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On a stable partnership problem with integer choice functions

Alexander V. Karzanov

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

On a stable partnership problem with integer choice functions

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 with nonnegative integer edge capacities , , in which for each vertex (``agent'') , the preferences on the set of its incident edges are given via a choice function acting on the vectors in 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 such that and is a set of pairwise edge-disjoint odd cycles in satisfying the following properties: if , then is a stable partnership, whereas if is nonempty, then a stable partnership does not exist, and in this case, the set 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

Paper Structure

This paper contains 11 sections, 8 theorems, 8 equations.

Table of Contents

  1. Introduction
  2. Definitions and settings
  3. Backgrounds in the bipartite case
  4. Rotations.
  5. Weights of rotations.
  6. Routes.
  7. Poset of rotations and closed functions.
  8. Complexities.
  9. Features of the poset in the symmetric bipartite case
  10. Transferring to the non-bipartite case
  11. Concluding remarks

Key Result

Lemma 4.1

Let $x,y\in{\cal S}$ be such that $x\prec_F y$ and $y=x+\lambda\chi^R$, where $R$ is a rotation applicable to $x$ and $\lambda$ is a feasible weight for $x,R$. Then $y^\ast\prec_F x^\ast$ and $x^\ast=y^\ast +\lambda\chi^{R^\ast}$ (where both $R,R^\ast$ are increasing w.r.t. $\prec_F$).

Theorems & Definitions (8)

  • Lemma 4.1
  • Lemma 4.2
  • Corollary 4.3
  • Lemma 4.4
  • Theorem 4.5
  • Theorem 5.1
  • Corollary 6.1
  • Theorem 6.2