Lattice-Valued Bottleneck Duality

TL;DR

This paper extends classical combinatorial duality theorems to order lattices, generalizing bottleneck path-cut and flow-cut dualities for networks with lattice-valued capacities, and applies these to various non-numerical network flow problems.

Contribution

It introduces a lattice-valued framework for duality theorems, broadening their applicability beyond numerical capacities and providing new insights into non-numerical network flows.

Findings

Generalized bottleneck dualities to lattice-valued capacities

Extended Dilworth's theorem to posets with lattice weights

Demonstrated applicability to diverse non-numerical network problems

Abstract

This note reformulates certain classical combinatorial duality theorems in the context of order lattices. For source-target networks, we generalize bottleneck path-cut and flow-cut duality results to edges with capacities in a distributive lattice. For posets, we generalize a bottleneck version of Dilworth's theorem, again weighted in a distributive lattice. These results are applicable to a wide array of non-numerical network flow problems, as shown. All results, proofs, and applications were created in collaboration with AI language models. An appendix documents their role and impact.