A characterization of strongly stable fractional matchings

Journal of the Operations Research Society of China, 2020

For the marriage model with indi¤erences, we de…ne an equivalence relation over the stable matching set. We identify a su¢ cient condition, the closing property, under which we can extend results of the classical model (without indi¤erences) to the equivalence classes of the stable matching set. This condition allows us to extend the lattice structure over classes of equivalences and the rural hospital theorem.

Discrete Mathematics, 2000

In the theory of two-sided matching markets there are two well-known mod- els: the marriage model (where no money is involved) and the assignment model (where payments are involved). Roth and Sotomayor (1990) asked for an expla- nation for the similarities in behavior between those two models. We address this question by introducing a common generalization that preserves the two

2020

In this paper we study the lattice structure of the random matching set, that is, all lotteries of stable matchings. We define the \textit{l.u.b.} (least upper bound) and the \textit{g.l.b.} (greatest lower bound) for both sides of the matching market, and we prove the that with these binary operations the set of random matchings has two dual lattices.

Journal of Statistical Mechanics: Theory and Experiment, 2018

We consider two formulations of the random-link fractional matching problem, a relaxed version of the more standard random-link (integer) matching problem. In one formulation, we allow each node to be linked to itself in the optimal matching configuration. In the other one, on the contrary, such a link is forbidden. Both problems have the same asymptotic average optimal cost of the random-link matching problem on the complete graph. Using a replica approach and previous results of Wästlund [1], we analytically derive the finitesize corrections to the asymptotic optimal cost. We compare our results with numerical simulations and we discuss the main differences between random-link fractional matching problems and the random-link matching problem.

arXiv (Cornell University), 2020

For a many-to-many matching market, we study the lattice structure of the set of random stable matchings. We define a partial order on the random stable set and present two intuitive binary operations to compute the least upper bound and the greatest lower bound for each side of the matching market. Then, we prove that with these binary operations the set of random stable matchings forms two dual lattices.

2021

We present a framework for deterministically rounding a dynamic fractional matching. Applying our framework in a black-box manner on top of existing fractional matching algorithms, we derive the following new results: (1) The first deterministic algorithm for maintaining a $(2-\delta)$-approximate maximum matching in a fully dynamic bipartite graph, in arbitrarily small polynomial update time. (2) The first deterministic algorithm for maintaining a $(1+\delta)$-approximate maximum matching in a decremental bipartite graph, in polylogarithmic update time. (3) The first deterministic algorithm for maintaining a $(2+\delta)$-approximate maximum matching in a fully dynamic general graph, in small polylogarithmic (specifically, $O(\log^4 n)$) update time. These results are respectively obtained by applying our framework on top of the fractional matching algorithms of Bhattacharya et al. [STOC'16], Bernstein et al. [FOCS'20], and Bhattacharya and Kulkarni [SODA'19]. Prior to o...

Lecture Notes in Computer Science, 2021

An instance of the super-stable matching problem with incomplete lists and ties is an undirected bipartite graph G = (A ∪ 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) ∈ E\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 been shown that super-stable matchings form a distributive lattice [1, 2] and the number of super-stable matchings can be exponential in the number of vertices. We give two compact representations of size O(m) that can be used to construct all super-stable matchings, where m denotes the number of edges in the graph. The construction of the second representation takes O(mn) time, where n denotes the number of vertices in the graph, and gives an explicit rotation poset similar to the rotation poset in the classical stable marriage problem. We also give a polyhedral characterisation of the set of all super-stable matchings and prove that the super-stable matching polytope is integral, thus solving an open problem stated in the book by Gusfield and Irving [3].

International Journal of Game Theory, 2008

We study the dynamics of stable marriage and stable roommates markets. Our main tool is the incremental algorithm of Roth and Vande Vate and its generalization by Tan and Hsueh. Beyond proposing alternative proofs for known results, we also generalize some of them to the nonbipartite case. In particular, we show that the lastcomer gets his best stable partner in both incremental algorithms. Consequently, we confirm that it is better to arrive later than earlier to a stable roommates market. We also prove that when the equilibrium is restored after the arrival of a new agent, some agents will be better off under any stable solution for the new market than at any stable solution for the original market. We also propose a procedure to find these agents.

Discrete Applied Mathematics, 2013

We give some upper bounds on the maximum number of stable matchings in the Gale-Shapley marriage model with n men and n women. We also characterize, with the use of some graph-theoretical notions, the exact number of such matchings, assuming that the preferences of men and women are given.

Graphs and Combinatorics, 1992

We study fractional matchings and covers in infinite hypergraphs, paying particular attention to the following questions: Do fractional matchings (resp. covers) of maximal (resp. minimal) size exist? Is there equality between the supremum of the sizes of fractional matchings and the infimum of the sizes of fractional covers? (This is called weak duality.) Are there a fractional matching and a fractional cover that satisfy the complementary slackness conditions of linear programming? (This is called strong duality.) In general, the answers to all these questions are negative, but for certain classes of infinite hypergraphs (classified according to edge cardinalities and vertex degrees) we obtain positive results. We also consider the question of the existence of optimal fractional matchings and covers that assume rational values. * Incumbent of the Robert Edward and Roselyn Rich Manson Career Development Chair.

2021

In a many-to-one matching model, we study the set of worker-quasi-stable matchings when firms’ preferences satisfy substitutability. Worker-quasi-stability is a relaxation of stability that allows blocking pairs involving a firm and an unemployed worker. We show that this set has a lattice structure and define a Tarski operator on this lattice that models a re-equilibration process and has the set of stable matchings as its fixed points. JEL classification: C78, D47.

Mathematics of Operations Research, 2006

Baïou and Balinski characterized the stable admissions polytope using a system of linear inequalities. The structure of feasible solutions to this system of inequalities—fractional stable matchings—is the focus of this paper. The main result associates a geometric structure with each fractional stable matching. This insight appears to be interesting in its own right, and can be viewed as a generalization of the lattice structure (for integral stable matchings) to fractional stable matchings. In addition to obtaining simple proofs of many known results, the geometric structure is used to prove the following two results: First, it is shown that assigning each agent their “median” choice among all stable partners results in a stable matching, which can be viewed as a “fair” compromise; second, sufficient conditions are identified under which stable matchings exist in a problem with externalities, in particular, in the stable matching problem with couples.

Review of Economic Design, 2001

This paper studies the structure of stable multipartner matchings in two-sided markets where choice functions are quotafilling in the sense that they satisfy the substitutability axiom and, in addition, fill a quota whenever possible. It is shown that (i) the set of stable matchings is a lattice under the common revealed preference orderings of all agents on the same side, (ii) the supremum (infimum) operation of the lattice for each side consists componentwise of the join (meet) operation in the revealed preference ordering of the agents on that side, and (iii) the lattice has the polarity, distributivity, complementariness and full-quota properties.

Lecture Notes in Computer Science

Let I be a stable matching instance with N stable matchings. For each man m, order his N stable partners from his most preferred to his least preferred. Denote the ith woman in his sorted list as pi(m). Let αi consist of the man-woman pairs where each man m is matched to pi(m). Teo and Sethuraman proved this surprising result: for i = 1 to N , not only is αi a matching, it is also stable. The αi's are called the generalized median stable matchings of I. In this paper, we present a new characterization of these stable matchings that is solely based on I's rotation poset. We then prove the following: when i = O(log n), where n is the number of men, αi can be found efficiently; but when i is a constant fraction of N , finding αi is NPhard. We also consider what it means to approximate the median stable matching of I, and present results for this problem.

Oper. Res. Lett., 2021

A bipartite graph consists of two disjoint vertex sets, where vertices of one set can only be joined with an edge to vertices in the opposite set. Hall's theorem gives a necessary and sufficient condition for a bipartite graph to have a saturating matching, meaning every vertex in one set is matched to some vertex in the other in a one-to-one correspondence. When we imagine vertices as agents and let them have preferences over other vertices, we have the classic stable marriage problem introduced by Gale and Shapley, who showed that one can always find a matching that is stable with respect to agent's preferences. These two results often clash: saturating matchings are not always stable, and stable matchings are not always saturating. I prove a simple necessary and sufficient condition for every stable matching being saturating for one side. I show that this result subsumes and generalizes some previous theorems in the matching literature. I find a necessary and sufficient c...