Stable Matching Research Articles

Bribery and Control in Stable Marriage

We initiate the study of external manipulations in Stable Marriage by considering several manipulative actions as well as several manipulation goals. For instance, one goal is to make sure that a given pair of agents is matched in a stable solution, and this may be achieved by the manipulative action of reordering some agents' preference lists. We present a comprehensive study of the computational complexity of all problems arising in this way. We find several polynomial-time solvable cases as well as NP-hard ones. For the NP-hard cases, focusing on the natural parameter "budget" (that is, the number of manipulative actions one is allowed to perform), we also conduct a parameterized complexity analysis and encounter mostly parameterized hardness results.

Effects of divorce in the church: a case study of selected CITAM churches in Kenya (CITAM Valley Road, CITAM Woodley, CITAM Ngong)

Purpose: The church is expected to contribute to the stable marriage and ensure that it has put measures in place that will reduce the rate of divorce as much as possible. The purpose of the paper is to determine the effects of divorce in the selected churches of CITAM in Kenya. The objectives is to establish whether divorce has any effect on the psychological wellbeing of children and spouses affected, some of the causes of divorce, and the role of CITAM in reducing divorce in the church.
 Methodology: The paper adopted qualitative research methods to obtain in-depth information on the situation. The study used a descriptive research design and the target population of the research was 100 church members who have undergone the divorce, from which the researcher was able to obtain a sample of 27 participants through the saturated sampling technique. The data was collected using interviews. The data was analyzed using QDA miner lite 2.0.7. The data was presented using tables and charts.
 Results: Some of the causes of divorce include infidelity, variation in expectations, communication breakdown, and lack of support system, finances, in-laws, abuse and wrong orientations to marriage. CITAM should enhance support to families and enhance premarital counselling. Divorce causes psychological problems to divorced spouses and children and also causes stigmatization.
 Recommendations: The church needs to diversify means in which its members are empowered on how to embrace God’s teaching much more than their tradition or culture. The church members are aware of the biblical teaching that discourages them from divorce and therefore the church needs to diversify means by which they live according to God’s word which is the pillar for every marriage. The church needs to improve its premarital counseling as this is the building block of marriage. The church should also intentionally have professional counselors who have experience in marital issues and who will not mislead those who come for help in the church. The church needs to develop programs that can support church members who find themselves in troubled marriages by coming up with counseling programs that identifies with the affected church members.

Myopic and Farsighted Stable Sets in 2-Player Strategic-Form Games

This paper revisits analysis of stable sets in two-player strategic-form games. Our two main contributions are (i) to establish a connection between myopic stable sets and stable matchings of an auxiliary two-sided matching problem and (ii) to identify a structural property of 2-player games, called the block partition property, which helps characterize strategy profiles that are indirectly dominated by a fixed profile. Our analysis also generalizes and unifies existing results on myopic and farsighted stable sets in 2-player games.

Coalitional permutation manipulations in the Gale-Shapley algorithm

In this paper, we consider permutation manipulations by any subset of women in the men-proposing version of the Gale-Shapley algorithm. This paper is motivated by the college admissions process in China. Our results also answer an open problem on what can be achieved by permutation manipulations. We present an efficient algorithm to find a strategy profile such that the induced matching is stable and Pareto-optimal (in the set of all achievable stable matchings) while the strategy profile itself is inconspicuous. Surprisingly, we show that such a strategy profile actually forms a Nash equilibrium of the manipulation game. In the end, we show that it is NP-complete to find a manipulation that is strictly better for all members of the coalition. This result demonstrates a sharp contrast between weakly better off outcomes and strictly better-off outcomes.

Preventing a loss of accuracy of the tennis serve under pressure.

Dynamically squeezing the left hand (left hand dynamic handgrip) has been shown to be effective in preventing choking under pressure in right-handers in a variety of sports. The current study assessed the effectiveness of the left hand dynamic handgrip in preventing a loss of accuracy of tennis serves in competitive situations. Twenty right-handed highly skilled junior athletes performed eight tennis serves at a target without pressure (pre-test), followed by eight serves under pressure (post-test). Ten of the participants conducted the left hand dynamic handgrip prior to the post-test, while the other ten performed an equivalent handgrip with their right hand. The serving accuracy of the group performing the handgrip with their right hand decreased significantly from pre- to post-test, while the accuracy of the left hand dynamic handgrip group remained stable. The results indicate the left hand dynamic handgrip to be effective in preventing reduced accuracy of the tennis serve in competition situations as a form of choking under pressure. This technique could easily be integrated into tennis players’ serving routines and promote stable match performance in competitions.

Application of edge computing and cloud service data system for doctors' psychological reconstruction after the epidemic

In response to the problems that the new crown pneumonia epidemic may bring to doctors in schools of Chinese medicine, such as information overload, traumatic stress, changes in mentality, crisis of trust, safety when going abroad, and employment choices, explain from the perspective of medical teachers in Chinese medicine schools. The understanding of the epidemic and the major events during the epidemic, combined with the experience of the psychological impact of the past epidemic on the doctors in the school, to highlight the outstanding contributions of medical staff in the fight against the epidemic, and to emphasize the strong leadership of the party and the unity of the people during the fight against the epidemic. The huge role played by Chinese medicine in the fight against the epidemic is the principle. In recent years, cloud computing technology has been continuously developed and improved. Traditional infrastructure as a service has been unable to meet users' needs for cloud computing delivery capabilities. They hope that more and more traditional IT software will be delivered in the form of cloud services. This paper mainly studies the utility optimization and stable matching strategy of UDRN based on EH and limited character input, divides the nodes into multiple clusters, and implements the utility optimization strategy within the clusters. Model users and relays as energy buyers and energy sellers. First, when users choose relays as their candidates, the golden section method is used to obtain the best relay power to maximize user utility. Secondly, based on the principle of maximizing the utility of matching, a mutual preference matrix between users and relays is established. Finally, based on the mutual preference matrix, an improved UDRN stable matching algorithm is proposed through the less complex GS algorithm. This algorithm will play a great role in the psychological reconstruction of doctors after the epidemic.

(Un)stable matchings with blocking costs

We introduce and study weighted bipartite matching problems under strict preferences where blocking edges can be paid for, thus imposing costs rather than constraints as in classical models. We focus on the setting in which the weight of an edge represents the benefit from including it in the matching and/or the cost if it is a blocking edge. We show that this setting encompasses interesting special cases that remain polynomially-solvable, although it becomes APX-hard even in a quite restricted case.

Inefficiencies in Residency Matching Associated with Gale–Shapley Algorithms

Abstract Objective This study aimed to investigate emerging trends and increasing costs in the National Residency Matching Program (NRMP) and San Francisco Residency and Fellowship Match Services (SF Match) associated with the current applicant/program Gale–Shapley-type matching algorithms. Design A longitudinal observational study of behavioral trends in national residency matching systems with modeling of match results with alternative parameters. Patients and Methods We analyzed publicly available data from the SF Match and NRMP websites from 1985 to 2020 for trends in the total number of applicants and available positions, as well the average number of applications and interviews per applicant for multiple specialties. To understand these trends and the algorithms' effect on the residency programs and applicants, we analyzed anonymized rank list and match data for ophthalmology from the SF Match between 2011 and 2019. Match results using current match parameters, as well as under conditions in which applicant and/or program rank lists were truncated with finalized rank lists, were analyzed. Results Both the number of applications and length of programs' rank lists have increased steadily throughout residency programs, particularly those with competitive specialties. Capping student rank lists at seven programs, or less than 80% of the average 8.9 programs currently ranked, results in a 0.71% decrease in the total number of positions filled. Similarly, capping program rank lists at seven applicants per spot, or less than 60% of the average 11.5 applicants ranked per spot, results in a 5% decrease in the total number of positions filled. Conclusion While the number of ophthalmology positions in the United States has increased only modestly, the number of applications under consideration has increased substantially over the past two decades. The current study suggests that both programs and applicants rank more choices than are required for a nearly complete and stable match, creating excess cost and work for both applicants and programs. “Stable-marriage” type algorithms induce applicants and programs to rank as many counterparties as possible to maximize individual chances of optimizing the match.

Relationship between metabolic syndrome and acquired premature ejaculation

Objectives The aim of the study was to investigate the relationship between metabolic syndrome (MetS) with its components and acquired premature ejaculation (APE). Background PE has recently received considerable attention. It adversely affects the quality of life of both partners. PE represents the most common sexual dysfunction in sexually active men. Studies proved higher incidence of erectile dysfunction among patients with MetS, but the relation of the latter to APE has not receive much attention yet. Patients and methods Our case–control study included 50 individuals who were classified into two groups, with 25 individuals each: MetS group and control group. Both groups are age matched (30–60 years old). All men in both groups were in continuous stable marriage with no marital conflict. They are free of diseases, taking no drugs, did not have operations before, and did not have congenital anomaly that might affect their sexual function (except the manifestations of MetS in the case group). All the participants underwent the intravaginal ejaculatory latency time (IELT) assessment and did MetS component measurements. Results The results of our study showed significantly lower IELT in the MetS group compared with the control (2.80 ± 1.384 vs. 4.52 ± 1.782 min, respectively, P = 0.0001). Arabic index of premature ejaculation score was found to be significantly lower in cases (18.32 ± 7.663) compared with controls (30.32 ± 6.46), confirming the presence of APE caused by MetS. IELT was found to be negatively correlated with BMI, fasting blood sugar, and triglycerides, but not high-density lipoprotein. Conclusion MetS can be considered as an independent predisposing factor for development of APE. Effective prevention and treatment of MetS could be important for APE prevention.

Research on Multi-objective Shared Parking Matching Decision Based on Two-side Matching

The purpose of this paper is to find a reasonable method for matching shared parking spaces. The travelers and the parking space sharing party are regarded as two separate subjects, and the matching of parking spaces is regarded as a two-side matching. The final parking space matching result belongs to the stable matching of the bipartite graph and the existence of Nash equilibrium is proved by the knowledge of graph theory. A multi-objective decision-making model for shared parking space matching based on two-side matching is established on the basis of considering the benefits of travelers and parking providers. The multi-objective model is transformed into a single-objective assignment model using weighting coefficients with adjustment parameters, and the corresponding solution algorithm is designed. Finally, an example is used to verify the rationality and feasibility of the established model. The results show that the parking space matching problem under sharing conditions belongs to the category of two-side matching. When the benefits of both parties are fully considered, the matching results obtained are relatively stable.

Saturating stable matchings

I relate bipartite graph matchings to stable matchings. I prove a necessary and sufficient condition for the existence of a saturating stable matching, where every agent on one side is matched, for all possible preferences. I extend my analysis to perfect stable matchings, where every agent on both sides is matched.

Balanced stable marriage: How close is close enough?

Balanced Stable Marriage (BSM) is a central optimization version of the classic Stable Marriage (SM) problem. We study BSM from the viewpoint of Parameterized Complexity. Informally, the input of BSM consists of n men, n women, and an integer k. Each person a has a (sub)set of acceptable partners, A(a), whom a ranks strictly; we use pa(b) to denote the position of b∈A(a) in a's preference list. The objective is to decide whether there exists a stable matching μ such that balance(μ)≜max⁡{∑(m,w)∈μpm(w),∑(m,w)∈μpw(m)}≤k. In SM, all stable matchings match the same set of agents, A⋆ which can be computed in polynomial time. As balance(μ)≥|A⋆|2 for any stable matching μ, BSM is trivially fixed-parameter tractable (FPT) with respect to k. Thus, a natural question is whether BSM is FPT with respect to k−|A⋆|2. With this viewpoint in mind, we draw a line between tractability and intractability in relation to the target value. This line separates additional natural parameterizations higher/lower than ours (e.g., we automatically resolve the parameterization k−|A⋆|2). The two extreme stable matchings are the man-optimal μM and the woman-optimal μW. Let OM=∑(m,w)∈μMpm(w), and OW=∑(m,w)∈μWpw(m). In this work, we prove that•BSM parameterized by t=k−min⁡{OM,OW} admits (1) a kernel where the number of people is linear in t, and (2) a parameterized algorithm whose running time is single exponential in t.•BSM parameterized by t=k−max⁡{OM,OW} is W[1]-hard.

Stable Matching for Selection of Intelligent Reflecting Surfaces in Multiuser MISO Systems

In this letter, we present an intelligent reflecting surface (IRS) selection strategy for multiple IRSs aided multiuser multiple-input single-output (MISO) systems. In particular, we pose the IRS selection problem as a stable matching problem. A two stage user-IRS assignment algorithm is proposed, where the main objective is to carry out a stable user-IRS matching which improves the sum rate of the system. The first stage of the proposed algorithm employs a well-known Gale Shapley matching algorithm. Due to interference in multiuser systems, the matching obtained after the first stage may not be stable. To overcome this issue, one-sided (i.e., only IRSs) blocking pairs (BPs) are identified in the second stage of the proposed algorithm, where the BP is a pair of IRSs which are better off after exchanging their partners. Thus, the second stage validates stable matching in the proposed algorithm. Numerical results show that the proposed assignment achieves better sum rate performance compared to distance-based and random matching algorithms.

An improved bound to manipulation in large stable matches

This paper builds on Kojima and Pathak (2009)'s result of vanishing manipulability in large stable mechanisms. We show that convergence toward truth-telling in stable mechanisms can be achieved much faster if colleges' preferences are independently drawn from an uniform distribution. Another novelty from our results is that they can be applied to competitive environments in which virtually all vacancies end up being filled. So this paper adds evidence to the fact that, though stable matching mechanisms are not entirely strategy-proof, in practice, when the number of participants in the market is sufficiently large, they can be treated as being effectively strategy-proof.

Mother’s Partnership Status and Allomothering Networks in the United Kingdom and United States

In high-income, low-fertility (HILF) settings, the mother’s partner is a key provider of childcare. However, it is not clear how mothers without partners draw on other sources of support to raise children. This paper reports the findings from a survey of 1532 women in the United Kingdom and the United States, in which women described who provided childcare for a focal child and how frequently they did so. We use multivariate Bayesian regression models to explore the drivers of support from partners, maternal kin, and other allomothers, as well as the potential impact of allomothering on women’s fertility. Relative to mothers who are in a stable first marriage or cohabitation, mothers who are unpartnered rely more heavily on fewer maternal kin, use more paid help, and have networks which include more non-kin helpers. Repartnered mothers received less help from their partners in the UK and less help from maternal kin in both countries, which US mothers compensated for by relying on other helpers. While repartnered mothers had higher age-adjusted fertility than women in a first partnership, allomaternal support was not clearly related to the mother’s fertility. These findings demonstrate the importance of partners but also of allomothering more broadly in HILF settings.

Fair Stable Matchings Under Correlated Preferences (Student Abstract)

Stable matching models are widely used in market design, school admission, and donor organ exchange. The classic Deferred Acceptance (DA) algorithm guarantees a stable matching that is optimal for one side (say men) and pessimal for the other (say women). A sex-equal stable matching aims at providing a fair solution to this problem. We demonstrate that under a class of correlated preferences, the DA algorithm either returns a sex-equal solution or has a very low sex-equality cost.

Almost mutually best in matching markets: rank gaps and size of the core

This paper studies the one-to-one two-sided marriage model (Gale and Shapley 1962). If agents’ preferences exhibit mutually best (i.e., each agent is most preferred by her/his most preferred matching partner), there is a unique stable matching without rank gaps (i.e., in each matched pair the agents assign one another the same rank). We study in how far this result is robust for matching markets that are “close” to mutually best. Without a restriction on preference profiles, we find that natural “distances” to mutually best neither bound the size of the core nor the rank gaps at stable matchings. However, for matching markets that satisfy horizontal heterogeneity, “local” distances to mutually best provide bounds for the size of the core and the rank gaps at stable matchings.

Improving efficiency in school choice under partial priorities

We provide a school choice model where the student priority orders for schools are allowed not to be total. We introduce a class of algorithms, each of which derives a student optimal stable matching once we have an initial stable matching, when the priorities are represented by partial orders. Since a method to derive a stable matching exists when the priorities are partial orders, we can use it to derive a student optimal stable matching. Moreover, we show that any student optimal stable matchings that Pareto dominate the starting stable one are obtained via an algorithm within this class. For the problem of improving efficiency by allowing some priorities to be violated, the algorithms can also be applied, with a weaker assumption on the violations than in the previous study. Finally, we suggest some specific rules of priorities that can be introduced by weakening the requirement of total ordering.

Quantile Stable Mechanisms

We introduce a new class of matching mechanisms—quantile stable mechanisms—that generate stable matchings that can be seen as a compromise between sides of a two-sided market. We show that responsiveness is a sufficient condition for the existence of such mechanisms and that all such mechanisms are distinct. We also analyze the manipulability of these mechanisms by market participants.

Strongly Stable and Maximum Weakly Stable Noncrossing Matchings

In IWOCA 2019, Ruangwises and Itoh introduced stable noncrossing matchings, where participants of each side are aligned on each of two parallel lines, and no two matching edges are allowed to cross each other. They defined two stability notions, strongly stable noncrossing matching (SSNM) and weakly stable noncrossing matching (WSNM), depending on the strength of blocking pairs. They proved that a WSNM always exists and presented an O(n^{2})-time algorithm to find one for an instance with n men and n women. They also posed open questions of the complexities of determining existence of an SSNM and finding a largest WSNM. In this paper, we show that both problems are solvable in polynomial time. Our algorithms are applicable to extensions where preference lists may include ties, except for one case which we show to be NP-complete. This NP-completeness holds even if each person's preference list is of length at most two and ties appear in only men's preference lists. To complement this intractability, we show that the problem is solvable in polynomial time if the length of preference lists of one side is bounded by one (but that of the other side is unbounded).