Characterization of the alldifferent kernel by Hall partitions and a calculation method

TL;DR

This paper introduces Hall partitions to characterize the alldifferent kernel in set-valued mappings, establishing their existence, uniqueness, and providing a calculation method for these structures.

Contribution

It defines Hall partitions for the first time, proves their uniqueness, and presents a method to compute the alldifferent kernel and Hall partitions.

Findings

Hall partitions are equivalent to the Hall condition.

Uniqueness of Hall partitions and alldifferent selections is established.

A practical calculation method for the alldifferent kernel is provided.

Abstract

We consider a set-valued mapping between two finite sets and define the alldifferent kernel which describes the submapping of alldifferent selections. This submapping is characterized by Hall partitions which are introduced in this paper. The existence of a Hall partition is equivalent to the Hall condition. The unicity of Hall partitions is proved and the unicity of an alldifferent selection is characterized. A calculation method for the determination of the Hall partition and the alldifferent kernel is presented.