无嫉妒的匹配与成本控制的配额
Envy-free matchings with cost-controlled quotas
摘要：
我们考虑的问题分配代理程序中存在的双边偏好，通常被称为医院居民的问题。在标准设置中，每个程序都有一个严格的上限，不能违反。出于应用程序中的配额是由资源的可用性，我们提出并研究了计算最佳匹配的问题与成本控制的配额-表示为CCQ设置。在CCQ设置中，我们有一个与每个程序相关联的成本，它表示将单个代理与程序相匹配的成本。我们的目标是计算匹配所有代理，尊重代理和程序的首选项列表，是最佳的成本标准。我们认为嫉妒免费作为一个概念的最优性和研究两个优化问题的成本-最小化总成本（MINSUM）和最小化最大成本在一个程序（MINMAX）。我们发现，有一个鲜明的对比，这两个问题的复杂性状态- MINMAX是多项式时间可解的，而MINSUM是NP难，很难近似在一个常数因子，除非P = NP，即使在严格的限制。在积极的一面，我们提出了近似算法的MINSUM的一般情况下，一个特殊的困难情况。我们通过一个技术上涉及的线性规划（LP）为基础的算法，实现了特殊情况下的近似保证。我们注意到，我们的LP是一般情况下的问题。
英文摘要：
We consider the problem of assigning agents to programs in the presence of two-sided preferences, commonly known as the Hospital Residents problem. In the standard setting each program has a rigid upper-quota which cannot be violated. Motivated by applications where quotas are governed by resource availability, we propose and study the problem of computing optimal matchings with cost-controlled quotas -- denoted as the CCQ setting. In the CCQ setting we have a cost associated with every program which denotes the cost of matching a single agent to the program. Our goal is to compute a matching that matches all agents, respects the preference lists of agents and programs and is optimal with respect to the cost criteria. We consider envy-freeness as a notion of optimality and study two optimization problems with respect to the costs -- minimize the total cost (MINSUM) and minimize the maximum cost at a program (MINMAX). We show that there is a sharp contrast in the complexity status of these two problems -- MINMAX is polynomial time solvable whereas MINSUM is NP-hard and hard to approximate within a constant factor unless P = NP even under severe restrictions. On the positive side, we present approximation algorithms for the MINSUM for the general case and a special hard case. We chieve the approximation guarantee for the special case via a technically involved linear programming (LP) based algorithm. We remark that our LP is for the general case of the problem.
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