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Balanced stable marriage: How close is close enough?
- Source :
-
Theoretical Computer Science . Sep2021, Vol. 883, p19-43. 25p. - Publication Year :
- 2021
-
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 balance (μ) ≥ | A ⋆ | 2 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 k − | A ⋆ | 2. 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 k − | A ⋆ | 2). The two extreme stable matchings are the man-optimal μ M and the woman-optimal μ W. Let O M = ∑ (m , w) ∈ μ M p m (w) , and O W = ∑ (m , w) ∈ μ W 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. [ABSTRACT FROM AUTHOR]
- Subjects :
- *MARRIAGE
*POLYNOMIAL time algorithms
*ALGORITHMS
Subjects
Details
- Language :
- English
- ISSN :
- 03043975
- Volume :
- 883
- Database :
- Academic Search Index
- Journal :
- Theoretical Computer Science
- Publication Type :
- Academic Journal
- Accession number :
- 151951911
- Full Text :
- https://doi.org/10.1016/j.tcs.2021.05.015