The Power of Unentanglement

The class QMA(k). introduced by Kobayashi et al., consists of all languages that can be verified using k unentangled quantum proofs. Many of the simplest questions about this class have remained embarrassingly open: for example, can we give any evidence that k quantum proofs are more powerful than one? Does QMA(k) = QMA(2) for k ≥ 2? Can QMA(k) protocols be amplified to exponentially small error? In this paper, we make progress on all of the above questions. * We give a protocol by which a verifier can be convinced that a 3SAT formula of size m is satisfiable, with constant soundness, given Õ (√m) unentangled quantum witnesses with O(log m) qubits each. Our protocol relies on the existence of very short PCPs. * We show that assuming a weak version of the Additivity Conjecture from quantum information theory, any QMA(2) protocol can be amplified to exponentially small error, and QMA(k) = QMA(2) for all k ≥ 2. * We prove the nonexistence of "perfect disentanglers" for simulating multiple Merlins with one

Discrete-Query Quantum Algorithm for NAND Trees

This is a comment on the article “A Quantum Algorithm for the Hamiltonian NAND Tree” by Edward Farhi, Jeffrey Goldstone, and Sam Gutmann, Theory of Computing 4 (2008) 169--190. That paper gave a quantum algorithm for evaluating NAND trees with running time O(√N) in the Hamiltonian query model. In this note, we point out that their algorithm can be converted into an algorithm using N^[1/2 + o(1)] queries in the conventional (discrete) quantum query model

On the nonlinearity of monotone Boolean functions

We first prove the truthfulness of a conjecture on the nonlinearity of monotone Boolean functions in even dimension, proposed in the recent paper ``Cryptographic properties of monotone Boolean functions , by D. Joyner, P. Stanica, D. Tang and the author, to appear in the Journal of Mathematical Cryptology. We prove then an upper bound on such nonlinearity, which is asymptotically much stronger than the conjectured upper bound and than the upper bound proved for odd dimension in this same paper. This bound shows a deep weakness of monotone Boolean functions; they are too closely approximated by affine functions for being usable as nonlinear components in cryptographic applications. We deduce a necessary criterion to be satisfied by a Boolean (resp. vectorial) function for being nonlinear

Otimização de sistemas intervalares não lineares acíclicos

Resumo: Intervalos permitem uma representação aproximada de números reais, com a qual podemos modelar matematicamente problemas do mundo real de uma forma menos restritiva que a modelagem sobre restrições reais. Assim, podemos definir problemas intervalares de decisão e otimização que são relaxamentos da Programação Não Linear usual. Recentemente, técnicas utilizadas em algoritmos para o problema da Satisfatibilidade Booleana foram aplicadas na solução de problemas intervalares de decisão, utilizando a álgebra intervalar para refinar intervalos e obter soluções que satisfaçam um conjunto de restrições sob uma precisão preestabelecida. Embora essa abordagem não resolva problemas de otimização, ela apresenta um método para extrair uma solução real de uma solução intervalar, se o problema apresentar determinadas características. Neste trabalho, estendemos esse método, definindo uma classe de problemas para os quais é possível a extração de uma solução real mesmo sem a garantia de todas as condições exigidas pelos resolvedores anteriores. Além disso, mostramos que o método estendido pode ser utilizado para resolver algumas classes de problemas de otimização

Multi-level facility location problems

We study of a class of discrete facility location problems, called multi-level facility location problems, that has received major attention in the last decade. These problems arise in several applications such as in production-distribution systems, telecommunication networks, freight transportation, and health care, among others. Moreover, they generalize well-known facility location problems which have been shown to lie at the heart of operations research due to their applicability and mathematical structure. We first present a comprehensive review of multi-level facility location problems where we formally define and categorize them based on the types of decisions involved. We also point out some gaps in the literature and present overviews of related applications, models and algorithms. We then concentrate our efforts on the development of solution methods for a general multi-level uncapacitated facility location problem. In particular, based on an alternative combinatorial representation of the problem whose objective function satisfies the submodularity property, we propose a mixed integer linear programming formulation. Using that same representation, we present approximation algorithms with constant performance guarantees for the problem and analyze some special cases where these worst-case bounds are sharper. Finally, we develop an exact algorithm based on Benders decomposition for a slightly more general problem where the activation of links between level of facilities is also considered part of the decision process. Extensive computational experiments are presented to assess the performance of the various models and algorithms studied. We show that the multi-level extension of some fundamental problems in operations research maintain certain structure that allows us to develop more efficient algorithms in practice