Cutoff stability under distributional constraints with an application to summer internship matching
aa r X i v : . [ c s . G T ] F e b Cutoff stability under distributional constraints withan application to summer internship matching ⋆ Haris Aziz · Anton Baychkov · P´eter Bir´o
Received: date / Accepted: date
Abstract
We introduce a new two-sided stable matching problem that de-scribes the summer internship matching practice of an Australian university.The model is a case between two models of Kamada and Kojima on matchingswith distributional constraints. We study three solution concepts, the strongand weak stability concepts proposed by Kamada and Kojima, and a new onein between the two, called cutoff stability. Kamada and Kojima showed that astrongly stable matching may not exist in their most restricted model with dis-joint 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 intern-ship problem but also for the general matching model with arbitrary heredityconstraints. We present an algorithm to compute a cutoff stable matching andshow that it runs in polynomial time in our special case of summer internshipmodel. However, we also show that finding a maximum size cutoff stable match-ing is NP-hard, but we provide a Mixed Integer Linear Program formulation forthis optimisation problem.
Keywords stable matching, distributional constraints, cutoff scores, NP-hardness, integer programming
Centralized two-sided matching market algorithms have received immensesuccess in several application domains, including matching students to schools, ⋆ This paper is the extension of our conference paper [Aziz et al., 2020]. The main additionis Section 5 on our new solution concept of cutoff stability and an algorithm to achieve it.Haris AzizUNSW Sydney and Data61 CSIRO, AustraliaE-mail: [email protected] BaychkovUniversity of Sydney, AustraliaE-mail: [email protected]´eter Bir´oCentre for Economic and Regional Studies, HungaryE-mail: [email protected] Haris Aziz, Anton Baychkov, and P´eter Bir´o residents to hospitals, and projects to workers. We present a novel matchingmarket model that we refer to as the summer internship matching market. Themodel captures the matching of student applicants to projects proposed bysupervisors in an internship program. A distinctive feature of the model is thatin order for an applicant to be assigned to any project, a certain amount ofmoney needs to be contributed from the project supervisors’ funds.Our problem is inspired by summer intern research programs in Australia. Itis common for undergraduate students to undertake research projects over thesummer. Each project is supervised by one or more members of the faculty, withmany offering multiple projects. Even though the projects may be discounted bycontributions from the faculty, supervisors are often required to contribute to thefunding of these positions, from their personal research budget. Alternatively,they could be constrained by the amount of time they can allocate to supervision.These supervisor-side constraints mean that not all projects can be funded.Just as the standard hospital resident matching models do not just applyto the matching of doctors to hospitals [Roth, 2008], our model also doesnot just apply to matching of student interns. It applies to any two-sidedmatching model in which very widely applicable budget requirements andbudget constraints are involved. For example, our problem also models hiringscenario in which different teams have their own budgets and they want to hireemployees. Certain employee roles could sit across various teams. In that case,multiple teams can pool in their money to fund joint positions.The budget constraints that we consider lead to interesting research chal-lenges. Firstly, applying standard matching algorithms such as the DeferredAcceptance Algorithm does not work as it do not deal with complex feasibilityconstraints. More critically, as we will discuss, there may exist no feasiblematching that satisfies stability as considered in seminal papers on matching(see, e.g., Gale and Shapley [1962b] and Abdulkadiro˘glu and S¨onmez [2003a]).1.1 ContributionsIn this paper, we formalize the summer internship problem with budgets,abbreviated as SIP. It falls within the class of models that matches applicants,projects, and supervisors. The main characteristic of our problem is thatsupervisors have budgets that they can spread across their projects. Ourmodel is more general than the widely-used hospital-resident matching modelin which the hospitals are partitioned into regions and regions have uppercapacities [Kamada and Kojima, 2015], a model that we abbreviate as REG.On the other hand, our model is a special case of the matching model underdistributional constraints of Kamada and Kojima [2017b], where the feasibilityof a matching is monotone in the number of applicants matched, the propertycalled heredity by Goto et al. [2017], that we abbreviate as HER.First, we study the concept of strong stability proposedin Kamada and Kojima [2015], where the authors showed that such solu-tion may not exist for REG. Here we prove that the problem of checking the For an overview of real-life matching markets, please see , and a recent survey [Bir´o, 2017].itle Suppressed Due to Excessive Length 3 existence of a strongly stable matching is NP-hard for REG (and thus also forany settings that contain REG as a special case, such as SIP and HER).Then we study weak stability, also introduced in Kamada and Kojima [2015]for REG and then studied in Kamada and Kojima [2017b] for HER. In the con-ference version of our paper [Aziz et al., 2020] we provided a strongly polynomialalgorithm for computing a weakly stable matching for our intermediate SIPmodel, based on the algorithm of Kamada and Kojima [2017b] for HER. Ouralgorithm uses as an oracle an algorithm based on network flows to repeatedlycheck whether a given matching is feasible or not. In this extended version of ourconference paper, we strengthen this result by presenting a general algorithmthat returns a matching satisfying cutoff stability. Fair matchings are exactlythose matchings that can be induced by a set of cutoff scores. Meanwhile, cutoffstable matchings are those that are induced by minimal cutoffs, i.e., where thedecrease of any cutoff would make the induced matching infeasible. We showthat cutoff stability is an intermediate notion between weak and strong stability,and the computation of a cutoff stable matching is always possible with ouralgorithm for HER. Since HER covers many well-studied settings includingrefugee matching (see, e.g., [Aziz et al., 2018, Delacr´etaz et al., 2016]), our re-sult have wide-applicability. We apply the algorithm in the context of SIP andshow that it can be implemented to run in polynomial time. However, we alsoshow that finding a maximum size cutoff stable matching is NP-hard even forREG (implying that is also NP-hard for SIP and HER). However, we formulatea Mixed Integer Linear Program (MILP) for finding such an optimal solutionfor SIP. Many of our general structural results for the HER model. Unlessspecified otherwise, we will assume that our formal statements apply to HER.Finally, we provide a normative criterion for egalitarian ex-post allocationof supervisor funding among projects, and a polynomial-time algorithm to findan egalitarian allocation. Combined with our polynomial-time algorithm forfinding a weakly stable matching, we present a compelling approach for findinga desirable solution for the summer internship problem that appeals to bothstability and fairness requirements.1.2 Related WorkThe literature on two-sided matching was inspired by the seminal paperof Gale and Shapley [1962a] who considered matching markets that matchstudents to schools and hospitals to residents. The paper has spawned richermatching models and resulted in new algorithmic work (see, e.g., Manlove[2013]). Our work is an extension of these models and falls in the generalumbrella of matching markets with various kinds of distributions constraints(see, e.g., Aziz et al. [2018], Kamada and Kojima [2017a], Fragiadakis et al.[2016], Fragiadakis and Troyan [2017], Kurata et al. [2017]).Our concept of budget-feasibility is a type of feasibility constraint, asdefined by Kamada and Kojima [2017b] and thus our problem falls under the See, for example, the discussion by Kamada and Kojima [2020] who point out thateven intra-project heredity constraints capture problems including college admissions withstudents with disabilities; refugee match, and daycare allocation. Haris Aziz, Anton Baychkov, and P´eter Bir´o umbrella of matching under feasibility constraints. Thus, the notions of strongand weak stability studied by Kamada and Kojima [2017b] also apply in ourmodel. In our paper, we focus on computational results such as establishingNP-completeness or polynomial-time solvability of stable matchings. Ourmodel also has the additional dimension of budget allocations for which weexplore fairness concepts as well as algorithms to divide the budget in anegalitarian manner. We also propose an intermediate concept called cutoffstability and prove that it can be achieved for the general matching model witharbitrary heredity constraints. Cutoff stability is a new notion for the HERmodel, though similar notions have been studied for different stable matchingmodels, such as the choice function model of Fleiner and Jank´o [2014]. Ourconcept of cutoff stability defined for general feasibility constraints is similarin spirit to the within-type envy-freeness concept of Echenique and Yenmez[2015] in the context of school choice with affirmative action.Goto et al. [2017] also considered two-sided matching under general feasi-bility constraints that satisfy the heredity property. Their main contributionis proposing an algorithm called Adaptive Deferred Acceptance that satisfiesstrategy-proofness, non-wastefulness, and a fairness property (that is weakerthan the weak stability concept of Kamada and Kojima [2017b] and hence alsocutoff stability). Kamada and Kojima [2020] also consider heredity constraintsthat apply to individual schools/hospitals.Our model bears some similarities with the hospital-resident matchingproblem with regional constraints [Kamada and Kojima, 2015, Biro et al.,2010, Goto et al., 2016, Aziz et al., 2019, Kamada and Kojima, 2018]. In theseregion-based problems, at most a certain number of students can be selectedfrom given regions. On the other hand, in the summer internship problem, a su-pervisor’s budget is divisible and can be spread partially over all of her projects.If the regions are disjoint (as studied by Kamada and Kojima [2015]), then theregion-based model is a special case of our model. The general setting with regionconstraints has not seen many positive results, and the more constrained hierar-chical regions as studied by Goto et al. [2016] and Kamada and Kojima [2018]can neither replicate, nor be replicated by a set of supervisors. In particular, theconcept of stability with regional priorities proposed by Kamada and Kojima[2018] cannot be directly applied to our setting. Our concept of cutoff stabilityis an alternative intermediate notion of stability and it can be applied to awider range of settings (those subsumed under the HER model).Abraham et al. [2007] considered a different model for student-project allo-cation. Notable differences in their model include: (1) no project has multiplesupervisors, (2) each supervisor has a universal priority list over students whichis not project specific, and (3) a supervisor has a rigid capacity constraint for thenumber of projects to supervise. Our model allows supervisors to explore moreefficient outcomes by pooling in their budgets to host a student. The universalpriority list of each supervisor makes the model of Abraham et al. [2007] muchmore restricted and different from our model.Two other recent models are similar to our setting. Goto et al. [2017] intro-duce the Student-Project-Room matching problem. Rooms are indivisible, andat most one room can be allocated to each project. Ismaili et al. [2018] extendthis to a more general Student-Project-Resource allocation problem. Resources itle Suppressed Due to Excessive Length 5 are still indivisible, but there is no longer a restriction imposed on the numberof resources that can be allocated to each project. Our model is distinct fromboth of these, as it allows the resources (in our case, supervisor budgets) to bedivisible. This divisibility allows for better computational results. For instance,verifying the feasibility of a matching, and finding a weakly stable (and thusnon-wasteful) matching can be done in polynomial time.There is also work on matching with budget constraints (see, e.g.,Kawase and Iwasaki [2017, 2018], Ismaili et al. [2019]). The models consideredin these papers are different in several respects. For example, hospitals haveadditive utilities and each hospital gives monetary compensation to doctors.Regarding the LP descriptions and IP techniques for two-sided stablematching problems, Ba¨ıou and Balinski [2000] gave the first description onthe stable admissions polytope. Integer programming techniques have beenused later for college admissions with special features [ ´Agoston et al., 2016],stable project allocation under distributional constraints [ ´Agoston et al.,2018], the hospital–resident problem with couples [Bir´o et al., 2014], and ties[Kwanashie and Manlove, 2014, Delorme et al., 2019].
First, we introduce our summer internship matching model (SIP), we showthat the checking the feasibility of a matching is polynomial time tractable,and finally we also show the relation of our model to the REG and HERmodels of Kamada and Kojima described in Kamada and Kojima [2015] andKamada and Kojima [2017b], respectively.2.1 ModelLet A be a finite set of applicants, and P a finite set of projects. Each applicant a ∈ A has a strict preference list ≻ a that ranks the subset of projects that a finds acceptable. Each project p ∈ P has a preference list ≻ p over the subset ofapplicants that p finds acceptable, and a maximum capacity c p .Furthermore, let S denote the set of project supervisors. Each supervisor s ∈ S has a list of projects P s that they supervise, and is endowed with abudget (e.g. quantity of funds) q s that they can allocate among those projects.We assume that these budgets are infinitely divisible and that each applicantrequires one unit of funding. Further, we assume that these endowments arepublicly known, and thus supervisors cannot strategise by misreporting theirbudgets. Additionally, denote the list of supervisors for project p by S p .We say that an applicant a ∈ A is matched to project p ∈ P if ( a,p ) ∈ M . A matching M is a subset of A × P that satisfies the following conditions: – Each applicant is matched to at most one project (for all a ∈ A , |{ ( a,p ) ∈ M : p ∈ P }| ≤ a finds the project they are matched to acceptable. – The number of applicants matched to any project does not exceed thatproject’s capacity (for all p ∈ P , |{ ( a,p ) ∈ M : a ∈ A }| ≤ c p ), and p finds allapplicants matched to it acceptable. Haris Aziz, Anton Baychkov, and P´eter Bir´o
We use M ( a ) to refer to the projectthat applicant a is matched to ( M ( a ) = ∅ if a is unmatched). Meanwhile, M ( p )denotes the set of applicants matched to p . Let x s,p be the amount of funds a supervisor s allocates to project p . We call amatching M feasible (or supervisor-feasible) if there exists a set { x s,p } s ∈ S,p ∈ P s that satisfies the following conditions: – x s,p ≥ s ∈ S,p ∈ P s – Every project receives one unit of funding for each applicant matched to it: P s ∈ S p x s,p = | M ( p ) | for all p ∈ P – Supervisors do not exceed their endowment: P p ∈ P s x s,p ≤ q s for all s ∈ S We call any set { x s,p } s ∈ S,p ∈ P s that satisfies the above conditions for matchingM a feasible funding allocation . The mathematical model presented exactlycaptures the student research internship program in our university: eachsupervisor can be part of multiple internship project proposals but does notnecessarily have the funding to contribute to all of them. Example 1 (Summer Internship Problem)
Consider the following instance of thesummer internship problem with 2 applicants 2 supervisors, and 2 projects. A = { a ,a } ≻ a : p ,p ≻ a : p ,p P = { p ,p } ≻ p : a ,a ≻ p : a ,a S = { s ,s } P s = { p ,p } P s = { p } q s = 0 . q s = 0 . c p = c p = 1The only three feasible matchings are the empty matching and the twomatchings in which some applicant is matched to project p . The reason no onecan be matched to project p is that p has a sole supervisor s who does nothave sufficient funding to fund p . On the other hand, the combined funding of s and s is more than 1 for project p so p can be funded. A feasible fundingallocation x is where s contributed half of the budget of project p and s contributes the rest: x s ,p = 0 . x s ,p = 0 . Define the funding flow graph G M associated with a matching M as follows: – V ( G M ) = { s ∗ }∪ S ∪ P ∪{ t ∗ } , where s ∗ is the source, and t ∗ is the sink – Arcs ( s,p ), for all supervisor-project pairs where p ∈ P s with capacity ∞ – Arcs ( s ∗ ,s ), for all s ∈ S , each with capacity q s – Arcs ( p,t ∗ ), for all p ∈ P , each with capacity | M ( p ) | Theorem 1
The feasibility of a matching can be checked in polynomial time O ((max {| S | , | P |} ) for the summer-internship problem. For an overview of network flows, see Ahuja et al. [1993].itle Suppressed Due to Excessive Length 7
Proof
Our first claim is that a matching M is feasible if and only if G M admitsa feasible s ∗ - t ∗ flow of size | M | .Suppose M is feasible. Define a flow f on G M as follows: – f ( s,p ) = x s,p , ∀ s ∈ S,p ∈ P s – f ( s ∗ ,s ) = P p ∈ P s x s,p , ∀ s ∈ S – f ( p,t ∗ ) = P s ∈ S p x s,p , ∀ p ∈ P It is easy to see that this is a feasible flow of size | M | . Now suppose that G M admits a feasible flow f of size | M | . Set x s,p = f ( s,p ), ∀ s ∈ S,p ∈ P s . We can thenshow that this { x s,p } satisfies the conditions of feasibility.Now that we have established the claim, we use the fact that the maximumflow problem can be solved in O ( V ) time, using for instance, the algorithmproposed by Malhotra et al. [1962]. V ( G M ) = | S | + | P | +2 and, given a matchingM, G M can be constructed in O (( | S | + | P | ) ) time. We can check whether M isfeasible in O (max {| S | , | P |} ) time by computing a maximum flow and verifyingwhether it equals | M | . ⊓⊔ Note that by the integer property of the network flow problem, if all thecapacities of the supervisors are integer and the flow is feasible then an integerfunding allocation exists.2.3 Connection with the models by Kamada and KojimaOur model satisfies the heredity property of Kamada and Kojima [2017b] andGoto et al. [2017], that says that if a matching is feasible then it remains feasibleif the numbers of applicants matched to each project weakly decreases. Moreformally, their matching model under distributional constraints that satisfyheredity (HER), can be represented by a feasibility function f : M → { , } that satisfies the following condition: – f ( ∅ ) = 1 and, for any two matchings M and M ′ , if | M ′ ( p ) | ≤ | M ( p ) | ∀ p ∈ P then f ( M ) = 1 ⇒ f ( M ′ ) = 1.A matching M is feasible if and only if f ( M ) = 1. Note that the feasibilityconstraints are ‘anonymous’ in the sense that they do not depend on theidentity of the applicants matched but only on their quantity.Kamada and Kojima [2015] studied a basic model with regional upper quotas,where the regions form a partition of the set of hospitals (REG). We can representthis in our model by replacing each region with a supervisor, endowed with abudget equal to that region’s capacity, and supervising a set of projects thatcorrespond to the hospitals in that region.Thus, REG is a special case of SIP, and SIP is a special case of HER.Therefore, every hardness result for REG implies the same hardness result forSIP and HER, and any easiness result for HER implies the same result for SIP,and an easiness result for SIP also holds for REG. Haris Aziz, Anton Baychkov, and P´eter Bir´o M is fair , if for every pair ( a,p ) / ∈ M , p ≻ a M ( a ) impliesthat a ′ ≻ p a for every a ′ ∈ M ( p ). This is a basic property for both strong andweak stability by Kamada and Kojima [2017b], and so also for our intermediatenotion of cutoff stability.In the classical Gale-Shapley model [Gale and Shapley, 1962a] a matching isstable if and only if it is fair and non-wasteful, where non-wastefulness meansthat there exists no project p with unfilled capacity and applicant a where p ≻ a M ( a ). In our context we can define different non-wastefulness notionsleading to weak and strong stability and a new intermediate stability concept,called cutoff stability. We will define and analyze these stability concepts inSections 3, 4 and 5, respectively.The concept of cutoff scores is closely related to fair matchings. Let d : P → [0 , ,... | A | + 1] be the cutoff score function, where d ( p ) is the cutoff atproject p . Without loss of generality we assume that each project p assigns ascore to each applicant a in accordance with its preference list, that is a has score | A |− k +1 if she is ranked k th by project p . Given cutoff scores d , applicant we saythat a is admissible to project p if her score achieves the cutoff. Cutoff scores d in-duce matching M , if every applicant is matched to the best project of her prefer-ence where she is admissible. The following observation is well-known (see, e.g.Lemma 3 in Fleiner and Jank´o [2014]), for completeness we give a short proof.
Proposition 1
A matching is fair if and only if it is induced by some cutoffscores.Proof
A matching induced by cutoffs is always fair by definition, since p ≻ a M ( a ) implies that a could not reach the cutoff score at p so everyoneassigned to p has a higher rank than her. In the other direction let M be a fairmatching. For every project p we set the cutoff d ( p ) to be equal to the score ofthe lowest ranked applicant in M ( p ). It is easy to see that d induces M . ⊓⊔ In the following sections we will consider the strong and weak stabilityconcepts of Kamada and Kojima [2017b], and subsequently introduce our newsolution concept of cutoff stability.
An applicant-project pair ( a,p ) is a blocking pair for matching M if: – applicant a prefers project p to the project they are currently matched to: p ≻ a M ( a ), and – either: – p is under capacity and finds a acceptable: | M ( p ) | < c p and a ≻ p ∅ ; or In the school choice literature this property is also called as justified envy-freeness , see,e.g., Abdulkadiro˘glu and S¨onmez [2003b]. In fact, in many college admission schemes only the cutoff scores are announced,the induced matchings are obvious for the participants involved, see, e.g. the cases ofHungary [ ´Agoston et al., 2016] and Australia [Artemov et al., 2017] and [Guillen et al., 2020].itle Suppressed Due to Excessive Length 9 – p prefers a to one of its currently matched applicants: ∃ a ′ ∈ M ( p ) suchthat a ≻ p a ′ Definition 1 (Strong Stability)
We call a matching M strongly stable iffor any blocking pair ( a,p ) for matching M , the following two conditions aresatisfied: – a ′ ≻ p a for all applicants a ′ ∈ M ( p ) – The matching M ′ = ( M ∪{ ( a,p ) } ) \{ ( a,M ( a )) } is not feasible.The first condition implies that we only allow the existence of blocking pairs in-volving a project p that is under its maximum capacity c p . The second conditionimplies that even if slot is free at project p , adding applicant a to project p willresult in a distributional constraint being violated so that M ′ = ( M ∪{ ( a,p ) } ) \{ ( a,M ( a )) } is not feasible. Thus, we allow blocking pairs that cannot be satisfiedwithout violating our feasibility constraint to exist in a strongly stable matching.Alternatively, we can define strong stability in terms of fairness and strongnon-wastefulness, as follows. We say that matching M is strongly non-wasteful if for every pair ( a,p ) / ∈ M , p ≻ a M ( a ) implies that ( M ∪{ ( a,p ) } ) \{ ( a,M ( a )) } isnot feasible or project p is at capacity. It is easy to see that M is strongly stableif and only if it is fair and strongly non-wasteful.Our first observation is that for our problem, a strongly stable matching doesnot exist. This follows from the observation that our model is more general thanthe hospital resident setting with disjoint regions [Kamada and Kojima, 2015].We provide an adaptation of an example (Example 1) of Kamada and Kojima[2017b] for the sake of completeness. Example 2 (A strongly stable matching does not necessarily exist for the summerinternship problem)
Consider the following instance of the summer internship problem: A = { a ,a } ≻ a : p ,p ≻ a : p ,p P = { p ,p } ≻ p : a ,a ≻ p : a ,a S = { s } P s = { p ,p } q s = c p = c p = 1It is easy to see that | M | ≤
1. If both applicants are unmatched then ( a ,p )forms a blocking pair. Suppose without loss of generality that a is matchedin the stable matching. If M = { ( a ,p ) } then ( a ,p ) is a blocking pair thatdoes not satisfy the second condition of strong stability. If M = { ( a ,p ) } then( a , p ) is a blocking pair that does not satisfy the first condition of strongstability. Thus, every feasible matching admits a blocking pair that is notpermitted under strong stability, and is therefore not strongly stable.Below we show that the problem of deciding the existence of a strongly stablematching is NP-complete. We reduce for Restricted MAX-SMTI, the problemof deciding whether there exists a complete stable matching for the stablemarriage problem with incomplete lists and ties under the restriction that thepreferences of the men are strict, and the preference list of each woman is eitherstrict or consists solely of a tie of length two [Manlove et al., 2002]. First, weintroduce an instance that will serve as the core of the construction imitatingan indifferent woman. Example 3 (An instance with two strongly stable matchings, covering differentagents)
The instance consists of a preference cycle involving four applicants andfour projects as follows. A = { a ,a ,a ,a }≻ a : p ,p ≻ a : p ,p ≻ a : p ,p ≻ a : p ,p P = { p ,p ,p ,p }≻ p : a ,a ≻ p : a ,a ≻ p : a ,a ≻ p : a ,a S = { s ,s ,s } P s = { p ,p } P s = { p } P s = { p } q s = q s = q s = 1 c p = c p = c p = c p = 1One can check that there are two strongly stable matchings: M = { ( a ,p ) , ( a ,p ) , ( a ,p ) } and M = { ( a ,p ) , ( a ,p ) , ( a ,p ) } . First, let usshow the stability of M (the argument is similar for M by symmetry). Theonly blocking pairs for M are ( a , p ) and ( a , p ), but adding any of thesepairs to M would make the matching infeasible. To show that M and M arethe only strongly stable matchings we observe first that a has to be matchedto p , and a has to be matched to p by symmetric reasons (so we prove onlythe former fact). Note that a cannot be unmatched, as she would block thematching with p . Suppose now for a contradiction that a is matched to p .Then p must be unfilled by feasibility, thus a has to be matched to p and a remains unmatched and so forms a blocking pair with p , a contradiction. Sowe showed that ( a ,p ) and ( a ,p ) must be contained in any strongly stablematching. Finally we shall note that both p and p cannot remain unfilled, as( a ,p ) would form a blocking pair. Therefore, in a strongly stable matching wemust also have either ( a ,p ) or ( a ,p ), resulting in M and M , respectively.Note that this example also shows that the Rural Hospitals’ Theorem of Roth[1986] does not hold for strongly stable matchings, which says that always thesame applicants are matched in every stable solution in a many-to-one college ad-missions problem. Actually, the same two matchings are also the only weakly sta-ble matchings, and thus the only cutoff stable matchings, by the same argument.Using the example as a gadget, we will show that deciding whether an instanceof our problem has a strongly stable solution is an NP-complete problem. Theorem 2
For the Summer Internship Problem (SIP), checking the existenceof a strongly stable matching is NP-complete, even if all the supervisors havecapacity one and each is responsible for at most two distinct projects.Proof
Given a solution, we can check whether it is strongly stable by consideringeach potential blocking pair in polynomial time, so the problem is in NP. Forproving NP-hardness, we reduce from the special version of the MAX-SMTIproblem [Manlove et al., 2002]. Here, we are given an instance of a stablemarriage problem with incomplete lists and ties, where U = { u ,u ,...,u n } isthe set of men, each having a strict preference list over the women acceptablefor him, and W = W s ∪ W t is the set of women, where W s = { w ,...,w k } are itle Suppressed Due to Excessive Length 11 the women with strict preference lists and W t = { w k +1 ,...,w n } are the women,where each has a single tie of length two in her preference list (i.e., she finds twomen acceptable, and she is indifferent between them). A matching is said tobe weakly stable if it is not blocked by a pair where both parties strictly prefereach other to their current partners. Let us denote the restricted instance ofSMTI by I , where the problem of deciding the existence of a complete weaklystable matching is NP-complete [Manlove et al., 2002].We construct the corresponding internship problem I ′ as follows. For everyman u i we create a project p i with a single supervisor s i each with capacity one.We denote this set of projects by P u . For every woman in w j ∈ W s we create anapplicant a j , and we denote this set of applicants by A s . Here the preferencelist of a j over the projects in I ′ is identical to the preference list of w j overthe men in I . Now, for each woman w j ∈ W t , we introduce a gadget G j that isidentical to the instance in Example 3, so let G j be constructed as follows. A j = { a j ,a j ,a j ,a j }≻ a j : p j ,p j ≻ a j : p j ,p j ≻ a j : p j ,p j ≻ a j : p j ,p j P j = { p j ,p j ,p j ,p j }≻ p j : a j ,a j ≻ p j : a j ,a j ≻ p j : a j ,a j ≻ p j : a j ,a j S = { s j ,s j ,s j } P s j = { p j ,p j } P s j = { p j } P s j = { p j } q s j = q s j = q s j = 1 c p j = c p j = c p j = c p j = 1Furthermore, if w j ∈ W t had preference list [ u i ,u i ] then we append p i to theend of the preference list of a j and we append p i to the end of the preferencelist of a j . Likewise, we adjust the preference list of p i ∈ P u as follows: for every w j ∈ W s , we replace w j with a j , whilst if w j ∈ W t then we replace w j witheither a j or a j according to whether u i was the first or the second man in the tieof w j . Finally, we add a gadget G ∗ , which is a copy of the unsolvable instancein Example 2, together with an additional applicant a ∗ who accepts one of theprojects, say p ∗ , in G ∗ . Let a ∗ be the most preferred applicant by the projects in G ∗ , so including this applicant in G ∗ will turn the instance solvable by assigning a ∗ to p ∗ and leaving the other applicants in G ∗ unmatched. To link G ∗ ∪ a ∗ withthe rest of I ′ we put all the projects in P u ahead of p ∗ in the preference list of a ∗ and we also append a ∗ to the end of the preference list of each p j ∈ P u .We will show that I has a complete weakly stable matching if and only if I ′ has a strongly stable matching. Let us suppose first that M is a complete stablematching of I , we construct a strongly stable matching M ′ of I ′ as follows. Forevery ( u i ,w j ) ∈ M , where w j ∈ W s we simply add the corresponding pair ( p i ,a j )to M ′ . Suppose now that ( u i ,w j ) ∈ M , where w j ∈ W t and u i is the first man inthe single tie of w j , so p i is linked with a j in I ′ . Let then include ( p i ,a j ) in M ′ ,together with the corresponding strongly stable matching M of G j that leaves a j unmatched, namely M j = { ( a j ,p j ) , ( a j ,p j ) , ( a j ,p j ) } . Similarly, if u i is thesecond man in the tie of w j , then we include ( p i ,a j ) in M ′ , together with thecorresponding strongly stable matching M of G j that leaves a j unmatched,namely M j = { ( a j , p j ) , ( a j , p j ) , ( a j , p j ) } . Finally, we add ( a ∗ , p ∗ ) to M ′ . Thestrong stability of M ′ is implied by the following reasons. Observe first that a ∗ cannot block with any project in P u , since all of these projects are assignedwith better applicants. The matching in gadget G ∗ is also stable internally, aswell as in each gadget G j . Finally, the weak stability of M implies the lack ofthe blocking pairs of form ( a j ,p i ) for M ′ , where a j ∈ A s and p i ∈ P u .In the other direction, let us assume that M ′ is a strongly stable matchingfor I ′ and we construct a complete weakly stable matching M for I as follows.First, we note that a ∗ must be matched to p ∗ in M ′ , as otherwise the separatedgadget G ∗ would cause instability. But if a ∗ is matched to p ∗ then each project p i ∈ P u must be assigned to an applicant better than a ∗ . Now, if ( p i ,a j ) ∈ M ′ for a j ∈ A s then we add ( u i ,w j ) to M (where w j ∈ W s ), and if ( p i ,a j ) or ( p i ,a j )belongs to M ′ then we add ( u i ,w j ) to M (where w j ∈ W t ). It is obvious that M is a complete matching in I , its weak stability is implied by the strong stabilityof M ′ as follows. Suppose for a contradiction that a pair ( u i , w j ) would beblocking for M (where w j ∈ W s ), then the corresponding pair ( p i , a j ) wouldalso block M ′ , a contradiction. This completes the proof. ⊓⊔ Our hardness result is strong because it holds for a very restricted settingand also implies NP-completeness for the setting of Kamada and Kojima [2015]that concerns disjoint regions (REG), even if at most two hospitals belong toeach region. The complexity of existence of strongly stable matching was alsoan open problem for the model of Kamada and Kojima [2015]. Corollary 1
For REG, checking the existence of a strongly stable matching isNP-complete.
Furthermore, this case can also occur when one hospital has a commonupper quota for two different types of jobs, e.g., daytime and night shifts, orsurgical and medical internship positions. Another motivating example is theHungarian college admission scheme, where students can be admitted to aprogramme under two contracts, state-funded and privately-funded, and thereis a common upper bound on them [Biro et al., 2010].
In view of the non-existence and NP-completeness of checking the existenceof strongly stable matchings, one can consider a weaker stability criterion.Kamada and Kojima [2017b] proposed a weak stability concept for a settingthat does not concern budgets but which has an abstract feasibility indicatorfunction for any given matching. We present the definition in our terminologyof applicants and projects. Most of the computational hardness results for distributional constraints concernoverlapping regions (see, e.g. Goto et al. [2016]).itle Suppressed Due to Excessive Length 13
Definition 2 (Weak Stability)
We call a matching M weakly stable if for anyblocking pair ( a,p ) for matching M , the following two conditions are satisfied. – a ′ ≻ p a for all applicants a ′ ∈ M ( p ) – M ∪{ ( a,p ) } is not feasible.Note the similarity in the definition of strong stability and weak stability.The only difference is that in the second condition, applicant a can have twocontracts: one with project M ( a ) and another with the project p she is blockingwith. One way to see this is that in order for applicant a to block with p , it mustsign the contract with p before it opts to annul its match with project M ( a ).We call any blocking pair that satisfies these condition permitted underweak stability . Note that due to the second condition, any empty matchingneed not be weakly stable.Alternatively, we can also define weak stability in terms of fairness and weaknon-wastefulness. We say that matching M is weakly non-wasteful if for everypair ( a,p ) / ∈ M , ≻ a M ( a ) implies that M ∪{ ( a,p ) } is not feasible or project p is atcapacity. It is easy to see that M is weakly stable if and only if it is fair and weaklynon-wasteful, for a short proof see Proposition 1 in Kamada and Kojima [2017b].We say that p is unconstrained for a feasible matching M if for any ( a,p ) / ∈ M , M ∪ { ( a,p ) } remains feasible, i.e., when one more applicant can be added toproject p by keeping the solution feasible. Proposition 2
A matching M is weakly stable if and only if it is induced bycutoff scores d such that for every unconstrained project p , d ( p ) = 0 .Proof Following Proposition 1, we only need to show that the condition of everyunconstrained project having zero cutoff is equivalent to weak non-wastefulness.Suppose first that M is weakly non-wasteful. This means that there cannotexist an unconstrained project p and an applicant a such that p ≻ a M ( a ), soindeed for all unconstrained project we can set the cutoff to be zero, and forthe constrained projects we just set the cutoff to be equal to the score of lowestranked applicant assigned. In the other direction, if we have cutoff scores satis-fying that every unconstrained project has cutoff score zero, then there cannotexist an applicant a such that p ≻ a M ( a ) for an unconstrained project p . ⊓⊔ In the conference version of our paper [Aziz et al., 2020] we presenteda polynomial-time algorithm that always returns a weakly stable match-ing based on the algorithm by Kamada and Kojima [2017b] (AppendixB.3 [Kamada and Kojima, 2017b]) In this extended version of our conferencepaper, we strengthen this result by showing that a so-called cutoff stablematching can also be computed efficiently for SIP, by designing an algorithmthat also works for HER.
In this section, we discuss cutoff stability that applies to any matching problemwith feasibility constraints.We say that M is cutoff non-wasteful if M is non-wasteful and for every pair( a,p ) / ∈ M , p ≻ a M ( a ) implies that either – M ∪{ ( a,p ) \ ( a,M ( a )) } is not feasible, or – there exists another applicant a ′ / ∈ M ( p ), such that a ′ ≻ p a , p ≻ a ′ M ( a ′ ) and( M ∪{ ( a ′ ,p ) } ) \{ ( a ′ ,M ( a ′ )) } is not feasible, or – project p is at capacityWe say a matching is cutoff stable if it is fair and cutoff non-wasteful.We also define the notion of a matching induced by minimal cutoff scores,as explored by Fleiner and Jank´o [2014] in a model without distributionalconstraints, and by Bir´o and Kiselgof [2015] for a college admission model withties. Let d − p denote the cutoff scores after decreasing the cutoff of p by one, andkeeping the other cutoffs the same, i.e., d − p ( p ) = d ( p ) −
1, and d − p ( p ′ ) = d ( p ′ )for every p ′ = p . We say that cutoffs d are minimal if we cannot decrease thecutoff score of any project without making the induced matching infeasible.More formally for every project p , either d ( p ) = 0 or the matching induced by d − p , which we call M − d , is not feasible. Proposition 3
A matching is cutoff stable if and only if it is induced byminimal cutoff scores.Proof
Due to Proposition 1, we only need to show that the minimality ofthe cutoff score is equivalent to cutoff non-wastefulness. Suppose first thatmatching M is induced by minimal cutoff scores d , which means that for anyproject p , if d ( p ) > d − p is not feasible. Consider project p with d ( p ) > a be the applicant who is assigned a score of d − p ( p ) by project p . M − p ,the matching induced d − p , is not feasible which means M − p = M . Therefore, M − p = ( M ∪ { ( a,p ) } ) \ { ( a,M ( a )) } and thus p ≻ a M ( a ), which in turn meansthat a is the highest ranked applicant who forms a blocking pair with p . Thus,the induced matching M is cutoff non-wasteful. Now suppose that M is acutoff stable matching and let d be some cutoff scores that induce M whichare minimal in that sense that no smaller cutoff scores can induce M . Considerproject p with d ( p ) >
0. The matching M − p induced by d − p must be differentfrom M . Since M is cutoff non-wasteful, M − p must be infeasible for everyproject p with d ( p ) > d are minimal. ⊓⊔ We can show the natural correspondence in between the three solutionconcepts as follows.
Proposition 4
Every strongly stable matching is cutoff stable, and everycutoff stable matching is weakly stable.Proof
By definition, strong non-wastefulness implies cutoff non-wastefulness,and cutoff non-wastefulness implies weak non-wastefulness. ⊓⊔ Furthermore, these notions do not coincide, as illustrated in the followingexample. itle Suppressed Due to Excessive Length 15
Example 4 (An instance where the sets of strongly stable, cutoff stable andweakly stable matchings are distinct) A = { a ,a ,a }≻ a : p ,p .p ≻ a : p ,p ≻ a : p P = { p ,p ,p } c p = c p = c p = 1 ≻ p : a ,a ≻ p : a ,a ≻ p : a ,a S = { s } P s = { p ,p ,p } q s = 2Due to the supervisor capacity every feasible matching has size at mosttwo, and any matching of size fewer that two is weakly wasteful. Thus, allthe relevant matchings for weak, cutoff and strong stability have size two, sowe consider all the matchings of size two below. M = { ( a ,p ) , ( a ,p ) } and M = { ( a ,p ) , ( a ,p ) } are strongly stable. M = { ( a ,p ) , ( a ,p ) } is not strongly stable, due to blocking pair ( a ,p ).However, M is cutoff stable, since a ≻ p a and M ∪{ ( a ,p ) } is not feasible M = { ( a ,p ) , ( a ,p ) } is not cutoff stable, due to blocking pairs ( a ,p ) and( a ,p ). However, M is weakly stable as adding an applicant to a results in aninfeasible matching. All other matchings of size two are not fair or not feasible.Note that for the classical college admission model these notions areequivalent to stability as formulated by Gale and Shapley [1962a]. Furthertheoretical findings about cutoff scores for this basic model are discussed byAzevedo and Leshno [2016].5.1 Algorithm for computing a cutoff stable matchingWe present an algorithm (Algorithm 1) that shows the existence of a cutoffstable matching for every instance of HER, and can compute a cutoff stablematching for an instance of SIP in strongly polynomial-time. Algorithm 1has similarities with the algorithm proposed by Kamada and Kojima [2017b]which finds a weakly stable matching for HER. In contrast to the algorithm ofKamada and Kojima [2017b], we do not explicitly work with a set of blockingof pairs, but with cutoff scores. Furthermore, a blocking pair ( a,p ) was satisfiedin Kamada and Kojima [2017b] whenever the new matching M ∪ { ( a, p ) } stayed feasible, leading to a weakly stable solution in the end, whilst we satisfya blocking pair ( a, p ) if ( M ∪ { ( a, p ) } ) \ { ( a, M ( a )) } stay feasible, leadingto a cutoff stable matching. Algorithm 1 can also be viewed as a modifiedversion of the Fleiner-Jank´o score-decreasing algorithm (see subsection 4.3in Fleiner and Jank´o [2014]). Whereas Fleiner and Jank´o [2014] do not con-sider distributional constraints, our goal is to achieve stability propertiesunder general heredity constraints. Finally, in independent and recent work,Kamada and Kojima [2020] consider a fixed-point approach based on cutoffs.However, their heredity constraints apply to individual hospitals/projects andcannot capture SIP or regional constraints.The idea of Algorithm 1 is simple. We start with maximum cutoffs thatinduce the empty matching and then we gradually decrease them until wecan no longer do so without making the induced matching infeasible. Note that for the classical college admission problem this process is equivalent tothe college proposing Gale-Shapley algorithm [Gale and Shapley, 1962a], thatwas observed in the Turkish college admission practice [Balinski and S¨onmez,1999], and also in the Hungarian college admission scheme [Bir´o and Kiselgof,2015] in a more general form, as ties have been present in the rankings. Input: lists ≻ p for all p ∈ P and ≻ a for all a ∈ A ; feasibility function f ; project order P ∗ =( p ,...,p k ) Output:
Matching M and corresponding cutoffs d M
1: Initialize M to empty and d M ( p )= | A | +1 for every project p .2: while Cutoff d M are not minimal do
3: Locate the first p j in the list P ∗ such that M − p j is feasible.4: Let M = M − p j and d M = d − p j M .Algorithm 2: Algorithm for matching with heredity constraints. Theorem 3
A cutoff stable matching always exists for a matching problemunder any set of distributional constraints that can be represented by a feasibilityfunction, i.e., for the HER model. Algorithm 1 produces one such matching.Proof
Note that M remains feasible and also fair during the algorithm since itis induced by cutoffs. The final matching is cutoff stable, due to Proposition 3,since the algorithm terminates when the cutoffs are minimal. ⊓⊔ We remark here that the argument for Theorem 3 does not require projectsto be selected in the order P ∗ .The next theorem shows that as long as the feasibility of a matching can betested in polynomial time, Algorithm 1 runs in polynomial time. Theorem 4
Suppose checking f ( w ) takes t time. Then, the running time ofAlgorithm 1 is O ( | A || P | t ) .Proof We decrease the cutoffs in at most ( | A | +1) | P | rounds. In every round wepotentially need to check the matching induced by decrementing the cutoff ofeach of | P | projects. To do this for project p j , we compute matching M − p j fromthe current matching M , by simply comparing whether the newly admittedapplicant a i with score d − p j M from p j prefers p j to her current match M ( a i ), andif so, whether the new matching ( M ∪ { ( a i ,p j ) } ) \ { ( a i ,M ( a i )) } is feasible. Byassuming that checking the feasibility of a matching takes t time, the overallrun time of the algorithm is O ( | A || P | t ). ⊓⊔ We note that Algorithm 1 can be applied to the summer internship problemwhere the feasibility function is based on the supervisor budgets. Therefore, forthe summer internship problem a cutoff stable matching exists. Furthermore,from Theorem 1 we know that the feasibility of a matching can be checked inpolynomial time, and thus a cutoff stable matching can be found in polynomialtime.Additionally, using the cutoff-decreasing algorithm we can find a naturalrelationship between weakly stable and cutoff stable matchings. itle Suppressed Due to Excessive Length 17
Proposition 5
Every weakly stable matching that is not cutoff stable isPareto-dominated by a cutoff stable matching for the applicants.Proof
Let M be a weakly stable matching that is not cutoff stable. Since M isfair, it can be induced by cutoff scores, but these cutoff scores are not minimal,since M is not cutoff stable. Therefore, we can gradually decrease these cutoffscores as described in the cutoff decreasing process until we reach a cutoff stablematching M ′ . Note that M ′ Pareto-dominates M since decreasing cutoff scorescan only make the applicants better off, and every change in the matchingmeans a strict improvement for at least one applicant. ⊓⊔ Although Algorithm 1 satisfies cutoff stability, it also has some drawbacks.Next we establish some properties of the algorithm.
Theorem 5
The following properties hold for Algorithm 1.1. Algorithm 1 is not strategyproof for the applicants.2. Algorithm 1 does not always find a strongly stable matching whenever oneexists.3. Changing the order of projects ordered after p by P ∗ can change p ’s allocation.4. There exist cutoff stable matchings that cannot be produced as a result ofAlgorithm 1 by changing the project order P ∗ .Proof We prove each of the statements separately.1. Consider the following instance. A = { a ,a } ≻ a : p ,p ≻ a : p P = { p ,p } ≻ p : a ,a ≻ p : a ,a S = { s } P s = { p ,p } q s = c p = c p = 1If we set P ∗ = ( p ,p ) then the algorithm outputs M = { ( a ,p ) } . However,if a were to modify their preferences to ≻ a : p ,p , then the algorithm willoutput M = { ( a ,p ) } , which is preferred by a . Thus, the algorithm is notstrategy-proof for applicants.2. For the example above, since ( a ,p ) is the only strongly stable matching,the algorithm does not find the strongly stable matching when one exists.3. Consider the following instance. A = { a ,a ,a }≻ a : p ,p ≻ a : p ,p ≻ a : p P = { p ,p ,p }≻ p : a ,a ≻ p : a ,a ≻ p : a S = { s } P s = Pc p = c p = 1 c p = 2 q s = 2Setting P ∗ = ( p , p , p ), the algorithm terminates in the matching M = { ( a ,p ) , ( a ,p ) } However, if we set P ∗ = { p ,p ,p } , a gets matchedto p and then a gets matched to p . At this point C p = ∅ since f ( { ( a ,p ) , ( a ,p ) , ( a ,p ) } ) = 0, so the algorithm terminates with matching M = { ( a ,p ) , ( a ,p ) } . Thisshows that changing the order of projects ordered after p by P ∗ can change p ’s allocation.
4. This is a consequence of the project-proposing nature of the algorithm.Consider the following instance. A = { a ,a } ≻ a : p ,p ≻ a : p ,p P = { p ,p } ≻ p : a ,a ≻ p : a ,a S = { s } P s = Pc p = c p = 1 q s = 2Both possible project orders produce M = { ( a , p ) , ( a , p ) } , but thematching M = { ( a , p ) , ( a , p ) } is also cutoff stable. Thus, there existweakly stable matchings that cannot be produced by the algorithm.This completes the proof. ⊓⊔ Since the examples above contain exactly one supervisor, Theorem 5 alsoapplies to REG and any setting that contains REG as a special case.5.2 Finding a maximum size cutoff stable matchingFirst, we show that this problem is NP-hard for the REG model by reducingit from MAX-SMTI. Then we provide a mixed integer linear programmingformulation.In the proof we will use a gadget equivalent to the problem in Example 2,where we have no strongly stable matching, but we have two cutoff stablematchings, namely { a ,p } and { a ,p } . Theorem 6
Finding a maximum size cutoff stable matching for regionalquotas is NP-hard.Proof
We reduce again from the special version of the MAX-SMTI prob-lem [Manlove et al., 2002]. Suppose that we have an instance I of MAX-SMTI.We will create an instance I ′ of SIP, such that the maximum size of the cutoffstable in I ′ is the same as the maximum size weakly stable matching in I . For I we use the same notation of W = W s ∪ W t to denote the set of women with strictpreferences and with a single tie, respectively, and let U denote the set of men.Every men in I will be replaced by an applicant in I ′ with essentially the samepreferences. Now, every woman in w j ∈ W s will be replaced with a single project p j with identical preferences. For every woman w j ∈ W t we create two projects, p j and p j that will correspond to the projects in Example 2. If u i and u k were thetwo men in the single tie of w j in I then let the two corresponding applicants a i and a k complete the instance of Example 2. So the preference lists of the projectsare ≻ p j : a i ,a k and ≻ p j : a k ,a i , whilst a i will have p j and p j in her list with prefer-ence for the former, and a k will also have them both with opposite preferences.For the reduction we have to show that if M is a weakly stable matching in I then there is a corresponding cutoff stable matching M ′ in I ′ with the same size,and vica versa. The correspondence between the matchings is as follows. – for every w j ∈ W s and u i ∈ U , ( u i ,w j ) ∈ M ⇐⇒ ( a i ,p j ) ∈ M ′ – for every w j ∈ W t and u i ∈ U , where u i is the first man in the tie of w j ,( u i ,w j ) ∈ M ⇐⇒ ( a i ,p j ) ∈ M ′ – for every w j ∈ W t and u k ∈ U , where u k is the second man in the tie of w j ,( u k ,w j ) ∈ M ⇐⇒ ( a k ,p j ) ∈ M ′ itle Suppressed Due to Excessive Length 19 We can observe that the sizes of M and M ′ are the same. ⊓⊔ Note that since MAX-SMTI is not approximable within a factor of 21/19unless P = N P [Halld´orsson et al., 2007], due to the one-to-one relation ofthe sizes of the matchings in our reduction, the same inapproximability resultapplies for our problem as well. Furthermore, we also note that the very sameresult applies for weak stability, since for Example 2 the cutoff stable matchingsare the same as the weakly stable matchings, and so the very same reductionworks also to show the NP-hardness of the problem of finding a maximum sizeweakly stable matching.MILP-formulationThe mixed integer linear program will have three main parts. The first set ofconstraints describe the feasibility of the funding allocation. The matchingin between applicants and projects are described with (0-1) binary variablesdefined for every mutually acceptable applicant-project pair, as follows, let y a,p = 1 ⇐⇒ ( a,p ) ∈ M . Let x s,p be a non-negative continuous variable denotingthe funding for project p provided by supervisor s . The following three sets ofconditions ensures the feasibility of the solution.Applicant-feasibility: X p ∈ P y a,p ≤ a ∈ A (1)Project-feasibility: X a ∈ A y a,p = X s ∈ S x s,p for each p ∈ P (2)Applicant-feasibility: X p ∈ P x s,p ≤ q s for each s ∈ S (3)The second set of constraints will describe the fairness of the matching bymeans of cutoff scores. Let z a,p be the score of applicant a at project p , agiven constant integer in the interval [0 , | A | ]. For every project p , let d ( p ) be aninteger variable in the range of [0 , | A | + 1] denoting the cutoff of project p . Wecan link the cutoff scores with the induced matching M by the following set ofconstraints, as also used in ´Agoston et al. [2016] and Delorme et al. [2019]. d ( p ) ≤ (1 − y a,p )( | A | +1)+ z a,p for each ( a,p ) (4)The above constraint enforces that if an applicant is assigned to a projectthen she reached the cutoff there. z a,p +1 ≤ d ( p )+ X p ′ (cid:23) a p y a,p ′ · ( | A | +1) for each ( a,p ) (5)This constraint implies that if the applicant is rejected from project p , soshe is not admitted there and nor to any better project of her preference (inwhich case the sum term is zero on the right hand side), then the cutoff mustbe higher at p than her score there. These two sets of conditions together implythat every student is admitted to her most preferred project where she achievedthe cutoff, which means that we get the matching induced by the cutoffs. For ensuring cutoff stability we have to provide cutoff non-wastefulness byenforcing that the cutoffs are minimal. This can be achieved by simply mini-mizing the sum of the cutoff scores in the objective function of the MILP. To seethis we just need to observe that if the matching would not be cutoff minimal,so we could decrease some of the cutoffs by keeping the solution feasible, butthen the solution was not optimal with respect to this objective function.Finally, if we want to maximize the number of applicants matched in thecutoff stable solution then we can use the following combined objective function,where W is a large enough constant.max X a ∈ A,p ∈ P W · y a,p − X p ∈ P d ( p ) (6)This completes the MILP for finding a maximum size cutoff stable matching. Given a feasible matching M , there can exist multiple ways to allocate super-visor budgets among projects to fund all applicants matched to them. Our goalis to find a method for the fairest such allocation. We have chosen to deal withfairness post-match, in order to find a solution that does not constrain the setof feasible matchings.For each supervisor s ∈ S , and each project they supervise p ∈ P s set 0 < t s,p ≤ p ∈ P , P s ∈ S p t s,p = 1. Call this the normative ‘target’ - howmuch we would want s to contribute to the funding of any applicant matchedto p . For instance, if we would ideally want the supervisors of any given projectto contribute equally to its funding, we would set: t s,p = | M ( p ) || S p | ∀ p ∈ P,s ∈ S p . However, given a feasible matching M , some supervisors may lack sufficientfunding to reach these targets. Therefore, we seek to find a funding allocationthat is closest to the target allocations. In order to define closest, we considerspecific lexicographic comparisons. Let X = { x s,p } s ∈ S,p ∈ P s be a feasible fundingallocation for matching M . Denote by φ X the vector corresponding to theweakly decreasing ordering of the set: n x s,p t s,p o s ∈ S,p ∈ P s . Denote by Φ M the set ofsuch vectors corresponding to all feasible funding allocations for matching M .The egalitarian feasible funding allocation for matching M is the fundingallocation corresponding to φ ∗ ∈ Φ M such that for all φ ∈ Φ M , φ ∗ ≺ lex φ , where ≺ lex refers to the well-known lexicographic order. This is exactly equivalentto finding the leximin optimum of the following set: n − x s,p t s,p o s ∈ S,p ∈ P s . Theegalitarian feasible funding allocation can be achieved via Algorithm 3, whichruns a series of linear programs.
Theorem 7
Given a feasible matching, the egalitarian funding allocationcan be computed in time O ( | S | | P | ) LP ( O ( | S | · | P | )) , where LP refers to therunning time of the linear programming algorithm used.Proof The algorithm solves a series of linear programs with O ( | S | · | P | ) con-straints. The while loop iterates at most | T | ≤ | S || P | times, as the algorithm adds itle Suppressed Due to Excessive Length 21 Input: T = { ( s,p ) | s ∈ S,p ∈ P s } , q s ∀ s ∈ S and { t s,p | ( s,p ) ∈ T } . Output:
Funding allocation x T tight ← ∅ while T tight = T do
3: Minimise the maximal component of n x s,p t s,p | ( s,p ) ∈ T o that is not yet tight by solvingthe following LP: λ ∗ ← Min λ s.t. x s,p ≥ ∀ ( s,p ) ∈ T X s ∈ S p x s,p = | T ( p ) | ∀ p ∈ P X p ∈ P s x s,p ≤ q s ∀ s ∈ Sx s,p t s.p ≤ λ ∀ ( s,p ) ∈ T \ T tight x s,p t s.p = λ s,p ∀ ( s,p ) ∈ T tight for ( s,p ) ∗ ∈ T \ T tight do
5: Determine whether the constraint corresponding to ( s,p ) ∗ is tight by solving thefollowing Auxiliary LP: ǫ ∗ ← Max ǫ s.t. x s,p ≥ ∀ ( s,p ) ∈ T X s ∈ S p x s,p = | T ( p ) | ∀ p ∈ P X p ∈ P s x s,p ≤ q s ∀ s ∈ Sx s,p t s.p ≤ λ ∗ ∀ ( s,p ) ∈ T \ T tight \{ ( s,p ) ∗ } x s,p t s.p = λ s,p ∀ ( s,p ) ∈ T tight x s,p t s.p + ǫ ≤ λ ∗ for ( s,p )=( s,p ) ∗ if ǫ ∗ =0 then
7: Add ( s,p ) ∗ to T tight λ s,p ← λ ∗ return { λ s,p | ( s,p ) ∈ T } Algorithm 3: Computing an egalitarian funding allocation for a given feasible matching at least one element to the set T tight in every iteration. The for loop also iteratesat most | T | times. Thus, the overall complexity is O ( | S | | P | ) LP ( O ( | S |·| P | )),where LP refers to the running time of the linear programming algorithm used.Since linear programs can be solved in polynomial time, the egalitarian fundingallocation can also be computed in polynomial time. ⊓⊔ We presented a novel matching model that captures many real-world scenarios.For the model, we presented a compelling solution that is polynomial-timeand satisfies stability and fairness properties. Our central algorithm computesa cutoff stable matching even for the general matching model with heredityconstraints. Hence, it applies to many applications such as refugee matchingthat involve heredity constraints.Several directions and problems arise as a result of our study. Our approachto finding a fair budget allocation was to first compute a cutoff stable matchingand then find an egalitarian budget allocation. It will be interesting to explorea fair outcome that is fairest in some global sense across all weakly stablematchings. We showed that the algorithm we consider is not strategyproof forapplicants. It is open whether there exists an algorithm that is strategyproofand satisfies weak stability or cutoff stability.
Acknowledgments
Aziz gratefully acknowledges support from Defence Science and Technology(DST). Baychkov’s work was supported by the CSIRO undergraduate vacationscholarship program, and he extends many thanks to Haris Aziz and GavinWalker for their invaluable mentorship. Bir´o is supported by the HungarianScientific Research Fund, OTKA, Grant No. K128611.
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