text: Many important stable matching problems are known to be NP-hard, even when strong restrictions are placed on the input. In this paper we seek to identify structural properties of instances of stable matching problems which will allow us to design efficient algorithms using elementary techniques. We focus on the setting in which all agents involved in some matching problem can be partitioned into k different types, where the type of an agent determines his or her preferences, and agents have preferences over types (which may be refined by more detailed preferences within a single type). This situation would arise in practice if agents form preferences solely based on some small collection of agents’ attributes. We also consider a generalisation in which each agent may consider some small collection of other agents to be exceptional, and rank these in a way that is not consistent with their types; this could happen in practice if agents have prior contact with a small number of candidates. We show that (for the case without exceptions), the well-known NP-hard matching problem Max SMTI (that of finding the maximum cardinality stable matching in an instance of stable marriage with ties and incomplete lists) belongs to the parameterised complexity class FPT when parameterised by the number of different types of agents needed to describe the instance. This tractability result can be extended to the setting in which each agent promotes at most one “exceptional” candidate to the top of his her list (when preferences within types are not refined), but the problem remains NP-hard if preference lists can contain two or more exceptions and the exceptional candidates can be placed anywhere in the preference lists.
  @cite_30: 
  @cite_14: Coalitions and cooperation are key topics in multi-agent systems (mas). They enable agents to achieve goals that they may not have been able to achieve independently. A range of previous studies have found that many problems in coalitional games tend to be computationally intractable - that is, the computational complexity grows rapidly as a function of the number of participating agents. However, these hardness results generally require that each agent is of a different type. Here, we observe that in many mas settings, while the number of agents may grow, the number of different types of agents remains small. We formally define the notion of agent types in cooperative games. We then re-examine the computational complexity of the different coalition formation problems when assuming that the number of agent types is fixed. We show that most of the previously hard problems become polynomial when the number of agent types is fixed. We consider multiple different game formulations and representations (characteristic function with subadditive utilities, crg, and graphical representations) and several different computational problems (including stability, core-emptiness, and Shapley value).
  @cite_3: We investigate the testable implications of the theory of stable matchings. We provide a characterization of the matchings that are rationalizable as stable matchings when agents' preferences are unobserved. The characterization is a simple nonparametric test for stability, in the tradition of revealed preference tests. We also characterize the observed stable matchings when monetary transfers are allowed and the stable matchings that are best for one side of the market: extremal stable matchings. We find that the theory of extremal stable matchings is observationally equivalent to requiring that there be a unique stable matching or that the matching be consistent with unrestricted monetary transfers.
  @cite_40: We study the behavior of random walks along the edges of the stable marriage lattice for various restricted families of allowable preference sets. In the "k-attribute model," each man is valued in each of k attributes, and each woman's ranking of the men is determined by a linear function, representing her relative ranking of those attributes; men's rankings of the women are determined similarly. We show that sampling with a random walk on the marriage lattice can take exponential time, even when k = 2. Moreover, we show that the marriage lattices arising in the k-attribute model are more restrictive than in the general setting; previously such a restriction had only been shown for the sets of preference lists. The second model we consider is the "k-range model," where each person lies in a position in [i, i + k - 1], for some i, on every preference list of the opposite sex. When k = 1 there is a unique stable marriage. When k = 2 there already can be an exponential number of stable marriages, but we show that a random walk on the stable marriage lattice always converges quickly to equilibrium. However, when k ≥ 5, there are preference sets such that the random walk on the lattice will require exponential time to converge. Lastly, we show that in the extreme case where each gender's rankings of the other are restricted to one of just a constant k possible preference lists, there are still instances for which the Markov chain mixes exponentially slowly, even when k = 4. This oversimplification of the general model helps elucidate why Markov chains based on spouse-swapping are not good approaches to sampling, even in specialized scenarios.
  @cite_16: We investigate the complexity of approximately counting stable roommate assignments in two models: (i) the k-attribute model, in which the preference lists are determined by dot products of ''preference vectors'' with ''attribute vectors'' and (ii) the k-Euclidean model, in which the preference lists are determined by the closeness of the ''positions'' of the people to their ''preferred positions''. Exactly counting the number of assignments is #P-complete, since Irving and Leather demonstrated #P-completeness for the special case of the stable marriage problem (Irving and Leather, 1986 [11]). We show that counting the number of stable roommate assignments in the k-attribute model (#k-attribute SR, k>=4) and the 3-Euclidean model (#k-Euclidean SR, k>=3) is interreducible, in an approximation-preserving sense, with counting independent sets (of all sizes) (#IS) in a graph, or counting the number of satisfying assignments of a Boolean formula (#SAT). This means that there can be no FPRAS for any of these problems unless NP=RP. As a consequence, we infer that there is no FPRAS for counting stable roommate assignments (#SR) unless NP=RP. Utilizing previous results by Chebolu, Goldberg and Martin (2010) [3], we give an approximation-preserving reduction from counting the number of independent sets in a bipartite graph (#BIS) to counting the number of stable roommate assignments both in the 3-attribute model and in the 2-Euclidean model. #BIS is complete with respect to approximation-preserving reductions in the logically-defined complexity class #RH@P"1. Hence, our result shows that an FPRAS for counting stable roommate assignments in the 3-attribute model would give an FPRAS for all #RH@P"1. We also show that the 1-attribute stable roommate problem always has either one or two stable roommate assignments, so the number of assignments can be determined exactly in polynomial time.
  @cite_25: This paper proposes and estimates a static transferable utility model of the marriage market. The model generates a nonparametric marriage matching function with spillover effects. It rationalizes the standard interpretation of marriage rate regressions and points out its limitations. The model was used to estimate U.S. marital behavior in 1971 72 and 1981 82. The marriage matching function estimates show that the gains to marriage for young adults fell substantially over the decade. Unlike contradictory marriage rate regression results, the marriage matching function estimates showed that the legalization of abortion had a significant quantitative impact on the fall in the gains to marriage for young men and women.
  @cite_11: We investigate the complexity of approximately counting stable matchings in the k-attribute model, where the preference lists are determined by dot products of ''preference vectors'' with ''attribute vectors'', or by Euclidean distances between ''preference points ''and ''attribute points''. Irving and Leather (1986) [14] proved that counting the number of stable matchings in the general case is #P-complete. Counting the number of stable matchings is reducible to counting the number of downsets in a (related) partial order [14] and is interreducible, in an approximation-preserving sense, to a class of problems that includes counting the number of independent sets in a bipartite graph (#BIS) ( (2004) [6]). It is conjectured that no FPRAS exists for this class of problems. We show this approximation-preserving interreducibility remains even in the restricted k-attribute setting when k>=3 (dot products) or k>=2 (Euclidean distances). Finally, we show it is easy to count the number of stable matchings in the 1-attribute dot-product setting.
summary: Settings in which agents are partitioned into different types, or derive their preferences based on a set of attributes assigned to each candidate, have been considered for the problems of sampling and counting stable matchings in instances of SM or SR (see, e.g., @cite_40 @cite_11 @cite_16 ). In @cite_3 , the authors study the problem of characterising matchings that are rationalisable as stable matchings when agents' preferences are unobserved. They focus on a restricted setting that translates into assigning each agent a type based on several attributes, and assuming that agents of the same type are identical and have identical preferences. They remark that empirical studies on marriage typically make such an assumption @cite_25 . Bounded agent types have been considered in @cite_30 @cite_14 to derive polynomial-time results for the coalition structure generation problem, an important issue in cooperative games when the goal is to partition the participants into exhaustive and disjoint coalitions in order to maximise the social welfare.