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6974 results in Algorithmics, Complexity, Computer Algebra, Computational Geometry

Page 99 of 349

  • 7 - Lower Bounds Techniques

  • Oded Goldreich, Weizmann Institute of Science, Israel
  • Book:
    Introduction to Property Testing
    Published online:
    13 November 2017
    Print publication:
    23 November 2017, pp 134-161

  • Summary

    Summary: We present and illustrate three techniques for proving lower bounds on the query complexity of property testers.

    1. Showing a pair of distributions, one on instances that have the property and the other on instances that are far from the property, such that an oracle machine of low query complexity cannot distinguish these two distributions.

    2. Showing a reduction from communication complexity. That is, showing that a communication complexity problem of high complexity can be solved within communication complexity that is related to the query complexity of the property testing task that we are interested in.

    3. Showing a reduction from another testing problem. That is, showing a “local” reduction of a hard testing problem to the testing problem that we are interested in.

    We also present simplifications of these techniques for the cases of onesided error probability testers and nonadaptive testers.

    The methodology of reducing from communication complexity was introduced by Blais, Brody, and Matulef [54], and our description of it is based on [136].

    Introduction

    Our perspective in this book is mainly algorithmic. Hence, we view complexity lower bounds mainly as justifications for the failure to provide better algorithms (i.e., algorithms of lower complexity). The lower bounds that we shall be discussing are lower bounds on the query complexity of testers. These lower bounds are of an information theoretic nature, and so they cannot (and do not) rely on computational assumptions.

    We start with two brief preliminary discussions. The first discussion is very abstract and vague: it concerns the difficulty of establishing lower bounds. The second discussion is very concrete: it highlights the fact that computational complexity considerations play no role in this chapter, a fact that is most evident in the avoidance of the uniformity condition.

    What Makes Lower Bounds Hard to Prove? Proving lower bounds is often more challenging than proving upper bounds, since one has to defeat all possible methods (or algorithms) rather than show that one of them works. Indeed, it seems harder to cope with a universal quantifier than with an existential one, but one should bear in mind that a second quantifier of opposite nature follows the first one.

Introduction to Property Testing
  • Oded Goldreich
  • Published online:
    13 November 2017
    Print publication:
    23 November 2017
  • Property testing is concerned with the design of super-fast algorithms for the structural analysis of large quantities of data. The aim is to unveil global features of the data, such as determining whether the data has a particular property or estimating global parameters. Remarkably, it is possible for decisions to be made by accessing only a small portion of the data. Property testing focuses on properties and parameters that go beyond simple statistics. This book provides an extensive and authoritative introduction to property testing. It provides a wide range of algorithmic techniques for the design and analysis of tests for algebraic properties, properties of Boolean functions, graph properties, and properties of distributions.

Complexity Dichotomies for Counting Problems
  • Volume 1, Boolean Domain
  • Jin-Yi Cai, Xi Chen
  • Published online:
    06 November 2017
    Print publication:
    16 November 2017
  • Complexity theory aims to understand and classify computational problems, especially decision problems, according to their inherent complexity. This book uses new techniques to expand the theory for use with counting problems. The authors present dichotomy classifications for broad classes of counting problems in the realm of P and NP. Classifications are proved for partition functions of spin systems, graph homomorphisms, constraint satisfaction problems, and Holant problems. The book assumes minimal prior knowledge of computational complexity theory, developing proof techniques as needed and gradually increasing the generality and abstraction of the theory. This volume presents the theory on the Boolean domain, and includes a thorough presentation of holographic algorithms, culminating in classifications of computational problems studied in exactly solvable models from statistical mechanics.

Page 99 of 349