The College Admissions problem with lower and common quotas

We study two generalised stable matching problems motivated by the current matching scheme used in the higher education sector in Hungary. The first problem is an extension of the College Admissions problem in which the colleges have lower quotas as well as the normal upper quotas. Here, we show that a stable matching may not exist and we prove that the problem of determining whether one does is NP-complete in general. The second problem is a different extension in which, as usual, individual colleges have upper quotas, but, in addition, certain bounded subsets of colleges have common quotas smaller than the sum of their individual quotas. Again, we show that a stable matching may not exist and the related decision problem is NP-complete. On the other hand, we prove that, when the bounded sets form a nested set system, a stable matching can be found by generalising, in non-trivial ways, both the applicant-oriented and college-oriented versions of the classical Gale–Shapley algorithm. Finally, we present an alternative view of this nested case using the concept of choice functions, and with the aid of a matroid model we establish some interesting structural results for this case

Strategyproof matching with regional minimum and maximum quotas

This paper considers matching problems with individual/regional minimum/maximum quotas. Although such quotas are relevant in many real-world settings, there is a lack of strategyproof mechanisms that take such quotas into account. We first show that without any restrictions on the regional structure, checking the existence of a feasible matching that satisfies all quotas is NP-complete. Then, assuming that regions have a hierarchical structure (i.e., a tree), we show that checking the existence of a feasible matching can be done in time linear in the number of regions. We develop two strategyproof matching mechanisms based on the Deferred Acceptance mechanism (DA), which we call Priority List based Deferred Acceptance with Regional minimum and maximum Quotas (PLDA-RQ) and Round-robin Selection Deferred Acceptance with Regional minimum and maximum Quotas (RSDA-RQ). When regional quotas are imposed, a stable matching may no longer exist since fairness and nonwastefulness, which compose stability, are incompatible. We show that both mechanisms are fair. As a result, they are inevitably wasteful. We show that the two mechanisms satisfy different versions of nonwastefulness respectively; each is weaker than the original nonwastefulness. Moreover, we compare our mechanisms with an artificial cap mechanism via simulation experiments, which illustrate that they have a clear advantage in terms of nonwastefulness and student welfare

Envy-free matchings with cost-controlled quotas

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 and these costs control the quotas. 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 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. The special hard case is theoretically challenging as well as practically motivated and we present a Linear Programming based algorithm for this case. We also establish the connection of our model with the stable extension problem in an apparently different two-round setting of the stable matching problem [Gajulapalli et al. FSTTCS 2020]. We show that our results in the CCQ setting generalize the stable extension problem.Comment: 23 pages, 7 figures, new results added, some changes in terminology for better clarit

Matchings, Critical Nodes, and Popular Solutions

We consider a matching problem in a marriage instance G. Every node has a strict preference order ranking its neighbors. There is a set C of prioritized or critical nodes and we are interested in only those matchings that match as many critical nodes as possible. Such matchings are useful in several applications and we call them critical matchings. A stable matching need not be critical. We consider a well-studied relaxation of stability called popularity. Our goal is to find a popular critical matching, i.e., a weak Condorcet winner within the set of critical matchings where nodes are voters. We show that popular critical matchings always exist in G and min-size/max-size such matchings can be efficiently computed

Towards a fair distribution mechanism for asylum

It has been suggested that the distribution of refugees over host countries can be made more fair or efficient if policy makers take into account not only numbers of refugees to be distributed but also the goodness of the matches between refugees and their possible host countries. There are different ways to design distribution mechanisms that incorporate this practice, which opens up a space for normative considerations. In particular, if the mechanism takes countries’ or refugees’ preferences into account, there may be trade-offs between satisfying their preferences and the number of refugees distributed. This article argues that, in such cases, it is not a reasonable policy to satisfy preferences. Moreover, conditions are given which, if satisfied, prevent the trade-off from occurring. Finally, it is argued that countries should not express preferences over refugees, but rather that priorities for refugees should be imposed, and that fairness beats efficiency in the context of distributing asylum. The framework of matching theory is used to make the arguments precise, but the results are general and relevant for other distribution mechanisms such as the relocations currently in effect in the European Unio

An 8/5 approximation algorithm for a hard variant of stable marriage

When ties and incomplete preference lists are permitted in the Stable Marriage problem, stable matchings can have different sizes. The problem of finding a maximum cardinality stable matching in this context is NP-hard, even under very severe restrictions on the number, size and position of ties. In this paper, we describe a polynomial-time 8/5-approximation algorithm for a variant in which ties are on one side only and at the end of the preference lists. This variant is motivated by important applications in large scale centralized matching schemes

"Almost-stable" matchings in the Hospitals / Residents problem with Couples

The Hospitals / Residents problem with Couples (hrc) models the allocation of intending junior doctors to hospitals where couples are allowed to submit joint preference lists over pairs of (typically geographically close) hospitals. It is known that a stable matching need not exist, so we consider min bp hrc, the problem of finding a matching that admits the minimum number of blocking pairs (i.e., is “as stable as possible”). We show that this problem is NP-hard and difficult to approximate even in the highly restricted case that each couple finds only one hospital pair acceptable. However if we further assume that the preference list of each single resident and hospital is of length at most 2, we give a polynomial-time algorithm for this case. We then present the first Integer Programming (IP) and Constraint Programming (CP) models for min bp hrc. Finally, we discuss an empirical evaluation of these models applied to randomly-generated instances of min bp hrc. We find that on average, the CP model is about 1.15 times faster than the IP model, and when presolving is applied to the CP model, it is on average 8.14 times faster. We further observe that the number of blocking pairs admitted by a solution is very small, i.e., usually at most 1, and never more than 2, for the (28,000) instances considered

Strategyproof and fair matching mechanism for ratio constraints

We introduce a new type of distributional constraints called ratio constraints, which explicitly specify the required balance among schools in two-sided matching. Since ratio constraints do not belong to the known well-behaved class of constraints called M-convex set, developing a fair and strategyproof mechanism that can handle them is challenging. We develop a novel mechanism called quota reduction deferred acceptance (QRDA), which repeatedly applies the standard DA by sequentially reducing artificially introduced maximum quotas. As well as being fair and strategyproof, QRDA always yields a weakly better matching for students compared to a baseline mechanism called artificial cap deferred acceptance (ACDA), which uses predetermined artificial maximum quotas. Finally, we experimentally show that, in terms of student welfare and nonwastefulness, QRDA outperforms ACDA and another fair and strategyproof mechanism called Extended Seat Deferred Acceptance (ESDA), in which ratio constraints are transformed into minimum and maximum quotas