Extreme Points of Base Polytope of Submodular Set Functions and Limit for Quotient Convergent Graph Sequence

Abstract: Submodular set functions are of great importance in mathematics and theoretical computer science, serving as fundamental tools in optimization, combinatorics, and economics due to their natural properties and wide-ranging applications. In 2023, Lov\'asz systematically extended the theory of submodular set functions from finite sets to general set algebras and proposed several open problems about the behavior of submodular functions in infinite settings, including the characterization of extreme points of the base polytope of submodular set functions. We characterize conditions under which the extreme points of the base polytope of a submodular function are restricting measures with respect to its majorizing measure. Applying this result, we characterize the core of increasing subadditive non-atomic games and provide a positive answer to a question of Krist\'of B\'erzi, M\'arton Borb\'enyi, L\'aszl\'o Lov\'asz and L\'aszl\'o M\'arton T\'oth regarding the rank function for graphing's cycle matroid. Furthermore, building on the limit theory for set functions, we prove that the limit of convergent sequence of bounded-degree graphs' cycle matroids can be represented as the cycle matroid of a graphing, analogous to the completeness result for local-global convergence.