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Estimating Parameters Associated with Monotone Properties

Authors Carlos Hoppen, Yoshiharu Kohayakawa, Richard Lang, Hanno Lefmann, Henrique Stagni

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Carlos Hoppen
Yoshiharu Kohayakawa
Richard Lang
Hanno Lefmann
Henrique Stagni

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Carlos Hoppen, Yoshiharu Kohayakawa, Richard Lang, Hanno Lefmann, and Henrique Stagni. Estimating Parameters Associated with Monotone Properties. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 60, pp. 35:1-35:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2016.35

Abstract

There has been substantial interest in estimating the value of a graph parameter, i.e., of a real function defined on the set of finite graphs, by sampling a randomly chosen 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 q_z=q_z(epsilon) of an estimable parameter z is the size of the random sample required to ensure that the value of z(G) may be estimated within error epsilon with probability at least 2/3. In this paper, we study the sample complexity of estimating two graph parameters associated with a monotone graph property, improving previously known results. To obtain our results, we prove that the vertex set of any graph that satisfies a monotone property 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 P}. Properties for which this holds are said to be recoverable, and the study of recoverable properties may be of independent interest.
Keywords
  • parameter estimation
  • parameter testing
  • edit distance to monotone graph properties
  • entropy of subgraph classes
  • speed of subgraph classes

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