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arXiv · 2406.08942

Quotient-convergence of Submodular Setfunctions

Kristóf Bérczi·Márton Borbényi·László Lovász·László Márton Tóth
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Abstract

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.

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Kristóf Bérczi, Márton Borbényi, László Lovász, László Márton Tóth. 2024-06-13. Quotient-convergence of Submodular Setfunctions. https://arxiv.org/abs/2406.08942

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