[Paper Review] Optimal Capacity Modification for Many-To-One Matching Problems

Jiehua Chen, Gergely Csáji|arXiv (Cornell University)|Feb 3, 2023

Game Theory and Voting SystemsEconomics, Econometrics and Finance3 citations

TL;DR

This paper studies optimal capacity increases in many-to-one matching problems to achieve stable and perfect or stable and Pareto-efficient matchings. It proves that while Min Max SE (minimizing maximum capacity increase for stable and efficient matchings) is polynomial-time solvable, all other variants—Min Sum SP, Min Sum SE, and Min Max SP—are NP-hard, with strong inapproximability results under standard complexity assumptions.

ABSTRACT

We consider many-to-one matching problems, where one side consists of students and the other side of schools with capacity constraints. We study how to optimally increase the capacities of the schools so as to obtain a stable and perfect matching (i.e., every student is matched) or a matching that is stable and Pareto-efficient for the students. We consider two common optimality criteria, one aiming to minimize the sum of capacity increases of all schools (abbrv. as MinSum) and the other aiming to minimize the maximum capacity increase of any school (abbrv. as MinMax). We obtain a complete picture in terms of computational complexity: Except for stable and perfect matchings using the MinMax criteria which is polynomial-time solvable, all three remaining problems are NP-hard. We further investigate the parameterized complexity and approximability and find that achieving stable and Pareto-efficient matchings via minimal capacity increases is much harder than achieving stable and perfect matchings.

Motivation & Objective

• To determine the computational complexity of modifying school capacities to achieve stable and perfect matchings.
• To analyze the complexity of achieving stable and Pareto-efficient matchings via capacity increases.
• To evaluate the optimality of capacity modifications under two criteria: minimizing total (Min Sum) and maximum (Min Max) capacity increase.
• To investigate parameterized complexity and approximability of the resulting problems.
• To provide a complete computational complexity picture for four variants of capacity modification in many-to-one matching.

Proposed method

• Reduces from 3-SAT to prove NP-hardness of Min Sum SP, Min Sum SE, and Min Max SP via gadget-based constructions.
• Uses a reduction from 3-SAT with bounded preference list lengths to show inapproximability: no O((n+m)^{1−ε})-approximation exists unless P=NP.
• Constructs a stable and efficient matching from a feasible capacity increase vector, ensuring consistency with truth assignments in the 3-SAT instance.
• Proves Min Max SE is polynomial-time solvable by showing it reduces to a maximum flow problem with capacity constraints.
• Analyzes parameterized complexity using parameters such as |U_un| (unmatched students), Δ_un (preference list length), and k^+ (number of schools with increased capacity).
• Employs a truth assignment mapping from stable matchings to satisfy 3-SAT instances, establishing equivalence between feasible solutions and satisfying assignments.

Experimental results

Research questions

• RQ1 Is the problem of minimizing the maximum capacity increase to achieve a stable and Pareto-efficient matching (Min Max SE) solvable in polynomial time?
• RQ2 Are the problems of minimizing total capacity increase for stable and perfect matchings (Min Sum SP) and stable and efficient matchings (Min Sum SE) NP-hard?
• RQ3 Can the Min Max SP and Min Sum SP problems be approximated within a constant factor or sub-polynomial factor?
• RQ4 What is the parameterized complexity of the capacity modification problems with respect to parameters like number of unmatched students or maximum preference list length?
• RQ5 Is there a non-trivial approximation algorithm for Min Sum SE and Min Max SE, or are they inapproximable under standard complexity assumptions?

Key findings

• Min Max SE (minimizing maximum capacity increase for stable and efficient matchings) is solvable in polynomial time.
• Min Sum SP (minimizing total capacity increase for stable and perfect matchings) is NP-hard.
• Min Sum SE (minimizing total capacity increase for stable and efficient matchings) is NP-hard.
• Min Max SP (minimizing maximum capacity increase for stable and perfect matchings) is NP-hard.
• There is no O((n+m)^{1−ε})-approximation algorithm for Min Sum SE or Min Max SE unless P = NP, even when preference lists are bounded to length 5.
• The problem remains paraNP-hard when parameterized by |U_un| (number of unmatched students) or by the sum of student and school preference list lengths.