11 research outputs found

    Testing Hereditary Properties of Sequences

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    A hereditary property of a sequence is one that is preserved when restricting to subsequences. We show that there exist hereditary properties of sequences that cannot be tested with sublinear queries, resolving an open question posed by Newman et al. This proof relies crucially on an infinite alphabet, however; for finite alphabets, we observe that any hereditary property can be tested with a constant number of queries

    Strong modeling limits of graphs with bounded tree-width

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    The notion of first order convergence of graphs unifies the notions of convergence for sparse and dense graphs. Ne\v{s}et\v{r}il and Ossona de Mendez [J. Symbolic Logic 84 (2019), 452--472] proved that every first order convergent sequence of graphs from a nowhere-dense class of graphs has a modeling limit and conjectured the existence of such modeling limits with an additional property, the strong finitary mass transport principle. The existence of modeling limits satisfying the strong finitary mass transport principle was proved for first order convergent sequences of trees by Ne\v{s}et\v{r}il and Ossona de Mendez [Electron. J. Combin. 23 (2016), P2.52] and for first order sequences of graphs with bounded path-width by Gajarsk\'y et al. [Random Structures Algorithms 50 (2017), 612--635]. We establish the existence of modeling limits satisfying the strong finitary mass transport principle for first order convergent sequences of graphs with bounded tree-width.Comment: arXiv admin note: text overlap with arXiv:1504.0812

    Limits of permutation sequences

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    A permutation sequence is said to be convergent if the density of occurrences of every fixed permutation in the elements of the sequence converges. We prove that such a convergent sequence has a natural limit object, namely a Lebesgue measurable function Z:[0,1]2→[0,1]Z:[0,1]^2 \to [0,1] with the additional properties that, for every fixed x∈[0,1]x \in [0,1], the restriction Z(x,⋅)Z(x,\cdot) is a cumulative distribution function and, for every y∈[0,1]y \in [0,1], the restriction Z(⋅,y)Z(\cdot,y) satisfies a "mass" condition. This limit process is well-behaved: every function in the class of limit objects is a limit of some permutation sequence, and two of these functions are limits of the same sequence if and only if they are equal almost everywhere. An ingredient in the proofs is a new model of random permutations, which generalizes previous models and might be interesting for its own sake.Comment: accepted for publication in the Journal of Combinatorial Theory, Series B. arXiv admin note: text overlap with arXiv:1106.166

    Limit structures and property testing

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    In the thesis, we study properties of large combinatorial objects. We analyze these objects from two different points of view. The first aspect is analytic - we study properties of limit objects of combinatorial structures. We investigate when graphons (limits of graphs) and permutons (limits of permutations) are finitely forcible, i.e., when they are uniquely determined by finitely many densities of their substructures. We give examples of families of permutons that are finitely forcible but the associated graphons are not and we disprove a conjecture of Lovasz and Szegedy on the dimension of the space of typical vertices of a finitely forcible graphon. In particular, we show that there exists a finitely forcible graphon W such that the topological spaces T(W) and T(W) have infinite Lebesgue covering dimension. We also study the dependence between densities of substructures. We prove a permutation analogue of the classical theorem of Erdos, Lovasz and Spencer on the densities of connected subgraphs in large graphs. The second aspect of large combinatorial objects we concentrate on is algorithmic|we study property testing and parameter testing. We show that there exists a bounded testable permutation parameter that is not finitely forcible and that every hereditary permutation property is testable (in constant time) with respect to the Kendall's tau distance, resolving a conjecture of Kohayakawa

    Testing permutation properties through subpermutations

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    There has been great interest in deciding whether a combinatorial structure satisfies some property, or in estimating the value of some numerical function associated with this combinatorial structure, by considering only a randomly chosen substructure of sufficiently large, but constant size. These problems are called property testing and parameter testing, where a property or parameter is said to be testable if it can be estimated accurately in this way. The algorithmic appeal is evident, as, conditional on sampling, this leads to reliable constant-time randomized estimators. Our paper addresses property testing and parameter testing for permutations in a subpermutation perspective; more precisely, we investigate permutation properties and parameters that can be well approximated based on a randomly chosen subpermutation of much smaller size. In this context, we use a theory of convergence of permutation sequences developed by the present authors [C. Hoppen, Y. Kohayakawa, C.G. Moreira, R.M. Sampaio, Limits of permutation sequences through permutation regularity, Manuscript, 2010, 34pp.] to characterize testable permutation parameters along the lines of the work of Borgs et al. [C. Borgs, J. Chayes, L Lovasz, V.T. Sos, B. Szegedy, K. Vesztergombi, Graph limits and parameter testing, in: STOC`06: Proceedings of the 38th Annual ACM Symposium on Theory of Computing, ACM, New York, 2006, pp. 261-270.] in the case of graphs. Moreover, we obtain a permutation result in the direction of a famous result of Alon and Shapira [N. Alon, A. Shapira, A characterization of the (natural) graph properties testable with one-sided error, SIAM J. Comput. 37 (6) (2008) 1703-1727.] stating that every hereditary graph property is testable. (C) 2011 Elsevier B.V. All rights reserved.Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)CNPq[484154/2010-9]CNPq[308509/2007-2]Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado do Rio Grande do Sul (FAPERGS)FAPERGS[10/0388-2]Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)FAPESP[2007/56496-3]FuncapFuncap[07.013.00/09
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