La obra de Leslie Valiant

Este año Leslie VALIANT cumple 65 años y nosotros queremos celebrar este importante aniversario con este trabajo en el que se analiza su obra. Centramos nuestra atención en aquellos de sus trabajos en los que una clara influencia de Volker STRASSEN puede ser detectada. Es patente la influencia de Strassen en la obra de Valiant, pero esto no quiere decir que el trabajo de Valiant, complejo y multifacético, sea un simple corolario a la obra del primero. Para citar este artículo: J. Andrés Montoya, The work of Leslie Valiant: alle die Strassen führen nach Strassen, Rev. Integr. Temas Mat. 32 (2014), no. 2, 153-168

Алгоритм комбинаторной оптимизации структуры логических связей децентрализованной системы

Предложена формальная математическая модель задачи оптимизации структуры логических связей децентрализованной системы. Показано, что оптимизация позволяет получить структуры, эффективные с точки зрения затрат на реализацию логических операций в заданном сетевом окружении. Рассмотрены типичные ограничения, позволяющие гарантировать масштабируемость и высокие показатели устойчивости системы при неоднородной нагрузке. Проведена оценка сложности задачи и предложен генетический алгоритм нахождения приближенного решения.Запропоновано формальну математичну модель задачі оптимізації структури логічних зв’язків децентралізованої системи. Показано, що оптимізація дозволяє отримати структури, які будуть ефективними з точку зору витрат на реалізацію логічних операцій у заданому мережевому середовищі. Розглянуто типові обмеження, які дозволяють гарантувати можливість масштабування та забезпечення високих показників стабільності роботи системи при нерівномірному завантаженні. Проведено оцінку складності задачі та запропоновано генетичний алгоритм знаходження наближеного розв’язку.A formal mathematical model of a problem for optimization of logical interlink structure of decentralized system is proposed. It is shown that optimization yields structures which are effective in regard to resources spent on logical operations in a predetermined network environment. Typical restrictions allowing to assure scalability and high robustness of the system under inhomogeneous load are discussed. Es92 timation of problem complexity is performed, and genetics algorithm of finding on approximate solution is proposed

Simple Constructions of Unique Neighbor Expanders from Error-correcting Codes

In this note, we give very simple constructions of unique neighbor expander graphs starting from spectral or combinatorial expander graphs of mild expansion. These constructions and their analysis are simple variants of the constructions of LDPC error-correcting codes from expanders, given by Sipser-Spielman\cite{SS96} (and Tanner\cite{Tanner81}), and their analysis. We also show how to obtain expanders with many unique neighbors using similar ideas. There were many exciting results on this topic recently, starting with Asherov-Dinur\cite{AD23} and Hsieh-McKenzie-Mohanty-Paredes\cite{HMMP23}, who gave a similar construction of unique neighbor expander graphs, but using more sophisticated ingredients (such as almost-Ramanujan graphs) and a more involved analysis. Subsequent beautiful works of Cohen-Roth-TaShma\cite{CRT23} and Golowich\cite{Golowich23} gave even stronger objects (lossless expanders), but also using sophisticated ingredients. The main contribution of this work is that we get much more elementary constructions of unique neighbor expanders and with a simpler analysis

Jointly Optimal Routing and Caching for Arbitrary Network Topologies

We study a problem of fundamental importance to ICNs, namely, minimizing routing costs by jointly optimizing caching and routing decisions over an arbitrary network topology. We consider both source routing and hop-by-hop routing settings. The respective offline problems are NP-hard. Nevertheless, we show that there exist polynomial time approximation algorithms producing solutions within a constant approximation from the optimal. We also produce distributed, adaptive algorithms with the same approximation guarantees. We simulate our adaptive algorithms over a broad array of different topologies. Our algorithms reduce routing costs by several orders of magnitude compared to prior art, including algorithms optimizing caching under fixed routing.Comment: This is the extended version of the paper "Jointly Optimal Routing and Caching for Arbitrary Network Topologies", appearing in the 4th ACM Conference on Information-Centric Networking (ICN 2017), Berlin, Sep. 26-28, 201

On the second eigenvalue and linear expansion of regular graphs.

Abstract. The spectral method is the best currently known technique to prove lower bounds on expansion. Ramanujan graphs, which have asymptotically optimal second eigenvalue, are the best-known explicit expanders. The spectral method yielded a lower bound of k\4 on the expansion of Iinear-sized subsets of k-regular Ramanujan graphs. We improve the lower bound ontheexpansion of Ramanujan graphs to approximately k/2, Moreover. we construct afamilyof k-regular graphs with asymptotically optimal second eigenvalue and linear expansion equal to k/2. This shows that k/2 is the best bound one can obtain using the second eigenwdue method. We also show an upper bound of roughly 1 +~on the average degree of linear-sized induced subgraphs of Ramanujan graphs. This compares positively with the classical bound 2~. As a byproduct, we obtain improved results on random walks on expanders and construct selection networks (respectively, extrovert graphs) of smaller size (respectively, degree) than was previously known