Capacitated House Allocation: Popularity and Perfectness

Core Concepts

Popular and perfect matchings in capacitated house allocation markets.

Abstract

The content discusses the complexities of popular and perfect matchings in capacitated house allocation markets. It addresses the NP-hardness of deciding if a popular matching exists, the optimization criteria for capacity changes, and the structural results of these markets. The paper provides algorithms and proofs for solving these problems efficiently.

1. Introduction to matching markets with capacities.
2. Two-sided matching problems with one-sided preferences.
3. Definitions and fundamentals of many-to-one matching.
4. Decision problems for Pareto-optimal and perfect matchings.
5. Structural results and algorithms for capacity modifications.
6. Complexity analysis and optimization criteria.
7. Application of the Bellman-Ford algorithm for verification.
8. Reduction from 3dm problem to MinSumpop-p.

Stats

"We consider two common optimality criteria, one aiming to minimize the sum of capacity increases of all houses (abbrv. as MinSum) and the other aiming to minimize the maximum capacity increase of any school (abbrv. as MinMax)."
"We show that if we are only allowed to increase the capacities, then minimizing the sum of changes is polynomial-time solvable."
"Surprisingly, if the maximum of the capacity changes is minimized, then both versions become NP-hard and also hard to approximate within a constant factor."

Quotes

"We consider two common optimality criteria, one aiming to minimize the sum of capacity increases of all houses (abbrv. as MinSum) and the other aiming to minimize the maximum capacity increase of any school (abbrv. as MinMax)."
"We show that if we are only allowed to increase the capacities, then minimizing the sum of changes is polynomial-time solvable."
"Surprisingly, if the maximum of the capacity changes is minimized, then both versions become NP-hard and also hard to approximate within a constant factor."

Key Insights Distilled From

Popularity and Perfectness in One-sided Matching Markets with Capacities

by Gerg... at arxiv.org 03-04-2024

https://arxiv.org/pdf/2403.00598.pdf

Deeper Inquiries

질문 1

용량 제약이 매칭 시장의 효율성에 어떤 영향을 미치나요?
답변 1 여기에

질문 2

인기 있는 매칭과 완벽한 매칭의 NP-난해성이 현실 세계 응용 프로그램에 미치는 영향은 무엇인가요?
답변 2 여기에

질문 3

이 연구 결과를 다른 유형의 매칭 시장에 어떻게 확장할 수 있을까요?
답변 3 여기에