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Estimating parameters associated with monotone properties

Part of: Theory of computing Discrete mathematics in relation to computer science Graph theory Algorithms - Computer Science

Published online by Cambridge University Press:  24 March 2020

Carlos Hoppen ,
Yoshiharu Kohayakawa ,
Richard Lang ,
Hanno Lefmann  and
Henrique Stagni
Carlos Hoppen
Affiliation:
Instituto de Matemática, UFRGS, Avenida Bento Gonçalves, 9500, 91501-970 Porto Alegre, RS, Brazil
Yoshiharu Kohayakawa
Affiliation:
Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, 05508-090 São Paulo, Brazil
Richard Lang
Affiliation:
Combinatorics and Optimization, University of Waterloo, N2L 3G1 Waterloo ON, Canada
Hanno Lefmann
Affiliation:
Fakultät für Informatik, Technische Universität Chemnitz, Straße der Nationen 62, 09111 Chemnitz, Germany
Henrique Stagni*
Affiliation:
Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, 05508-090 São Paulo, Brazil
*
*Corresponding author. Email: stagni@gmail.com
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Abstract

There has been substantial interest in estimating the value of a graph parameter, i.e. of a real-valued function defined on the set of finite graphs, by querying a randomly sampled substructure whose size is independent of the size of the input. Graph parameters that may be successfully estimated in this way are said to be testable or estimable, and the sample complexity qz = qz(ε) of an estimable parameter z is the size of a random sample of a graph G required to ensure that the value of z(G) may be estimated within an error of ε with probability at least 2/3. In this paper, for any fixed monotone graph property $\mathcal{P}= \text{Forb}\!(\mathcal{F}),$ we study the sample complexity of estimating a bounded graph parameter z that, for an input graph G, counts the number of spanning subgraphs of G that satisfy$\mathcal{P}$. To improve upon previous upper bounds on the sample complexity, we show that the vertex set of any graph that satisfies a monotone property $\mathcal{P}$ may be partitioned equitably into a constant number of classes in such a way that the cluster graph induced by the partition is not far from satisfying a natural weighted graph generalization of $\mathcal{P}$. Properties for which this holds are said to be recoverable, and the study of recoverable properties may be of independent interest.

MSC classification

Primary: 68W20: Randomized algorithms
Secondary: 68W25: Approximation algorithms 68Q25: Analysis of algorithms and problem complexity 68R10: Graph theory (including graph drawing) 05C35: Extremal problems 05C85: Graph algorithms

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 29 , Issue 4 , July 2020 , pp. 616 - 632
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© Cambridge University Press 2020

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