For completeness, sublogarithmic space is no space

It is shown that for any class C closed under linear-time reductions, the complete sets for C under sublogarithmic reductions are also complete under 2DFA reductions, and thus are isomorphic under first-order reductions

Complexity Theory

Coordinated Science Laboratory was formerly known as Control Systems LaboratoryNational Science Foundation / CCR-9315696Originally published July 1995; revision published November 1995 with the same tech report number

Complejidad algorítmica : Cuestiones y aplicaciones Notables

Tesis Univ. Complutense de Madrid. Fac. CC. Mat., Dir. por Francisco Cano Sevilla, leída en Madrid, el 13 de julio de 1982.Depto. de Estadística e Investigación OperativaFac. de Ciencias MatemáticasTRUEProQuestpu

Approximation Complexity of Optimization Problems : Structural Foundations and Steiner Tree Problems

In this thesis we study the approximation complexity of the Steiner Tree Problem and related problems as well as foundations in structural complexity theory. The Steiner Tree Problem is one of the most fundamental problems in combinatorial optimization. It asks for a shortest connection of a given set of points in an edge-weighted graph. This problem and its numerous variants have applications ranging from electrical engineering, VLSI design and transportation networks to internet routing. It is closely connected to the famous Traveling Salesman Problem and serves as a benchmark problem for approximation algorithms. We give a survey on the Steiner tree Problem, obtaining lower bounds for approximability of the (1,2)-Steiner Tree Problem by combining hardness results of Berman and Karpinski with reduction methods of Bern and Plassmann. We present approximation algorithms for the Steiner Forest Problem in graphs and bounded hypergraphs, the Prize Collecting Steiner Tree Problem and related problems where prizes are given for pairs of terminals. These results are based on the Primal-Dual method and the Local Ratio framework of Bar-Yehuda. We study the Steiner Network Problem and obtain combinatorial approximation algorithms with reasonable running time for two special cases, namely the Uniform Uncapacitated Case and the Prize Collecting Uniform Uncapacitated Case. For the general case, Jain's algorithms obtains an approximation ratio of 2, based on the Ellipsoid Method. We obtain polynomial time approximation schemes for the Dense Prize Collecting Steiner Tree Problem, Dense k-Steiner Problem and the Dense Class Steiner Tree Problem based on the methods of Karpinski and Zelikovsky for approximating the Dense Steiner Tree Problem. Motivated by the question which parameters make the Steiner Tree problem hard to solve, we make an excurs into Fixed Parameter Complexity, focussing on structural aspects of the W-Hierarchy. We prove a Speedup Theorem for the classes FPT and SP and versions if Levin's Lower Bound Theorem for the class SP as well as for Randomized Space Complexity. Starting from the approximation schemes for the dense Steiner Tree problems, we deal with the efficiency of polynomial time approximation schemes in general. We separate the class EPTAS from PTAS under some reasonable complexity theoretic assumption. The same separation was achieved by Cesaty and Trevisan under some assumtion from Fixed Parameter Complexity. We construct an oracle under which our assumtion holds but that of Cesati and Trevisan does not, which implies that using relativizing proof techniques one cannot show that our assumption implies theirs

Introducción a la Teoría de la Complejidad Computacional

Esre Trabajo de Fin de Grado consiste en una introducción a la Teoría de la Complejidad Computacional de Tiempo, para lo cual se comienza definiendo la máquina de Turing como modelo computacional básico, antes de adentrarnos en las diferentes clases de complejidad tanto deterministas como indeterministas, haciendo una revisión más profunda de las clases P y NP como protagonistas de esta área. Se estudian las reducciones, herramientas fundamentales en el manejo de las clases de complejidad, y la NP-completitud como consecuencia directa de lo anterior. Como colofón, se termina con un capítulo dedicado en exclusiva la famosa Conjetura de Cook, uno de los problemas del milenio. Debido a su utilidad como referencia inicial en Complejidad Computacional, la selección de resultados y demostraciones pretende además familiarizar al lector con las técnicas propias de esta rama de las Matemáticas.Grado en Matemática

Computational complexity and the P vs NP problem

Orientador: Arnaldo Vieira MouraDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação CientíficaResumo: A teoria de complexidade computacional procura estabelecer limites para a eficiência dos algoritmos, investigando a dificuldade inerente dos problemas computacionais. O problema P vs NP é uma questão central em complexidade computacional. Informalmente, ele procura determinar se, para uma classe importante de problemas computacionais, a busca exaustiva por soluções é essencialmente a melhor alternativa algorítmica possível. Esta dissertação oferece tanto uma introdução clássica ao tema, quanto uma exposição a diversos teoremas mais avançados, resultados recentes e problemas em aberto. Em particular, o método da diagonalização é discutido em profundidade. Os principais resultados obtidos por diagonalização são os teoremas de hierarquia de tempo e de espaço (Hartmanis e Stearns [54, 104]). Apresentamos uma generalização desses resultados, obtendo como corolários os teoremas clássicos provados por Hartmanis e Stearns. Essa é a primeira vez que uma prova unificada desses resultados aparece na literaturaAbstract: Computational complexity theory is the field of theoretical computer science that aims to establish limits on the efficiency of algorithms. The main open question in computational complexity is the P vs NP problem. Intuitively, it states that, for several important computational problems, there is no algorithm that performs better than a trivial exhaustive search. We present here an introduction to the subject, followed by more recent and advanced results. In particular, the diagonalization method is discussed in detail. Although it is a classical technique in computational complexity, it is the only method that was able to separate strong complexity classes so far. Some of the most important results in computational complexity theory have been proven by diagonalization. In particular, Hartmanis and Stearns [54, 104] proved that, given more resources, one can solve more computational problems. These results are known as hierarchy theorems. We present a generalization of the deterministic hierarchy theorems, recovering the classical results proved by Hartmanis and Stearns as corollaries. This is the first time that such unified treatment is presented in the literatureMestradoTeoria da ComputaçãoMestre em Ciência da Computaçã