, by the first and the third authors of the present paper. Under the differential ratio, bin-packing has a natural formulation as the problem of maximizing the number of unused bins. We first show that two basic fit bin-packing algorithms, the first-fit and the best-fit, admit differential approximation ratios 1/2. Next, we show that slightly improved versions of them achieve ratios 2/3. Refining our analysis we show that the famous first-fit-decreasing and best-fit decreasing algorithms achieve differential approximation ratio 3/4. Finally, we show that first-fit-decreasing achieves asymptotic differential approximation ratio 7/9

Variational autoencoder with weighted samples for high-dimensional non-parametric adaptive importance sampling

Probability density function estimation with weighted samples is the main foundation of all adaptive importance sampling algorithms. Classically, a target distribution is approximated either by a non-parametric model or within a parametric family. However, these models suffer from the curse of dimensionality or from their lack of flexibility. In this contribution, we suggest to use as the approximating model a distribution parameterised by a variational autoencoder. We extend the existing framework to the case of weighted samples by introducing a new objective function. The flexibility of the obtained family of distributions makes it as expressive as a non-parametric model, and despite the very high number of parameters to estimate, this family is much more efficient in high dimension than the classical Gaussian or Gaussian mixture families. Moreover, in order to add flexibility to the model and to be able to learn multimodal distributions, we consider a learnable prior distribution for the variational autoencoder latent variables. We also introduce a new pre-training procedure for the variational autoencoder to find good starting weights of the neural networks to prevent as much as possible the posterior collapse phenomenon to happen. At last, we explicit how the resulting distribution can be combined with importance sampling, and we exploit the proposed procedure in existing adaptive importance sampling algorithms to draw points from a target distribution and to estimate a rare event probability in high dimension on two multimodal problems.Comment: 20 pages, 5 figure

A Mechanism Design Approach to Climate Agreements *

Abstract: We analyze environmental agreements in contexts with voluntary participation by sovereign countries, incentives problems and possible limits on enforcement and commitment. Taking a mechanism design perspective, we study how countries may agree on effort targets and compensations to take into account multilateral externalities. The optimal mechanism unveils an important trade-off between solving a free riding problem in effort provision at the intensive margin for participating countries and another free riding problem at the extensive margin to ensure that all countries participate. This mechanism can easily be approximated by means of simple menus with attractive implementation and robustness properties. However, limits on enforcement and commitment might hinder its performances making the "business as usual" scenario more attractive. Keywords: public goods, incentive constraints, mechanism design, global warming. JEL Codes: Q54, D82, H23. * We thank workshop participants at Paris School of Economics, the Paris Environmental and Energy Economics Seminar, the CIRPEE (UQAM/HEC Montréal, Laval University)'s annual conference, CREST-LEI Paris, Frankfort, GREQAM-Marseille, the Congress of the Canadian Economic Association in Calgary, the Workshop on the Economics of Climate Change CDC Paris, Collège de France Paris, and ETH Zürich, but also Antoine Bommier Jean-Marc Bourgeon, Renaud Bourlès, Gabrielle Demange, Pierre Fleckinger, Hans Gersbach, Bard Harstad, Jérôme Pouyet, Jean-Charles Rochet, François Salanié and Alain Trannoy for helpful comments on an earlier draft. We also thank Daniel Coublucq and Perrin Lefèbvre for outstanding research assistance. All errors are ours

Mélangeur double-grille et amplificateur filtrant MMIC à 5.8 GHz

International audienceCet article présente la réalisation, en technologie MMIC sur substrat d'AsGa, de deux circuits , un mélangeur double grille et un amplificateur filtrant à 5. 8 GHz (bande ISM). Pour chaque circuit, un compromis a été recherché entre simplicité et performances dans le but d'obtenir des dispositifs compacts, faciles à intégrer en MMIC et faciles à tester, avec des caractéristiques suffisantes pour une intégration dans une chaîne de réception RF

Des équations à diffusion rapide aux inégalités de Sobolev sur les modèles de la géométrie

TOULOUSE3-BU Sciences (315552104) / SudocSudocFranceF

Une introduction aux problèmes combinatoires inverses

This chapter is an introduction to inverse combinatorial optimization. Given an instance of a problem and a fixed feasible solution, the aim is to optimally (w.r.t. a given norm) modify the cost system of the instance (or its structure) to make this solution optimal. We first give the main known examples of polynomial inverse problems (essentially based on linear programming), then we give some complexity results in this area.no

Oxygen and sulphurisotopic compositions of authigenic gypsum from methane seeps of the south-west african continental margin

International audienc

Improved Gagliardo-Nirenberg-Sobolev inequalities on manifolds with positive curvature

International audienc