11 research outputs found

    A bipartite analogue of Dilworth's theorem for multiple partial orders

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    AbstractLet r be a fixed positive integer. It is shown that, given any partial orders <1, …, <r on the same n-element set P, there exist disjoint subsets A,B⊂P, each with at least n1−o(1) elements, such that one of the following two conditions is satisfied: (1) there is an i(1≤i≤r) such that every element of A is larger than every element of B in the partial order <i, or (2) no element of A is comparable with any element of B in any of the partial orders <1, …, <r. As a corollary, we obtain that any family C of n convex compact sets in the plane has two disjoint subfamilies A,B⊂C, each with at least n1−o(1) members, such that either every member of A intersects all members of B, or no member of A intersects any member of B

    Min-max results in combinatorial optimization

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    Packing and covering in combinatorics

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    Detecting stochastic dominance for poset-valued random variables as an example of linear programming on closure systems

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    In this paper we develop a linear programming method for detecting stochastic dominance for random variables with values in a partially ordered set (poset) based on the upset-characterization of stochastic dominance. The proposed detection-procedure is based on a descriptively interpretable statistic, namely the maximal probability-difference of an upset. We show how our method is related to the general task of maximizing a linear function on a closure system. Since closure systems are describable via their valid formal implications, we can use here ingredients of formal concept analysis. We also address the question of inference via resampling and via conservative bounds given by the application of Vapnik-Chervonenkis theory, which also allows for an adequate pruning of the envisaged closure system that allows for the regularization of the test statistic (by paying a price of less conceptual rigor). We illustrate the developed methods by applying them to a variety of data examples, concretely to multivariate inequality analysis, item impact and differential item functioning in item response theory and to the analysis of distributional differences in spatial statistics. The power of regularization is illustrated with a data example in the context of cognitive diagnosis models

    Property testing : theory and applications

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    Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2003.Includes bibliographical references (p. 107-111).(cont.) We show upper and lower bounds for the general problem and for specific partial orders. A few of our intermediate results are of independent interest. 1. If strings with a property form a vector space, adaptive 2-sided error tests for the property have no more power than non-adaptive 1-sided error tests. 2. Random LDPC codes with linear distance and constant rate are not locally testable. 3. There exist graphs with many edge-disjoint induced matchings of linear size. In the final part of the thesis, we initiate an investigation of property testing as applied to images. We study visual properties of discretized images represented by n x n matrices of binary pixel values. We obtain algorithms with the number of queries independent of n for several basic properties: being a half-plane, connectedness and convexity.Property testers are algorithms that distinguish inputs with a given property from those that are far from satisfying the property. Far means that many characters of the input must be changed before the property arises in it. Property testing was introduced by Rubinfeld and Sudan in the context of linearity testing and first studied in a variety of other contexts by Goldreich, Goldwasser and Ron. The query complexity of a property tester is the number of input characters it reads. This thesis is a detailed investigation of properties that are and are not testable with sublinear query complexity. We begin by characterizing properties of strings over the binary alphabet in terms of their formula complexity. Every such property can be represented by a CNF formula. We show that properties of n-bit strings defined by 2CNF formulas are testable with O([square root of]n) queries, whereas there are 3CNF formulas for which the corresponding properties require Q(n) queries, even for adaptive tests. We show that testing properties defined by 2CNF formulas is equivalent, with respect to the number of required queries, to several other function and graph testing problems. These problems include: testing whether Boolean functions over general partial orders are close to monotone, testing whether a set of vertices is close to one that is a vertex cover of a specific graph, and testing whether a set of vertices is close to a clique. Testing properties that are defined in terms of monotonicity has been extensively investigated in the context of the monotonicity of a sequence of integers and the monotonicity of a function over the m-dimensional hypercube (1,... , a)m. We study the query complexity of monotonicity testing of both Boolean and integer functions over general partial orders.by Sofya Raskhodnikova.Ph.D

    Monotonic functions of finite posets

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    Classes of monotone functions from finite posets to chains are studied. These include order-preserving and strict order-preserving maps. When the maps are required to be bijective they are called linear extensions. Techniques for handling the first two types are closely related; whereas for linear extensions quite distinct methods are often necessary, which may yield results for order-preserving injections. First, many new fundamental properties and inequalities of a combinatorial nature are established for these maps. Quantities considered here include the range, height, depth and cardinalities of subposets. In particular we study convexity in posets and similarly pre-images of intervals in chains. The problem of extending a map defined on a subposet to the whole poset is discussed. We investigate positive correlation inequalities, having implications for the complexity of sorting algorithms. These express monotonicity properties for probabilities concerning sets of relations in posets. New proofs are given for existing inequalities and we obtain corresponding negative correlations, along with a generalization of the xyz inequality. The proofs involve inequalities in distributive lattices, some of which arose in physics. A characterization is given for posets satisfying necessary conditions for correlation properties under linear extensions. A log concavity type inequality is proved for the number of strict or non-strict order-preserving maps of an element. We define an explicit injection whereas the bijective case is proved in the literature using inequalities from the theory of mixed volumes. These results motivate a further group of such inequalities. But now we count numbers of strict or non-strict order-preserving maps of subposets to varying heights in the chain. Lastly we consider the computational cost of producing certain posets which can be associated with classical sorting and selection problems. A lower bound technique is derived for this complexity, incorporating either a new distributive lattice inequality, or the log concavity inequalities
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