Stable Roommates Problem Research Articles

Popularity on the roommate diversity problem

A recently introduced restricted variant of the multidimensional stable roommates problem is the roommate diversity problem: each agent belongs to one of two types (e.g., red and blue), and the agents' preferences over the rooms solely depend on the fraction of agents of their own type among their roommates. We study this variant with the notion of popularity.We show that in the roommate diversity problem with the room size fixed to 2, the problem becomes tractable. Particularly, a popular partitioning of agents is guaranteed to exist and can be computed in polynomial time. Additionally, a mixed popular partitioning of agents is always guaranteed to exist in any roommate diversity game. By contrast, when there are no restrictions on the room size of a roommate diversity game, a popular partitioning may fail to exist and the problem becomes intractable.

Marriage and Roommate

This paper has two objectives. One is to give a linear time algorithm that solves the stable roommates problem (i.e., obtains one stable matching) using the stable marriage problem. The idea is that a stable matching of a roommate instance [Formula: see text] is a stable matching (that however must satisfy a certain condition) of some marriage instance [Formula: see text]. [Formula: see text] is obtained just by making two copies of [Formula: see text], one for the men’s table and the other for the women’s table. The second objective is to investigate the possibility of reducing the roommate problem to the marriage problem (with one-to-one correspondence between their stable matchings) in polynomial time. For a given [Formula: see text], we construct the rotation POSET [Formula: see text] of [Formula: see text] and then we “halve” it to obtain [Formula: see text], by which we can forget the above condition and can use all the closed subsets of [Formula: see text] for all the stable matchings of [Formula: see text]. Unfortunately this approach works (runs in polynomial time) only for restricted instances.

Enhancement of Irving’s Algorithm with Autocomplete Feature

Irving?s algorithm is the solution Robert W. Irving proposed to Gale and Shapley?s Stable Roommate problem. It determines if there is a possible stable match in each set using a list of preferences, and if it is possible, finds the stable matches. However, incomplete preference lists would increase the chances of having one participant become unmatched. This paper introduces an enhanced version of this algorithm that uses the addition of descriptive lists to fill in the vacancies in the incomplete preference list. The conclusion is that it is more efficient to fill the vacancies according to approximate preference rather than leaving the lists as incomplete to create more stable matches.

Knowledge-Based Stable Roommates Problem: A Real-World Application

AbstractThe Stable Roommates problem with Ties and Incomplete lists (SRTI) is a matching problem characterized by the preferences of agents over other agents as roommates, where the preferences may have ties or be incomplete. SRTI asks for a matching that is stable and, sometimes, optimizes a domain-independent fairness criterion (e.g. Egalitarian). However, in real-world applications (e.g. assigning students as roommates at a dormitory), we usually consider a variety of domain-specific criteria depending on preferences over the habits and desires of the agents. With this motivation, we introduce a knowledge-based method to SRTI considering domain-specific knowledge and investigate its real-world application for assigning students as roommates at a university dormitory.

A General Framework for Stable Roommates Problems using Answer Set Programming

AbstractThe Stable Roommates problem (SR) is characterized by the preferences of agents over other agents as roommates: each agent ranks all others in strict order of preference. A solution to SR is then a partition of the agents into pairs so that each pair shares a room, and there is no pair of agents that would block this matching (i.e., who prefers the other to their roommate in the matching). There are interesting variations of SR that are motivated by applications (e.g., the preference lists may be incomplete (SRI) and involve ties (SRTI)), and that try to find a more fair solution (e.g., Egalitarian SR). Unlike the Stable Marriage problem, every SR instance is not guaranteed to have a solution. For that reason, there are also variations of SR that try to find a good-enough solution (e.g., Almost SR). Most of these variations are NP-hard. We introduce a formal framework, called SRTI-ASP, utilizing the logic programming paradigm Answer Set Programming, that is provable and general enough to solve many of such variations of SR. Our empirical analysis shows that SRTI-ASP is also promising for applications.

Stable roommates with narcissistic, single-peaked, and single-crossing preferences

The classical Stable Roommates problem is to decide whether there exists a matching of an even number of agents such that no two agents which are not matched to each other would prefer to be with each other rather than with their respectively assigned partners. We investigate Stable Roommates with complete (i.e., every agent can be matched with any other agent) or incomplete preferences, with ties (i.e., two agents are considered of equal value to some agent) or without ties. It is known that in general allowing ties makes the problem NP-complete. We provide algorithms for Stable Roommates that are, compared to those in the literature, more efficient when the input preferences are complete and have some structural property, such as being narcissistic, single-peaked, and single-crossing. However, when the preferences are incomplete and have ties, we show that being single-peaked and single-crossing does not reduce the computational complexity—Stable Roommates remains NP-complete.

Subjective Homophily and the Fixtures Problem

The Stable Fixtures problem (Irving and Scott (2007)) is a generalized matching model that nests the well-known Stable Roommates, Stable Marriage, and College Admissions problems as special cases. This paper extends a result of the Stable Roommates problem to demonstrate that a class of homophilic preferences with an appealing psychological interpretation is sufficient to ensure that starting from an arbitrary matching, a decentralized process of allowing the sequential matching of randomly chosen blocking pairs will converge to a pairwise-stable matching with probability one. Strategic implications of this class of preferences are examined and further possible generalizations and directions for future research are discussed.

The Stable Roommates Problem with Unranked Entries

The Stable Roommates Problem with Unranked Entries

On random stable partitions

It is well known that the one-sided stable matching problem (“stable roommates problem”) does not necessarily have a solution. We had found that, for the independent, uniformly random preference lists, the expected number of solutions converges to $$e^{1/2}$$ as n, the number of members, grows, and with Rob Irving we proved that the limiting probability of solvability is below $$e^{1/2}/2$$ , at most. Stephan Mertens’s extensive numerics compelled him to conjecture that this probability is of order $$n^{-1/4}$$ . Jimmy Tan introduced a notion of a stable cyclic partition, and proved existence of such a partition for every system of members’ preferences, discovering that presence of odd cycles in a stable partition is equivalent to absence of a stable matching. In this paper we show that the expected number of stable partitions with odd cycles grows as $$n^{1/4}$$ . However the standard deviation of that number is of order $$n^{3/8}\gg n^{1/4}$$ , i.e. too large to conclude that the odd cycles exist with probability $$1-o(1)$$ . Still, as a byproduct, we show that with probability $$1-o(1)$$ the fraction of members with more than one stable “predecessor” is of order $$n^{-1/2+o(1)}$$ . Furthermore, with probability $$1-o(1)$$ the average rank of a predecessor in every stable partition is of order $$n^{1/2}$$ . The likely size of the largest stable matching is $$n/2-O(n^{1/4+o(1)})$$ , and the likely number of pairs of unmatched members blocking the optimal complete matching is $$O(n^{3/4+o(1)})$$ .

On random exchange-stable matchings

Consider the group of n men and n women, each with their own preference list for a potential marriage partner. The stable marriage is a bipartite matching such that no unmatched pair (man, woman) prefer each other to their partners in the matching. Its non-bipartite version, with an even number n of members, is known as the stable roommates problem. Jose Alcalde introduced an alternative notion of exchange-stable, one-sided, matching: no two members prefer each other’s partners to their own partners in the matching. Katarina Cechlárová and David Manlove showed that the e-stable matching decision problem is NP-complete for both types of matchings. We prove that the expected number of e-stable matchings is asymptotic to πn21∕2 for two-sided case, and to e1∕2 for one-sided case. However, the standard deviation of this number exceeds 1.13n, (1.06n resp.). As an obvious byproduct, there exist instances of preference lists with at least 1.13n (1.06n resp.) e-stable matchings. The probability that there is no matching which is stable and e-stable is at least 1−e−n1∕2−o(1), (1−O(2−n∕2) resp.).

The Stable Roommates Problem with Short Lists

We consider two variants of the classical Stable Roommates problem with Incomplete (but strictly ordered) preference lists (sri) that are degree constrained, i.e., preference lists are of bounded length. The first variant, egald-sri, involves finding an egalitarian stable matching in solvable instances of sri with preference lists of length at most d. We show that this problem is NP-hard even if d = 3. On the positive side we give a frac {2d+3}{7}-approximation algorithm for d ∈{3,4,5} which improves on the known bound of 2 for the unbounded preference list case. In the second variant of sri, called d-srti, preference lists can include ties and are of length at most d. We show that the problem of deciding whether an instance of d-srti admits a stable matching is NP-complete even if d = 3. We also consider the “most stable” version of this problem and prove a strong inapproximability bound for the d = 3 case. However for d = 2 we show that the latter problem can be solved in polynomial time.

Hardness results for stable exchange problems

In this paper, we study variants of the stable exchange problem which can be viewed as a model for kidney exchange. The b-way stable l-way exchange problem is a generalization of the stable roommates problem. For b=l=3, Biró and McDermid (2010) [1] proved that the problem is NP-complete and asked whether a polynomial time algorithm exists for b=2, l=3. We prove that the problem is NP-complete and it is W[1]-hard with the number of 3-cycles in the exchange as a parameter. We answer a question of Biró (2007) [2] by proving that it is NP-hard to maximize the number of covered nodes in a stable exchange. We also prove some related results on strong stability and approximation.

Simulation approach for investigating dynamics of passenger matching problem in smart ridesharing system

Abstract: Smart ridesharing system (SRS) is a ridesharing system integrated with information and communication technology. We formulate a passenger matching problem by considering user preference in SRS. This study aims to investigate day-to-day dynamics and characteristics of passenger matching problem in SRS. A modified stable roommates problem is adopted to represent this matching problem. The characteristics of day-to-day dynamics (e.g., existence, uniqueness, and stability of equilibria) of SRS are understood and revealed by a simulation of formulated model. Through the simulations, the existence of multiple day-to-day equilibria is investigated under some conditions. One of the existing equilibria is an undesirable equilibrium where SRS is not feasible to be operated as no one rideshares, while the existence of a desirable stable equilibrium, where SRS is feasible to be operated, is realized under some conditions.

Hardness results for stable exchange problems

In this paper, we study variants of the stable exchange problem which can be viewed as a model for kidney exchange. The b-way stable l-way exchange problem is a generalization of the stable roommates problem. For b=l=3, Biró and McDermid proved that the problem is NP-complete and asked whether a polynomial time algorithm exists for b=2, l=3. We prove that the problem is NP-complete and it is W[1]-hard with the number of 3-cycles in the exchange as a parameter. We answer a question of Biró by proving that it is NP-hard to maximize the number of covered nodes in a stable exchange. We also prove some related results on strong stability, approximation and variants of the problem where we allow chains.

Planar Matchings for Weighted Straight Skeletons

We introduce planar matchings on directed pseudo-line arrangements, which yield a planar set of pseudo-line segments such that only matching-partners are adjacent. By translating the planar matching problem into a corresponding stable roommates problem we show that such matchings always exist. Using our new framework, we establish, for the first time, a complete, rigorous definition of weighted straight skeletons, which are based on a so-called wavefront propagation process. We present a generalized and unified approach to treat structural changes in the wavefront that focuses on the restoration of weak planarity by finding planar matchings.

Stable Matching with Network Externalities

We study the stable roommates problem in networks where players are embedded in a social context and may incorporate positive externalities into their decisions. Each player is a node in a social network and strives to form a good match with a neighboring player. We consider the existence, computation, and inefficiency of stable matchings from which no pair of players wants to deviate. We characterize prices of anarchy and stability, which capture the ratio of the total profit in the optimum matching over the total profit of the worst and best stable matching, respectively. When the benefit from a match (which we model by associating a reward with each edge) is the same for both players, we show that externalities can significantly improve the price of stability, while the price of anarchy remains unaffected. Furthermore, a good stable matching achieving the bound on the price of stability can be obtained in polynomial time. We extend these results to more general matching rewards, when players matched to each other may receive different benefits from the match. For this more general case, we show that network externalities (i.e., "caring about your friends") can make an even larger difference and greatly reduce the price of anarchy. We show a variety of existence results and present upper and lower bounds on the prices of anarchy and stability for various structures of matching benefits. All our results on stable matchings immediately extend to the more general case of fractional stable matchings.

A Passengers Matching Problem in Ridesharing Systems by Considering User Preference

Ridesharing system has been recognized as an efficient transport mode to solve the environment, energy, traffic congestion issues. In recent researches, user preference was revealed as an important factor to enhance the performance and reliability of ridesharing systems. With the development in information and communication technologies (ICT), internet-enabled devices and social network enable us to obtain real-time travel information and users' personality. Therefore, user preference based on their personality is considered in this study. The aims of this study are to formulate a matching model between passengers by considering user preference and to investigate how user preference affects ridesharing system's performance. The algorithm for solving stable roommates problem is modified for one-on-one passengers matching problem. The factors of user preference are personality and steadiness of user personality. The results of numerical experiments showed that steadiness of user personality affected to both individual and system success rate, while personality only affected to system success rate.

Strategies for utility maximization in social groups with preferential exploration

is group of interconnected nodes interested in obtaining common content (Scott, in network analysis, 2012). groups are observed in many networks for example, cellular network assisted Device-to-Device network (Fodor et al., in IEEE Commun Mag 50:170---177, 2012, Lei et al., in Wirel Commun 19:96---104, 2012), hybrid Peer-to-Peer content distribution (Christos Gkantsidis and Miller, in 5th International Workshop on Peer-to-Peer Systems, 2006, Vakali and Pallis, in IEEE Internet Comput 7:68---74, 2003) etc. In this paper, we consider a Social Group of networked nodes, seeking a of data segments for maximizing their individual utilities. Each node in social group has a subset of the universe, and access to an expensive link for downloading data. Nodes can also acquire the universe by exchanging copies of data segments among themselves, at low cost, using inter-node links. While exchanges over inter-node links ensure minimum or negligible cost, some nodes in the group try to exploit the system by indulging in collusion, identity fraud etc. We term such nodes as `non-reciprocating nodes' and prohibit such behavior by proposing the Give-and-Take criterion, where exchange is allowed iff each participating node provides at least one segment to the node which is unavailable with the node. While complying with this criterion, each node wants to maximize its utility, which depends on the node's segment set available with the node. Link activation between pair of nodes requires mutual consent of the participating nodes. Each node tries to find a pairing partner by preferentially exploring nodes for link formation. Unpaired nodes download data segments using the expensive link with pre-defined probability (defined as segment aggressiveness probability). We present various linear complexity decentralized algorithms based on the Stable Roommates Problem that can be used by nodes for choosing the best strategy based on available information. We present a decentralized randomized algorithm that is asymptotically optimal in the number of nodes. We define Price of Choice for benchmarking the performance of social groups consisting of non-aggressive nodes (i.e. nodes not downloading data segments from the expensive link) only. We evaluate performances of various algorithms and characterize the behavioral regime that will yield best results for nodes and social groups, spending the least on the expensive link. The proposed algorithms are compared with the optimal. We find that the Link For Sure algorithm performs nearly optimally.

Small random instances of the stable roommates problem

Let pn denote the probability that a random instance of the stable roommates problem of size n admits a solution. We derive an explicit formula for pn and compute exact values of pn for n ⩽ 12.

Stable roommates problem with random preferences

The stable roommates problem with n agents has worst case complexity O(n2) in time and space. Random instances can be solved faster and with less memory, however. We introduce an algorithm that has average time and space complexity for random instances. We use this algorithm to simulate large instances of the stable roommates problem and to measure the probabilty pn that a random instance of size n admits a stable matching. Our data support the conjecture that pn = Θ(n−1/4).