Abstract

Stable matching problems with lower quotas are fundamental in academic hiring and ensuring operability of rural hospitals. Only few tractable (polynomial-time solvable) cases of stable matching with lower quotas have been identified; most such problems are mathsf {NP}-hard and also hard to approximate (Hamada et al. in Algorithmica 74(1):440–465, 2016). We therefore consider stable matching problems with lower quotas under a relaxed notion of tractability, namely fixed-parameter tractability. By cloning hospitals we focus on the case when all hospitals have upper quota equal to 1, which generalizes the setting of “arranged marriages” first considered by Knuth (Mariages stables et leurs relations avec d’autres problèmes combinatoires, Les Presses de l’Université de Montréal, Montreal, 1976). We investigate how a set of natural parameters, namely the maximum length of preference lists for men and women, the number of distinguished men and women, and the number of blocking pairs allowed determine the computational tractability of this problem. Our main result is a complete complexity trichotomy: for each choice of parameters we either provide a polynomial-time algorithm, or an mathsf {NP}-hardness proof and fixed-parameter algorithm, or mathsf {NP}-hardness proof and mathsf {W}[1]-hardness proof. As corollary, we negatively answer a question by Hamada et al. (Algorithmica 74(1):440–465, 2016) by showing fixed-parameter intractability parameterized by optimal solution size. We also classify all cases of one-sided constraints where only women may be distinguished.

Highlights

  • The Stable Marriage (SM) problem is a fundamental problem first studied by Gale and Shapley [19]

  • We provide an extensive algorithmic analysis of the Stable Marriage with Covering Constraints (SMC) problem and its special case SMC-1

  • We apply the framework of parameterized complexity, which deals with computationally hard problems and focuses on how certain parameters of a problem instance influence its tractability; for background, we refer to the book by Cygan et al [11]

Read more

Summary

Introduction

The Stable Marriage (SM) problem is a fundamental problem first studied by Gale and Shapley [19]. An instance of SM consists of a set M of men, a set W of women, and a preference list for each person ordering members of the opposite sex. Gale and Shapley proved [19] that any instance of SM admits at least one stable matching, and gave a polynomial-time algorithm, known as the Gale–Shapley algorithm, to find one. Gale and Shapley considered the many-to-one extension of SM, known as the Hospitals/Residents (HR) problem. In HR, the two sets H and R that correspond to men and women in the SM problem are residents and hospitals, respectively. For HR it still holds true that a stable matching always exists, and can be found efficiently

Objectives
Results
Conclusion
Full Text
Published version (Free)

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call