We show that the ratio of matched individuals to blocking pairs grows linearly with the number of propose--accept rounds executed by the Gale--Shapley algorithm for the stable marriage problem. Consequently, the participants can arrive at an almost stable matching even without full information about the problem instance; for each participant, knowing only its local neighbourhood is enough. In distributed-systems parlance, this means that if each person has only a constant number of acceptable partners, an almost stable matching emerges after a constant number of synchronous communication rounds. This holds even if ties are present in the preference lists.
We apply our results to give a distributed $(2+ε)$-approximation algorithm for maximum-weight matching in bicoloured graphs and a centralised randomised constant-time approximation scheme for estimating the size of a stable matching.
We study the problem of finding "fair" stable matchings in the Stable Marriage problem with Incomplete lists (SMI). For an instance $I$ of SMI there may be many stable matchings, providing significantly different outcomes for the sets of men and women. We introduce two new notions of fairness in SMI. Firstly, a regret-equal stable matching minimises the difference in ranks of a worst-off man and a worst-off woman, among all stable matchings. Secondly, a min-regret sum stable matching minimises the sum of ranks of a worst-off man and a worst-off woman, among all stable matchings. We present two…
We study a natural generalization of stable matching to the maximum weight stable matching problem and we obtain a combinatorial polynomial time algorithm for it by reducing it to the problem of finding a maximum weight ideal cut in a DAG. We give the first polynomial time algorithm for the latter problem; this algorithm is also combinatorial. The combinatorial nature of our algorithms not only means that they are efficient but also that they enable us to obtain additional structural and algorithmic results: - We show that the set, $\cal M'$, of maximum weight stable matchings forms a sublatti…
A super-stable matching, which was introduced by Irving, is a solution concept in a variant of the stable matching problem in which the preferences may contain ties. Irving proposed a polynomial-time algorithm for the problem of finding a super-stable matching if a super-stable matching exists. In this paper, we consider a matroid generalization of a super-stable matching. We call our generalization of a super-stable matching a super-stable common independent set. This can be considered as a generalization of the matroid generalization of a stable matching for strict preferences proposed by Fl…
We study the problem of finding solutions to the stable matching problem that are robust to errors in the input and we obtain a polynomial time algorithm for a special class of errors. In the process, we also initiate work on a new structural question concerning the stable matching problem, namely finding relationships between the lattices of solutions of two "nearby" instances. Our main algorithmic result is the following: We identify a polynomially large class of errors, $D$, that can be introduced in a stable matching instance. Given an instance $A$ of stable matching, let $B$ be the random…
Despite significant research efforts, the state-of-the-art algorithm for maintaining an approximate matching in fully dynamic graphs has a polynomial {worst-case} update time, even for very poor approximation guarantees. In a recent breakthrough, Bhattacharya, Henzinger and Nanongkai showed how to maintain a constant approximation to the minimum vertex cover, and thus also a constant-factor estimate of the maximum matching size, with polylogarithmic worst-case update time. Later (in SODA'17 Proc.) they improved the approximation factor all the way to $2+ε$. Nevertheless, the longstanding funda…
An instance of the super-stable matching problem with incomplete lists and ties is an undirected bipartite graph $G = (A \cup B, E)$, with an adjacency list being a linearly ordered list of ties. Ties are subsets of vertices equally good for a given vertex. An edge $(x,y) \in E \backslash M$ is a blocking edge for a matching $M$ if by getting matched to each other neither of the vertices $x$ and $y$ would become worse off. Thus, there is no disadvantage if the two vertices would like to match up. A matching $M$ is super-stable if there is no blocking edge with respect to $M$. It has previously…
The stable roommates problem can admit multiple different stable matchings. We have different criteria for deciding which one is optimal, but computing those is often NP-hard. We show that the problem of finding generous or rank-maximal stable matchings in an instance of the roommates problem with incomplete lists is NP-hard even when the preference lists are at most length 3. We show that just maximising the number of first choices or minimising the number of last choices is NP-hard with the short preference lists. We show that the number of $R^{th}$ choices, where $R$ is the minimum-regret o…
Gale and Shapley's stable assignment problem has been extensively studied, applied, and extended. In the context of school choice, mechanisms often aim at finding an assignment that is more favorable to students. We investigate two extensions introduced in this framework -- legal assignments and the EADAM algorithm -- through the lens of classical theory of stable matchings. In any instance, the set ${\cal L}$ of legal assignments is known to contain all stable assignments. We prove that ${\cal L}$ is exactly the set of stable assignments in another instance. Moreover, we show that essentially…
We introduce the problem of adapting a stable matching to forced and forbidden pairs. Specifically, given a stable matching $M_1$, a set $Q$ of forced pairs, and a set $P$ of forbidden pairs, we want to find a stable matching that includes all pairs from $Q$, no pair from $P$, and that is as close as possible to $M_1$. We study this problem in four classical stable matching settings: Stable Roommates (with Ties) and Stable Marriage (with Ties). As our main contribution, we employ the theory of rotations for Stable Roommates to develop a polynomial-time algorithm for adapting Stable Roommates m…
We study the problem of finding "fair" stable matchings in the Stable Marriage problem with Incomplete lists (SMI). In particular, we seek stable matchings that are optimal with respect to profile, which is a vector that indicates the number of agents who have their first-, second-, third-choice partner, etc. In a rank maximal stable matching, the maximum number of agents have their first-choice partner, and subject to this, the maximum number of agents have their second-choice partner, etc., whilst in a generous stable matching $M$, the minimum number of agents have their $d$th-choice partner…
We study (coalitional) exchange stability, which Alcalde [Economic Design, 1995] introduced as an alternative solution concept for matching markets involving property rights, such as assigning persons to two-bed rooms. Here, a matching of a given Stable Marriage or Stable Roommates instance is called coalitional exchange-stable if it does not admit any exchange-blocking coalition, that is, a subset S of agents in which everyone prefers the partner of some other agent in S. The matching is exchange-stable if it does not admit any exchange-blocking pair, that is, an exchange-blocking coalition o…
Let $G$ be a graph and suppose we are given, for each $v \in V(G)$, a strict ordering of the neighbors of $v$. A set of matchings ${\cal M}$ of $G$ is called internally stable if there are no matchings $M,M' \in {\cal M}$ such that an edge of $M$ blocks $M'$. The sets of stable (à la Gale and Shapley) matchings and of von Neumann-Morgenstern stable matchings are examples of internally stable sets of matching. In this paper, we study, in both the marriage and the roommate case, inclusionwise maximal internally stable sets of matchings. We call those sets internally closed. By building on known …
While the stable marriage problem and its variants model a vast range of matching markets, they fail to capture complex agent relationships, such as the affiliation of applicants and employers in an interview marketplace. To model this problem, the existing literature on matching with externalities permits agents to provide complete and total rankings over matchings based off of both their own and their affiliates' matches. This complete ordering restriction is unrealistic, and further the model may have an empty core. To address this, we introduce the Dichotomous Affiliate Stable Matching (DA…
Stable matching in a community consisting of men and women is a classical combinatorial problem that has been the subject of intense theoretical and empirical study since its introduction in 1962 in a seminal paper by Gale and Shapley, who designed the celebrated ``deferred acceptance'' algorithm for the problem. In the input, each participant ranks participants of the opposite type, so the input consists of a collection of permutations, representing the preference lists. A bipartite matching is unstable if some man-woman pair is blocking: both strictly prefer each other to their partner in th…
An instance $I$ of the Stable Matching Problem (SMP) is given by a bipartite graph with a preference list of neighbors for every vertex. A swap in $I$ is the exchange of two consecutive vertices in a preference list. A swap can be viewed as a smallest perturbation of $I$. Boehmer et al. (2021) designed a polynomial-time algorithm to find the minimum number of swaps required to turn a given maximal matching into a stable matching. We generalize this result to the many-to-many version of SMP. We do so first by introducing a new representation of SMP as an extended bipartite graph and subsequentl…
