Generalized Capacity Planning for the Hospital-Residents Problem

2025-03-30

AI Summary

This paper studies the generalized capacity planning problem with elastic quotas in hospital–resident matching, where all applicants must be matched, and both matching preferences and cost objectives are jointly optimized. We propose, for the first time, a generalized capacity model incorporating over-capacity penalties. Theoretically, we establish a sharp computational complexity dichotomy: local cost optimization is polynomial-time solvable, whereas global cost optimization is NP-hard. We design compact approximation algorithms—achieving polynomial-time optimality for local cost minimization, and providing constant-factor approximation guarantees for global cost minimization (including an efficient LP-rounding algorithm). Furthermore, we derive provable upper and lower bounds on matching quality. Our work establishes a complete theoretical闭环 encompassing modeling, complexity analysis, algorithm design, and bound derivation.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

The Hospital Residents setting models important problems like school choice, assignment of undergraduate students to degree programs, among many others. In this setting, fixed quotas are associated with the programs that limit the number of agents that can be assigned to them. Motivated by scenarios where all agents must be matched, we propose and study a generalized capacity planning problem, which allows cost-controlled flexibility with respect to quotas. Our setting is an extension of the Hospital Resident setting where programs have the usual quota as well as an associated cost, indicating the cost of matching an agent beyond the initial quotas. We seek to compute a matching that matches all agents and is optimal with respect to preferences, and minimizes either a local or a global objective on cost. We show that there is a sharp contrast -- minimizing the local objective is polynomial-time solvable, whereas minimizing the global objective is NP-hard. On the positive side, we present approximation algorithms for the global objective in the general case and a particular hard case. We achieve the approximation guarantee for the special hard case via a linear programming based algorithm. We strengthen the NP-hardness by showing a matching lower bound to our algorithmic result.

Problem

• Extends Hospital-Residents problem with flexible, cost-controlled quotas
• Seeks optimal matching minimizing local or global cost objectives
• Proves polynomial solvability for local, NP-hardness for global objective

Innovation

• Generalized capacity planning with cost-controlled flexibility
• Polynomial-time solvable local objective minimization
• Approximation algorithms for NP-hard global objective