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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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References

  1. Noga Alon, Richard A. Duke, Hanno Lefmann, Vojtěch Rödl, and Raphael Yuster. The algorithmic aspects of the regularity lemma. J. Algorithms, 16(1):80-109, 1994. URL: http://dx.doi.org/10.1006/jagm.1994.1005.
  2. Noga Alon, Eldar Fischer, Michael Krivelevich, and Mario Szegedy. Efficient testing of large graphs. Combinatorica, 20(4):451-476, 2000. URL: http://dx.doi.org/10.1007/s004930070001.
  3. Noga Alon, Eldar Fischer, Ilan Newman, and Asaf Shapira. A combinatorial characterization of the testable graph properties: it’s all about regularity. SIAM J. Comput., 39(1):143-167, 2009. URL: http://dx.doi.org/10.1137/060667177.
  4. Noga Alon and Asaf Shapira. A characterization of the (natural) graph properties testable with one-sided error. SIAM J. Comput., 37(6):1703-1727, 2008. URL: http://dx.doi.org/10.1137/06064888X.
  5. Noga Alon and Asaf Shapira. Every monotone graph property is testable. SIAM J. Comput., 38(2):505-522, 2008. URL: http://dx.doi.org/10.1137/050633445.
  6. Noga Alon, Asaf Shapira, and Benny Sudakov. Additive approximation for edge-deletion problems. Ann. of Math. (2), 170(1):371-411, 2009. URL: http://dx.doi.org/10.4007/annals.2009.170.371.
  7. Jean-Pierre Barthélémy and Bernard Monjardet. The median procedure in cluster analysis and social choice theory. Math. Social Sci., 1(3):235-267, 1980/81. URL: http://dx.doi.org/10.1016/0165-4896(81)90041-X.
  8. Béla Bollobás. Hereditary properties of graphs: asymptotic enumeration, global structure, and colouring. Doc. Math., pages 333-342 (electronic), 1998. Extra Vol. III. Google Scholar
  9. Christian Borgs, Jennifer T. Chayes, László Lovász, Vera T. Sós, and Katalin Vesztergombi. Convergent sequences of dense graphs. I. Subgraph frequencies, metric properties and testing. Adv. Math., 219(6):1801-1851, 2008. URL: http://dx.doi.org/10.1016/j.aim.2008.07.008.
  10. Irène Charon and Olivier Hudry. An updated survey on the linear ordering problem for weighted or unweighted tournaments. Ann. Oper. Res., 175:107-158, 2010. URL: http://dx.doi.org/10.1007/s10479-009-0648-7.
  11. David Conlon and Jacob Fox. Bounds for graph regularity and removal lemmas. Geom. Funct. Anal., 22(5):1191-1256, 2012. URL: http://dx.doi.org/10.1007/s00039-012-0171-x.
  12. Paul Erdős, Péter Frankl, and Vojtěch Rödl. The asymptotic number of graphs not containing a fixed subgraph and a problem for hypergraphs having no exponent. Graphs and Combinatorics, 2(1):113-121, 1986. URL: http://dx.doi.org/10.1007/BF01788085.
  13. Paul Erdős, Daniel J. Kleitman, and Bruce L. Rothschild. Asymptotic enumeration of K_n-free graphs. In Colloquio Internazionale sulle Teorie Combinatorie (Rome, 1973), Tomo II, pages 19-27. Atti dei Convegni Lincei, No. 17. Accad. Naz. Lincei, Rome, 1976. Google Scholar
  14. Eldar Fischer and Ilan Newman. Testing versus estimation of graph properties. SIAM J. Comput., 37(2):482-501 (electronic), 2007. URL: http://dx.doi.org/10.1137/060652324.
  15. Jacob Fox. A new proof of the graph removal lemma. Ann. of Math. (2), 174(1):561-579, 2011. URL: http://dx.doi.org/10.4007/annals.2011.174.1.17.
  16. Alan Frieze and Ravi Kannan. Quick approximation to matrices and applications. Combinatorica, 19(2):175-220, 1999. URL: http://dx.doi.org/10.1007/s004930050052.
  17. Z. Füredi. Extremal hypergraphs and combinatorial geometry. In S. D. Chatterji, editor, Proceedings of the International Congress of Mathematicians: August 3-11, 1994 Zürich, Switzerland, pages 1343-1352. Birkhäuser Basel, 1995. URL: http://dx.doi.org/10.1007/978-3-0348-9078-6_65.
  18. Oded Goldreich, editor. Property Testing - Current Research and Surveys [outgrow of a workshop at the Institute for Computer Science ITCS) at Tsinghua University, January 2010], volume 6390 of Lecture Notes in Computer Science. Springer, 2010. URL: http://dx.doi.org/10.1007/978-3-642-16367-8.
  19. Oded Goldreich, Shafi Goldwasser, and Dana Ron. Property testing and its connection to learning and approximation. J. ACM, 45(4):653-750, 1998. URL: http://dx.doi.org/10.1145/285055.285060.
  20. Oded Goldreich and Luca Trevisan. Three theorems regarding testing graph properties. Random Structures Algorithms, 23(1):23-57, 2003. URL: http://dx.doi.org/10.1002/rsa.10078.
  21. William T. Gowers. Lower bounds of tower type for Szemerédi’s uniformity lemma. Geom. Funct. Anal., 7(2):322-337, 1997. URL: http://dx.doi.org/10.1007/PL00001621.
  22. László Lovász and Balázs Szegedy. Szemerédi’s lemma for the analyst. Geom. Funct. Anal., 17(1):252-270, 2007. URL: http://dx.doi.org/10.1007/s00039-007-0599-6.
  23. Michal Parnas, Dana Ron, and Ronitt Rubinfeld. Tolerant property testing and distance approximation. J. Comput. System Sci., 72(6):1012-1042, 2006. URL: http://dx.doi.org/10.1016/j.jcss.2006.03.002.
  24. Endre Szemerédi. Regular partitions of graphs. In Problèmes combinatoires et théorie des graphes (Colloq. Internat. CNRS, Univ. Orsay, Orsay, 1976), volume 260 of Colloq. Internat. CNRS, pages 399-401. CNRS, Paris, 1978. Google Scholar
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