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Quotient-convergence of Submodular Setfunctions
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We introduce the concept of quotient-convergence for sequences of submodular set functions, providing, among others, a new framework for the study of convergence of matroids through their rank functions. Extending the limit theory of bounded degree graphs, which analyzes graph sequences via neighborhood sampling, we address the challenge posed by the absence of a neighborhood concept in matroids. We show that any bounded set function can be approximated by a sequence of finite set functions that quotient-converges to it. In addition, we explicitly construct such sequences for increasing, submodular, and upper continuous set functions, and prove the completeness of the space under quotient-convergence.
Forward citations
Cited by 2 Pith papers
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The model theory of metric lattices: pseudofinite partition lattices
Infinite pseudofinite limits of partition lattices have a definable Boolean sublattice of modular elements, and each element is determined by its set of modular selectors.
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Convergent sequences of combinatorial submodular setfunctions
The paper proves quotient-convergence for sequences of matroid and submodular set functions from finite linear spaces, cut capacities, and cycle matroids near positive graphons, and constructs a dense counterexample f...
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