## Abstract

BALANCED STABLE MARRIAGE (BSM) is a central optimization version of the classic STABLE MARRIAGE (SM) problem. We study BSM from the viewpoint of Parameterized Complexity. Informally, the input of BSM consists of n men, n women, and an integer k. Each person a has a (sub)set of acceptable partners, A(a), whom a ranks strictly; we use p_{a}(b) to denote the position of b∈A(a) in a's preference list. The objective is to decide whether there exists a stable matching μ such that balance(μ)≜max{∑_{(m,w)∈μ}p_{m}(w),∑_{(m,w)∈μ}p_{w}(m)}≤k. In SM, all stable matchings match the same set of agents, A^{⋆} which can be computed in polynomial time. As [Formula presented] for any stable matching μ, BSM is trivially fixed-parameter tractable (FPT) with respect to k. Thus, a natural question is whether BSM is FPT with respect to [Formula presented]. With this viewpoint in mind, we draw a line between tractability and intractability in relation to the target value. This line separates additional natural parameterizations higher/lower than ours (e.g., we automatically resolve the parameterization [Formula presented]). The two extreme stable matchings are the man-optimal μ_{M} and the woman-optimal μ_{W}. Let O_{M}=∑_{(m,w)∈μM}p_{m}(w), and O_{W}=∑_{(m,w)∈μW}p_{w}(m). In this work, we prove that • BSM parameterized by t=k−min{O_{M},O_{W}} admits (1) a kernel where the number of people is linear in t, and (2) a parameterized algorithm whose running time is single exponential in t. • BSM parameterized by t=k−max{O_{M},O_{W}} is W[1]-hard.

Original language | English |
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Pages (from-to) | 19-43 |

Number of pages | 25 |

Journal | Theoretical Computer Science |

Volume | 883 |

DOIs | |

State | Published - 3 Sep 2021 |

## Keywords

- Kernelization
- Keywords balanced stable marriage
- Parameterized complexity

## ASJC Scopus subject areas

- Theoretical Computer Science
- General Computer Science