We give a $\frac{3}{2}$ -approximation algorithm for finding stable matchings that runs in O(m) time. The previous most well-known algorithm, by McDermid, has the same approximation ratio but runs in O(n^3/2m) time, where n denotes the number of people andm is the total length of the preference lists in a given instance. In addition, the algorithm and the analysis are much simpler. We also give the extension of the algorithm for computing stable many-to-many matchings. Full article

We build an abstract model, closely related to the stable marriage problem and motivated by Hungarian college admissions. We study different stability notions and show that an extension of the lattice property of stable marriages holds in these more general settings, even if the choice function on one side is not path independent. We lean on Tarski’s fixed point theorem and the substitutability property of choice functions. The main virtue of the work is that it exhibits practical, interesting examples, where non-path independent choice functions play a role, and proves various stability-related results. Full article

We describe a flow model related to ordinary network flows the same way as stable matchings are related to maximum matchings in bipartite graphs. We prove that there always exists a stable flow and generalize the lattice structure of stable marriages to stable flows. Our main tool is a straightforward reduction of the stable flow problem to stable allocations. For the sake of completeness, we prove the results we need on stable allocations as an application of Tarski’s fixed point theorem. Full article

A key property of overlay networks is the overlay nodes’ ability to establish connections (or be matched) to other nodes by preference, based on some suitability metric related to, e.g., the node’s distance, interests, recommendations, transaction history or available resources. When there are no preference cycles among the nodes, a stable matching exists in which nodes have maximized individual satisfaction, due to their choices, however no such guarantees are currently being given in the generic case. In this work, we employ the notion of node satisfaction to suggest a novel modeling for matching problems, suitable for overlay networks. We start by presenting a simple, yet powerful, distributed algorithm that solves the many-to-many matching problem with preferences. It achieves that by using local information and aggregate satisfaction as an optimization metric, while providing a guaranteed convergence and approximation ratio. Subsequently, we show how to extend the algorithm in order to support and adapt to changes in the nodes’ connectivity and preferences. In addition, we provide a detailed experimental study that focuses on the levels of achieved satisfaction, as well as convergence and reconvergence speed. Full article

The stable matching problem (also known as the stable marriage problem) is a well-known problem of matching men to women, so that no man and woman, who are not married to each other, both prefer each other. Such a problem has a wide variety of practical applications, ranging from matching resident doctors to hospitals, to matching students to schools or, more generally, to any two-sided market. In the classical stable marriage problem, both men and women express a strict preference order over the members of the other sex, in a qualitative way. Here, we consider stable marriage problems with weighted preferences: each man (resp., woman) provides a score for each woman (resp., man). Such problems are more expressive than the classical stable marriage problems. Moreover, in some real-life situations, it is more natural to express scores (to model, for example, profits or costs) rather than a qualitative preference ordering. In this context, we define new notions of stability and optimality, and we provide algorithms to find marriages that are stable and/or optimal according to these notions. While expressivity greatly increases by adopting weighted preferences, we show that, in most cases, the desired solutions can be found by adapting existing algorithms for the classical stable marriage problem. We also consider the manipulability properties of the procedures that return such stable marriages. While we know that all procedures are manipulable by modifying the preference lists or by truncating them, here, we consider if manipulation can occur also by just modifying the weights while preserving the ordering and avoiding truncation. It turns out that, by adding weights, in some cases, we may increase the possibility of manipulating, and this cannot be avoided by any reasonable restriction on the weights. Full article

The stable marriage (SM) problem has a wide variety of practical applications, ranging from matching resident doctors to hospitals, to matching students to schools or, more generally, to any two-sided market. In the classical formulation, n men and n women express their preferences (via a strict total order) over the members of the other sex. Solving an SM problem means finding a stable marriage where stability is an envy-free notion: no man and woman who are not married to each other would both prefer each other to their partners or to being single. We consider both the classical stable marriage problem and one of its useful variations (denoted SMTI (Stable Marriage with Ties and Incomplete lists)) where the men and women express their preferences in the form of an incomplete preference list with ties over a subset of the members of the other sex. Matchings are permitted only with people who appear in these preference lists, and we try to find a stable matching that marries as many people as possible. Whilst the SM problem is polynomial to solve, the SMTI problem is NP-hard. We propose to tackle both problems via a local search approach, which exploits properties of the problems to reduce the size of the neighborhood and to make local moves efficiently. We empirically evaluate our algorithm for SM problems by measuring its runtime behavior and its ability to sample the lattice of all possible stable marriages. We evaluate our algorithm for SMTI problems in terms of both its runtime behavior and its ability to find a maximum cardinality stable marriage. Experimental results suggest that for SM problems, the number of steps of our algorithm grows only as O(n log(n)), and that it samples very well the set of all stable marriages. It is thus a fair and efficient approach to generate stable marriages. Furthermore, our approach for SMTI problems is able to solve large problems, quickly returning stable matchings of large and often optimal size, despite the NP-hardness of this problem. Full article

In this paper, the notion of stability is extended to network flows over time. As a useful device in our proofs, we present an elegant preflow-push variant of the Gale-Shapley algorithm that operates directly on the given network and computes stable flows in pseudo-polynomial time, both in the static flow and the flow over time case. We show periodical properties of stable flows over time on networks with an infinite time horizon. Finally, we discuss the influence of storage at vertices, with different results depending on the priority of the corresponding holdover edges. Full article

We consider a two-sided market under incomplete preference lists with ties, where the goal is to find a maximum size stable matching. The problem is APX-hard, and a 3/2-approximation was given by McDermid [1]. This algorithm has a non-linear running time, and, more importantly needs global knowledge of all preference lists. We present a very natural, economically reasonable, local, linear time algorithm with the same ratio, using some ideas of Paluch [2]. In this algorithm every person make decisions using only their own list, and some information asked from members of these lists (as in the case of the famous algorithm of Gale and Shapley). Some consequences to the Hospitals/Residents problem are also discussed. Full article

Motivated by the observation that most companies are more likely to consider job applicants referred by their employees than those who applied on their own, Arcaute and Vassilvitskii modeled a job market that integrates social networks into stable matchings in an interesting way. We call their model HR+SN because an instance of their model is an ordered pair (I, G) where I is a typical instance of the Hospital/Residents problem (HR) and G is a graph that describes the social network (SN) of the residents in I. A matching p, of hospitals and residents has a local blocking pair (h, r) if (h, r) is a blocking pair of ii, and there is a resident r' such that r' is simultaneously an employee of h in the matching and a neighbor of r in G. Such a pair is likely to compromise the matching because the participants have access to each other through r': r can give her resume to r' who can then forward it to h. A locally stable matching is a matching with no local blocking pairs. The cardinality of the locally stable matchings of I can vary. This paper presents a variety of results on computing a locally stable matching with maximum cardinality. Full article

In the stable marriage problem, any instance admits the so-called man-optimal stable matching, in which every man is assigned the best possible partner. However, there are instances for which all men receive low-ranked partners even in the man-optimal stable matching. In this paper we consider the problem of improving the man-optimal stable matching by changing only one man’s preference list. We show that the optimization variant and the decision variant of this problem can be solved in time O(n³) and O(n²), respectively, where n is the number of men (women) in an input. We further extend the problem so that we are allowed to change k men’s preference lists. We show that the problem is W[1]-hard with respect to the parameter k and give O(n^2k+1)-time and O(n^k+1)-time exact algorithms for the optimization and decision variants, respectively. Finally, we show that the problems become easy when k = n; we give O(n^2.5 log n)-time and O(n²)-time algorithms for the optimization and decision variants, respectively. Full article

We extend the stable flow model of Fleiner to multicommodity flows. In addition to the preference lists of agents on trading partners for each commodity, every trading pair has a preference list on the commodities that the seller can sell to the buyer. A blocking walk (with respect to a certain commodity) may include saturated arcs, provided that a positive amount of less preferred commodity is traded along the arc. We prove that a stable multicommodity flow always exists, although it is PPAD-hard to find one. Full article