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A probabilistic approach to the stable matching problem has been identified as an important research area with several important open problems. When considering random matchings, ex-post stability is a fundamental stability concept. A prominent open problem is characterizing ex-post stability and establishing its computational complexity. We investigate the computational complexity of testing ex-post stability. Our central result is that when either side has ties in the preferences/priorities, testing ex-post stability is NP-complete. The result even holds if both sides have dichotomous preferences. On the positive side, we give an algorithm using an integer programming approach, that can determine a decomposition with a maximum probability of being weakly stable. We also consider stronger versions of ex-post stability (in particular robust ex-post stability and ex-post strong stability) and prove that they can be tested in polynomial time.

In kidney exchange programmes (KEP) patients may swap their incompatible donors leading to cycles of kidney transplants. Nowadays, countries try to merge their national patient-donor pools leading to international KEPs (IKEPs). As shown in the literature, long-term stability of an IKEP can be achieved through a credit-based system. In each round, every country is prescribed a "fair" initial allocation of kidney transplants. The initial allocation, which we obtain by using solution concepts from cooperative game theory, is adjusted by incorporating credits from the previous round, yielding the target allocation. The goal is to find, in each round, an optimal solution that closely approximates this target allocation. There is a known polynomial-time algorithm for finding an optimal solution that lexicographically minimizes the country deviations from the target allocation if only $2$-cycles (matchings) are permitted. In practice, kidney swaps along longer cycles may be performed. However, the problem of computing optimal solutions for maximum cycle length $\ell$ is NP-hard for every $\ell\geq 3$. This situation changes back to polynomial time once we allow unbounded cycle length. However, in contrast to the case where $\ell=2$, we show that for $\ell=\infty$, lexicographical minimization is only polynomial-time solvable under additional conditions (assuming P $\neq$ NP). Nevertheless, the fact that the optimal solutions themselves can be computed in polynomial time if $\ell=\infty$ still enables us to perform a large scale experimental study for showing how stability and total social welfare are affected when we set $\ell=\infty$ instead of $\ell=2$.

We introduce partitioned matching games as a suitable model for international kidney exchange programmes, where in each round the total number of available kidney transplants needs to be distributed amongst the participating countries in a "fair" way. A partitioned matching game $(N,v)$ is defined on a graph $G=(V,E)$ with an edge weighting $w$ and a partition $V=V_1 \cup \dots \cup V_n$. The player set is $N = \{1, \dots, n\}$, and player $p \in N$ owns the vertices in $V_p$. The value $v(S)$ of a coalition $S \subseteq N$ is the maximum weight of a matching in the subgraph of $G$ induced by the vertices owned by the players in $S$. If $|V_p|=1$ for all $p\in N$, then we obtain the classical matching game. Let $c=\max\{|V_p| \; |\; 1\leq p\leq n\}$ be the width of $(N,v)$. We prove that checking core non-emptiness is polynomial-time solvable if $c\leq 2$ but co-NP-hard if $c\leq 3$. We do this via pinpointing a relationship with the known class of $b$-matching games and completing the complexity classification on testing core non-emptiness for $b$-matching games. With respect to our application, we prove a number of complexity results on choosing, out of possibly many optimal solutions, one that leads to a kidney transplant distribution that is as close as possible to some prescribed fair distribution.

Matching games naturally generalize assignment games, a well-known class of cooperative games. Interest in matching games has grown recently due to some breakthrough results and new applications. This state-of-the-art survey provides an overview of matching games and extensions, such as $b$-matching games and partitioned matching games; the latter originating from the emerging area of international kidney exchange. In this survey we focus on computational complexity aspects of various game-theoretical solution concepts, such as the core, nucleolus and Shapley value, when the input is restricted to a matching game or one if its variants.

In a multiple partners matching problem the agents can have multiple partners up to their capacities. In this paper we consider both the two-sided many-to-many stable matching problem and the one-sided stable fixtures problem under lexicographic preferences. We study strong core and Pareto-optimal solutions for this setting from a computational point of view. First we provide an example to show that the strong core can be empty even under these severe restrictions for many-to-many problems, and that deciding the non-emptiness of the strong core is NP-hard. We also show that for a given matching checking Pareto-optimality and the strong core properties are co-NP-complete problems for the many-to-many problem, and deciding the existence of a complete Pareto-optimal matching is also NP-hard for the fixtures problem. On the positive side, we give efficient algorithms for finding a near feasible strong core solution, where the capacities are only violated by at most one unit for each agent, and also for finding a half-matching in the strong core of fractional matchings. These polynomial time algorithms are based on the Top Trading Cycle algorithm. Finally, we also show that finding a maximum size matching that is Pareto-optimal can be done efficiently for many-to-many problems, which is in contrast with the hardness result for the fixtures problem.

We study housing markets as introduced by Shapley and Scarf (1974). We investigate the computational complexity of various questions regarding the situation of an agent $a$ in a housing market $H$: we show that it is $\mathsf{NP}$-hard to find an allocation in the core of $H$ where (i) $a$ receives a certain house, (ii) $a$ does not receive a certain house, or (iii) $a$ receives a house other than her own. We prove that the core of housing markets respects improvement in the following sense: given an allocation in the core of $H$ where agent $a$ receives a house $h$, if the value of the house owned by $a$ increases, then the resulting housing market admits an allocation in its core in which $a$ receives either $h$, or a house that $a$ prefers to $h$; moreover, such an allocation can be found efficiently. We further show an analogous result in the Stable Roommates setting by proving that stable matchings in a one-sided market also respect improvement., Comment: 33 pages

To overcome incompatibility issues, kidney patients may swap their donors. In international kidney exchange programmes (IKEPs), countries merge their national patient-donor pools. We consider a recently introduced credit system. In each round, countries are given an initial ``fair'' allocation of the total number of kidney transplants. This allocation is adjusted by a credit function yielding a target allocation. The goal is to find a solution that approaches the target allocation as closely as possible, to ensure long-term stability of the international pool. As solutions, we use maximum matchings that lexicographically minimize the country deviations from the target allocation. We perform, for the first time, a computational study for a {\it large} number of countries. For the initial allocations we use two easy-to-compute solution concepts, the benefit value and the contribution value, and four classical but hard-to-compute concepts, the Shapley value, nucleolus, Banzhaf value and tau value. By using state-of-the-art software we show that the latter four concepts are now within reach for IKEPs of up to fifteen countries. Our experiments show that using lexicographically minimal maximum matchings instead of ones that only minimize the largest deviation from the target allocation (as previously done) may make an IKEP up to 54% more balanced.

After the closure of the schools in Hungary from March 2020 due to the pandemic, many students were left at home with no or not enough parental help for studying, and in the meantime some people had more free time and willingness to help others in need during the lockdown. In this paper we describe the optimisation aspects of a joint NGO project for allocating voluntary mentors to students using a web-based coordination mechanism. The goal of the project has been to form optimal pairs and study groups by taking into the preferences and the constraints of the participants. In this paper we present the optimisation concept, and the integer programming techniques used for solving the allocation problems. Furthermore, we conducted computation simulations on real and generated data for evaluate the performance of this dynamic matching scheme under different parameter settings.

We introduce a new two-sided stable matching problem that describes the summer internship matching practice of an Australian university. The model is a case between two models of Kamada and Kojima on matchings with distributional constraints. We study three solution concepts, the strong and weak stability concepts proposed by Kamada and Kojima, and a new one in between the two, called cutoff stability. Kamada and Kojima showed that a strongly stable matching may not exist in their most restricted model with disjoint regional quotas. Our first result is that checking its existence is NP-hard. We then show that a cutoff stable matching exists not just for the summer internship problem but also for the general matching model with arbitrary heredity constraints. We present an algorithm to compute a cutoff stable matching and show that it runs in polynomial time in our special case of summer internship model. However, we also show that finding a maximum size cutoff stable matching is NP-hard, but we provide a Mixed Integer Linear Program formulation for this optimisation problem., Comment: Extended version of our AAMAS 2020 paper "Summer Internship Matching with Funding Constraints"

In a housing market of Shapley and Scarf, each agent is endowed with one indivisible object and has preferences over all objects. An allocation of the objects is in the (strong) core if there exists no (weakly) blocking coalition. In this paper we show that in the case of strict preferences the unique strong core allocation (or competitive allocation) respects improvement: if an agent's object becomes more attractive for some other agents, then the agent's allotment in the unique strong core allocation weakly improves. We obtain a general result in case of ties in the preferences and provide new integer programming formulations for computing (strong) core and competitive allocations. Finally, we conduct computer simulations to compare the game-theoretical solutions with maximum size and maximum weight exchanges for markets that resemble the pools of kidney exchange programmes.

Kidney exchange programs (KEP's) represent an additional possibility of transplant for patients suffering from end stage kidney disease. If a patient has a willing living donor with whom the patient is not compatible, the pair patient--donor can join a pool of incompatible pairs and, if compatibility between patient and donor in two our more pairs exists, organs can be exchanged between them. The problem can be modeled as an integer program that, in general, aims at finding the pairs that should be selected for transplant such that maximum number of transplants is performed. In this paper we consider that for each patient there may exist a preference order over the organs that he/she can receive, since a patient may be compatible with several donors but may have a better fit over some than over others. Under this setting, the aim is to find the maximum cardinality stable exchange, a solution where no blocking cycle exists. For this purpose we propose three novel integer programming models based on the well-known edge and cycle formulations. These formulations are adjusted for both finding stable and strongly stable exchanges under strict preferences and for the case when ties in preferences may exist. Furthermore, we study a situation when the stability requirement can be relaxed by addressing the trade-off between maximum cardinality versus number of blocking cycles allowed in a solution. The effectiveness of the proposed models is assessed through extensive computational experiments on a wide set of instances.

In a two-sided matching market when agents on both sides have preferences the stability of the solution is typically the most important requirement. However, we may also face some distributional constraints with regard to the minimum number of assignees or the distribution of the assignees according to their types. These two requirements can be challenging to reconcile in practice. In this paper we describe two real applications, a project allocation problem and a workshop assignment problem, both involving some distributional constraints. We used integer programming techniques to find reasonably good solutions with regard to the stability and the distributional constraints. Our approach can be useful in a variety of different applications, such as resident allocation with lower quotas, controlled school choice or college admissions with affirmative action.

We consider the two-sided stable matching setting in which there may be uncertainty about the agents' preferences due to limited information or communication. We consider three models of uncertainty: (1) lottery model --- in which for each agent, there is a probability distribution over linear preferences, (2) compact indifference model --- for each agent, a weak preference order is specified and each linear order compatible with the weak order is equally likely and (3) joint probability model --- there is a lottery over preference profiles. For each of the models, we study the computational complexity of computing the stability probability of a given matching as well as finding a matching with the highest probability of being stable. We also examine more restricted problems such as deciding whether a certainly stable matching exists. We find a rich complexity landscape for these problems, indicating that the form uncertainty takes is significant., Comment: A preliminary version of this paper has been accepted for publication in the proceedings of the 9th International Symposium on Algorithmic Game Theory (SAGT 2016)

Reallocating resources to get mutually beneficial outcomes is a fundamental problem in various multi-agent settings. While finding an arbitrary Pareto optimal allocation is generally easy, checking whether a particular allocation is Pareto optimal can be much more difficult. This problem is equivalent to checking that the allocated objects cannot be reallocated in such a way that at least one agent prefers her new share to his old one, and no agent prefers her old share to her new one. We consider the problem for two related types of preference relations over sets of objects. In the first part of the paper we focus on the setting in which agents express additive cardinal utilities over objects. We present computational hardness results as well as polynomial-time algorithms for testing Pareto optimality under different restrictions such as two utility values or lexicographic utilities. In the second part of the paper we assume that agents express only their (ordinal) preferences over single objects, and that their preferences are additively separable. In this setting, we present characterizations and polynomial-time algorithms for possible and necessary Pareto optimality.

We generalize two well-known game-theoretic models by introducing multiple partners matching games, defined by a graph $G=(N,E)$, with an integer vertex capacity function $b$ and an edge weighting $w$. The set $N$ consists of a number of players that are to form a set $M\subseteq E$ of 2-player coalitions $ij$ with value $w(ij)$, such that each player $i$ is in at most $b(i)$ coalitions. A payoff vector is a mapping $p: N \times N \rightarrow {\mathbb R}$ with $p(i,j)+p(j,i)=w(ij)$ if $ij\in M$ and $p(i,j)=p(j,i)=0$ if $ij\notin M$. The pair $(M,p)$ is called a solution. A pair of players $i,j$ with $ij\in E\setminus M$ blocks a solution $(M,p)$ if $i,j$ can form, possibly only after withdrawing from one of their existing 2-player coalitions, a new 2-player coalition in which they are mutually better off. A solution is stable if it has no blocking pairs. We give a polynomial-time algorithm that either finds that a given multiple partners matching game has no stable solution, or obtains a stable solution for it. We characterize the set of stable solutions of a multiple partners matching game in two different ways and show how this leads to simple proofs for a number of known results of Sotomayor (1992,1999,2007) for multiple partners ssignment games and to generalizations of some of these results to multiple partners matching games. We also perform a study on the core of the corresponding cooperative game, where coalitions of any size may be formed. In particular we show that the standard relation between the existence of a stable solution and the non-emptiness of the core, which holds in the other models with payments, is no longer valid for our (most general) model. We also prove that the problem of deciding if an allocation belongs to the core jumps from being polynomial-time solvable for $b\leq 2$ to NP-complete for $b\equiv 3$.

We develop Integer Programming (IP) solutions for some special college admission problems arising from the Hungarian higher education admission scheme. We focus on four special features, namely the solution concept of stable score-limits, the presence of lower and common quotas, and paired applications. We note that each of the latter three special feature makes the college admissions problem NP-hard to solve. Currently, a heuristic based on the Gale-Shapley algorithm is being used in the application. The IP methods that we propose are not only interesting theoretically, but may also serve as an alternative solution concept for this practical application, and also for other ones.