On a stable partnership problem with integer choice functions

TL;DR

This paper generalizes the stable partnership problem to non-bipartite graphs with integer capacities and choice functions, providing a solvability criterion and an algorithm to find stable solutions or certify their non-existence.

Contribution

It introduces a comprehensive framework for the stable partnership problem with integer choice functions, extending previous bipartite results and providing a canonical set of odd cycles when no stable solution exists.

Findings

Provides a solvability criterion for SPPIC

Develops an algorithm to find stable partnerships or certify non-existence

Characterizes the structure of solutions using odd cycles

Abstract

We consider a far generalization of the well-known stable roommates and non-bipartite stable allocation problems. In its setting, one is given a finite non-bipartite graph $G=(V,E)$ with nonnegative integer edge capacities $b(e)\in{\mathbb Z}_+$, $e\in E$, in which for each vertex (``agent'') $v\in V$, the preferences on the set $E_v$ of its incident edges are given via a choice function $C_v$ acting on the vectors in ${\mathbb Z}_+^{E_v}$ bounded by the capacities and obeying the standard axioms of substitutability and size monotonicity. We refer to the related stability problem as the stable partnership problem with integer choice functions, or SPPIC for short. Extending well-known results for particular cases, we give a solvability criterion for SPPIC and develop an algorithm of finding a stable solution, called a stable partnership, or establishing that there is none. Moreover, in general the algorithm constructs a pair $(x,{\cal K})$ such that $x\in {\mathbb Z}_+^E$ and ${\cal K}$ is a set of pairwise edge-disjoint odd cycles in $G$ satisfying the following properties: if ${\cal K}=\emptyset$, then $x$ is a stable partnership, whereas if ${\cal K}$ is nonempty, then a stable partnership does not exist, and in this case, the set ${\cal K}$ is determined canonically. Our constructions essentially use earlier author's results on the corresponding bipartite counterpart of SPPIC. Keywords: stable marriage problem, stable roommates problem, stable partition, stable allocation, choice function