Stochastic Budget Optimization in Internet Advertising

Internet advertising is a sophisticated game in which the many advertisers "play" to optimize their return on investment. There are many "targets" for the advertisements, and each "target" has a collection of games with a potentially different set of players involved. In this paper, we study the problem of how advertisers allocate their budget across these "targets". In particular, we focus on formulating their best response strategy as an optimization problem. Advertisers have a set of keywords ("targets") and some stochastic information about the future, namely a probability distribution over scenarios of cost vs click combinations. This summarizes the potential states of the world assuming that the strategies of other players are fixed. Then, the best response can be abstracted as stochastic budget optimization problems to figure out how to spread a given budget across these keywords to maximize the expected number of clicks. We present the first known non-trivial poly-logarithmic approximation for these problems as well as the first known hardness results of getting better than logarithmic approximation ratios in the various parameters involved. We also identify several special cases of these problems of practical interest, such as with fixed number of scenarios or with polynomial-sized parameters related to cost, which are solvable either in polynomial time or with improved approximation ratios. Stochastic budget optimization with scenarios has sophisticated technical structure. Our approximation and hardness results come from relating these problems to a special type of (0/1, bipartite) quadratic programs inherent in them. Our research answers some open problems raised by the authors in (Stochastic Models for Budget Optimization in Search-Based Advertising, Algorithmica, 58 (4), 1022-1044, 2010).Comment: FINAL versio

Sticky Brownian Rounding and its Applications to Constraint Satisfaction Problems

Semidefinite programming is a powerful tool in the design and analysis of approximation algorithms for combinatorial optimization problems. In particular, the random hyperplane rounding method of Goemans and Williamson has been extensively studied for more than two decades, resulting in various extensions to the original technique and beautiful algorithms for a wide range of applications. Despite the fact that this approach yields tight approximation guarantees for some problems, e.g., Max-Cut, for many others, e.g., Max-SAT and Max-DiCut, the tight approximation ratio is still unknown. One of the main reasons for this is the fact that very few techniques for rounding semidefinite relaxations are known. In this work, we present a new general and simple method for rounding semi-definite programs, based on Brownian motion. Our approach is inspired by recent results in algorithmic discrepancy theory. We develop and present tools for analyzing our new rounding algorithms, utilizing mathematical machinery from the theory of Brownian motion, complex analysis, and partial differential equations. Focusing on constraint satisfaction problems, we apply our method to several classical problems, including Max-Cut, Max-2SAT, and MaxDiCut, and derive new algorithms that are competitive with the best known results. To illustrate the versatility and general applicability of our approach, we give new approximation algorithms for the Max-Cut problem with side constraints that crucially utilizes measure concentration results for the Sticky Brownian Motion, a feature missing from hyperplane rounding and its generalization

Sticky Brownian rounding and its applications to constraint satisfaction problems

Semidefinite programming is a powerful tool in the design and analysis of approximation algorithms for combinatorial optimization problems. In particular, the random hyperplane rounding method of Goemans and Williamson [23] has been extensively studied for more than two decades, resulting in various extensions to the original technique and beautiful algorithms for a wide range of applications. Despite the fact that this approach yields tight approximation guarantees for some problems, e.g., Max-Cut, for many others, e.g., Max-SAT and Max-DiCut, the tight approximation ratio is still unknown. One of the main reasons for this is the fact that very few techniques for rounding semidefinite relaxations are known. In this work, we present a new general and simple method for rounding semi-definite programs, based on Brownian motion. Our approach is inspired by recent results in algorithmic discrepancy theory. We develop and present tools for analyzing our new rounding algorithms, utilizing mathematical machinery from the theory of Brownian motion, complex analysis, and partial differential equations. Focusing on constraint satisfaction problems, we apply our method to several classical problems, including Max-Cut, Max-2SAT, and Max-DiCut, and derive new algorithms that are competitive with the best known results. To illustrate the versatility and general applicability of our approach, we give new approximation algorithms for the Max-Cut problem with side constraints that crucially utilizes measure concentration results for the Sticky Brownian Motion, a feature missing from hyperplane rounding and its generalizations

Estimation et réduction du coût d’algorithmes quantiques

Plusieurs algorithmes ont été marquants dans le développement de l’informatique quantique : on peut penser notamment à l’algorithme de Shor ou à l’algorithme de Grover. Bien que pour ce dernier, par exemple, nous ayons une accélération quadratique prouvée, il n’est pas clair que ce sera suffisant pour offrir un avantage pratique par rapport aux algorithmes classiques. D’abord, celui-ci est formulé en termes d’oracle, une boîte noire qui cache une sous-routine non comprise dans le calcul du coût. Ensuite, lorsque l’on prend en considération le surcoût engendré notamment par la correction d’erreur quantique, il est possible de perdre l’accélération promise. Mais également, la durée d’une porte logique quantique est considérablement plus long que son homologue classique. Quel est le coût réel d’un algorithme quantique lorsque l’on prend en considération toutes les sous-routines ? À quand un ordinateur quantique utile qui surpassera les performances d’un superordinateur ? Dans cette thèse, nous présenterons trois projets visant tous à estimer et réduire le coût d’algorithmes ou sous-routines quantiques. Les algorithmes abordés sont issus d’une discrétisation de l’évolution adiabatique. Le premier se concentre sur un algorithme de préparation d’état d’un système à N corps par une évolution adiabatique via l’effet Zénon. Le second porte sur une version quantique des algorithmes de marches aléatoires et de recuit simulé pouvant, par exemple, préparer un état stationnaire. Le dernier décrit un nouvel algorithme : une évolution adiabatique basée sur la réflexion. Celui-ci permet, entre autres, de résoudre des problèmes MAX-kSAT, une classe de problèmes NP-difficile. Avec ces projets, nous voulons, d’une part, proposer des algorithmes efficaces ainsi que leur implémentation de A à Z et, d’autre part, estimer les caractéristiques nécessaires à un ordinateur quantique utile (p. ex. taille, résistance au bruit, vitesse d’opération). Les résultats présentés démontrent le coût élevé associé aux algorithmes tolérants aux fautes. Bien qu’on s’attende à avoir une accélération par rapport au classique, lorsque l’on prend en considération le nombre de qubits physiques, le nombre d’opérations physiques et la durée de chacune de ces opérations, en incluant la correction d’erreur notamment, la taille des instances offrant un avantage réel est loin d’être atteignable pour les processeurs quantiques à court terme. Toutefois, en combinant des méthodes astucieuses et au moyen de différents procédés d’optimisation, il est possible de réduire considérablement le coût des algorithmes quantiques, et donc de réduire le délai pour atteindre la suprématie quantique