Lattice-weighted versions of other combinatorial duality theorems

Determine whether lattice-weighted versions exist of Kőnig's Theorem on bipartite graphs and the Matroid Intersection Theorem, by formulating duality statements where the relevant optimization quantities are valued in a distributive lattice and the dual characterization holds via lattice joins and meets analogous to the lattice-valued bottleneck duality established for flow networks and posets in this work.

Background

The paper extends bottleneck duality from real-valued capacities to capacities and weights taking values in distributive lattices, proving lattice-valued bottleneck path–cut duality and a lattice-weighted bottleneck version of Dilworth’s theorem. These results show that joins and meets in distributive lattices can replace min/max over reals in duality identities.

Beyond these specific results, the authors note the broader landscape of combinatorial duality theorems (e.g., Kőnig’s theorem and the Matroid Intersection Theorem) and explicitly raise the question of whether analogous lattice-weighted formulations exist. Such formulations would potentially generalize classical dualities by replacing scalar cardinalities or weights with lattice-valued quantities while preserving the duality structure.