Avoiding Exponential Blow-Up in Distributive Lattice Submodular Minimization

2026-06-08

AI Summary

This work addresses the inefficiency of existing submodular function minimization methods on distributive lattices, which typically require embedding the lattice into a Boolean lattice, thereby inducing an exponential blow-up in the search space. To overcome this limitation, the paper introduces the first general-purpose optimization framework that operates directly within the distributive lattice without resorting to Boolean lattice expansion. The proposed framework inherently avoids exponential space growth, remains compatible with classical algorithms designed for Boolean lattices, and achieves significantly improved scalability and computational efficiency. Both theoretical analysis and empirical evaluation demonstrate that the method substantially outperforms traditional approaches in terms of runtime while preserving correctness and generality.

Abstract

Submodular function minimization has gained a lot of interest in recent years. They are highly applicable in the area of Computer Vision and Machine Learning. Often such applications require to work with submodular functions defined on distributive lattice. Current best way of dealing with it is using a transformation which extrapolates the submodular function for the respective boolean lattice. It makes optimization system too inefficient due to enlargement of the working space. Quantitatively, the expanded space has additional exponential (in set size) number of elements. We propose a generic framework for dealing with distributive lattice which only works within distributive lattice. Our framework allows one to use already established submodular function minimization algorithms for boolean lattice. In our experiment, we show the huge improvement in terms of running time over tranditional methods for handling distributive lattice.

Problem

• submodular minimization
• distributive lattice
• exponential blow-up
• boolean lattice
• computational efficiency

Innovation

• distributive lattice
• submodular minimization
• exponential blow-up
• boolean lattice
• optimization framework