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Stable Matchings with Covering Constraints: A Complete Computational Trichotomy.

Matthias Mnich1,2, Ildikó Schlotter3

  • 11Universität Bonn, Bonn, Germany.

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|March 28, 2020
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Summary
This summary is machine-generated.

This study analyzes stable matching with lower quotas, crucial for academic hiring and rural hospitals. It introduces fixed-parameter tractability, providing a complete complexity trichotomy for various parameters.

Keywords:
Fixed-parameter algorithmsLower quotasStable marriage

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Area of Science:

  • Computer Science
  • Discrete Mathematics
  • Algorithmic Game Theory

Background:

  • Stable matching problems with lower quotas are computationally challenging, often NP-hard and difficult to approximate.
  • Tractability has been identified in only a few specific cases, necessitating alternative approaches for broader applicability.

Purpose of the Study:

  • To investigate the computational complexity of stable matching problems with lower quotas using fixed-parameter tractability.
  • To analyze the impact of parameters like preference list length, distinguished individuals, and allowed blocking pairs on problem solvability.

Main Methods:

  • The study employs a cloning technique for hospitals, simplifying the problem to a setting generalizing 'arranged marriages' with upper quotas of 1.
  • It systematically examines various parameter combinations to determine their effect on computational tractability.

Main Results:

  • A complete complexity trichotomy is established: problems are either polynomial-time solvable, NP-hard with fixed-parameter algorithms, or NP-hard with W[1]-hardness.
  • Fixed-parameter intractability is proven for the parameter of optimal solution size, answering a key question in the field.

Conclusions:

  • The research provides a comprehensive understanding of the computational landscape for stable matching with lower quotas under parameterized complexity.
  • It offers precise classifications for one-sided constraints, advancing the theoretical foundations of matching algorithms.