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From One Stable Marriage to the Next: How Long Is the Way?

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Abstract

The diameter of the stable matching (stable marriage) polytope is bounded from above by [n/2], where n is the number of men (or women) involved in the matching; this bound is attainable.

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... diameter results for matchings, TSP, or network flows and transportation in [2,7,11,21,24,4,6,3,5,26]). For the stable marriage polytope, which we call P SM here, Eirinakis et al. [8] proved a diameter upper bound of ⌊n/4⌋, where n := |M ∪ W |. The authors also show the existence of instances for which this bound holds tight. ...
... Our result generalizes what is known for P SM , meaning that if all preference orderings are strict, then it recovers the bound given in [8]. However, it relies on different and new ingredients, which we are going to describe next. ...
... A key tool used in [8] to bound the diameter of P SM is the so-called stable marriage graph, introduced in [17]. The stable marriage graph is an auxiliary graph that one can construct (in polynomial time) for a given instance of the standard stable marriage problem. ...
Preprint
The stable marriage problem with ties is a well-studied and interesting problem in game theory. We are given a set of men and a set of women. Each individual has a preference ordering on the opposite group, which can possibly contain ties. A stable marriage is given by a matching between men and women for which there is no blocking pair, i.e., a men and a women who strictly prefer each other to their current partner in the matching. In this paper, we study the diameter of the polytope given by the convex hull of characteristic vectors of stable marriages, in the setting with ties. We prove an upper bound of $\lfloor \frac{n}{3}\rfloor$ on the diameter, where $n$ is the total number of men and women, and give a family of instances for which the bound holds tight. Our result generalizes the bound on the diameter of the standard stable marriage polytope (i.e., the well-known polytope that describes the setting without ties), developed previously in the literature.
... [math.CO] 9 Jun 2018 of the diameter of polytopes that correspond to the set of feasible solutions of classical combinatorial optimization problems. Just to mention a few, such problems include matchings [3,8], TSP [19,20], edge cover [13], fractional stable set [18], network flows and transportation problems [2,5,6,7], stable marriage [12], and many more. ...
... Clearly invariant (12) holds for w, because of the definition of witnesses. The algorithm selects one cycle in C ∈ C \ C y at the time, and performs a move which involves at most one path-component of the target graph. ...
... Invariant (12) holds trivially, since if we use a token of a witness node v k to pay for moving from to¯ , then v k was a single witness and the cycle witnessed by v k is in C¯ ∩ C y . ...
Preprint
The (combinatorial) diameter of a polytope $P \subseteq \mathbb R^d$ is the maximum value of a shortest path between a pair of vertices on the 1-skeleton of $P$, that is the graph where the nodes are given by the $0$-dimensional faces of $P$, and the edges are given the 1-dimensional faces of $P$. The diameter of a polytope has been studied from many different perspectives, including a computational complexity point of view. In particular, [Frieze and Teng, 1994] showed that computing the diameter of a polytope is (weakly) NP-hard. In this paper, we show that the problem of computing the diameter is strongly NP-hard even for a polytope with a very simple structure: namely, the \emph{fractional matching} polytope. We also show that computing a pair of vertices at maximum shortest path distance on the 1-skeleton of this polytope is an APX-hard problem. We prove these results by giving an \emph{exact characterization} of the diameter of the fractional matching polytope, that is of independent interest.
... To accomplish this, we characterise the neighbouring vertices of the order polytope, as an addition to the work of [4]. As a corollary, we explain why a bound on the diameter of the MM polytope is obtainable as for the SM polytope in [14]. ...
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A kernel of a directed graph D is a set of vertices which is both independent and absorbant. In 1983, Berge and Duchet conjectured that an undirected graph G is perfect if and only if the following condition is satisfied: “If D is any orientation of G such that every clique of D has a kernel, then D has a kernel.” We prove here that the conjecture holds when G is the line-graph of another graph H, i.e., G represents the incidence between the edges of H.
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The stable marriage problems is a well-known problem of matching n men to n women to achieve a certain type of 'stability; ' the O(n**2) time Gale-Shapley algorithm for finding two particular, but extreme, stable marriages (out of a possibly exponential number of stable marriages) is also well known. In this paper we consider four problems concerned with finding information about the set of all stable marriages, and with finding stable marriages other than those obtained by the Gale-Shapley algorithm. In particular, we give an O(n**2) time algorithm which, for any problem instance of n men and n women, finds every man-woman pair that is contained in at least one stable marriage; we show that the same algorithm finds all the 'rotations' for the problem instance in O(n**2) time; we give an O(n**2 plus n vertical S vertical ) time and O(n**2) space bounded algorithm to enumerate all stable marriages, where S is the set of them; and we give an O(n**2) time algorithm to find the minimum regret stable marriage.
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We describe a fixed-point based approach to the theory of bipartite stable matchings. By this, we provide a common framework that links together seemingly distant results, like the stable marriage theorem of Gale and Shapley, the Mendelsohn-Dulmage theorem, the Kundu-Lawler theorem, Tarski's fixed-point theorem, the Cantor-Bernstein theorem, Pym's linking theorem, or the monochromatic path theorem of Sands et al. In this framework, we formulate a matroid-generalization of the stable marriage theorem and study the lattice structure of generalized stable matchings. Based on the theory of lattice polyhedra and blocking polyhedra, we extend results of Vande Vate and Rothblum on the bipartite stable matching polytope.
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The stable marriage problem is a game theoretic model introduced by Gale and Shapley (1962). It involves two sets of players referred to as men and women. A marriage is a set of disjoint pairs, where each pair consists of a woman and a man. Each individual has a strict linear order of preference over the set of opposite sex. A marriage is called stable if there is no man and woman who both prefer being matched to each other over the outcome they obtained in the marriage.Conway showed that the set of stable marriages can be ordered as a lattice. Vande Vate (1989) described the polytope whose extreme points are the set of stable marriages. Rothblum simplified and extended this result.In this paper the marriage problem is reformulated in terms of a marriage market graph. A stable marriage is characterized as a kernel of the graph. Equivalent marriage graphs are those having the same sets of stable marriages, and it is shown how a “simplest” such graph can be obtained. The faces of the polytope are characterized in terms of the lattice of stable marriages.
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The paper continues recent work that introduced algebraic methods for studying the stable marriage problem of Gale and Shapley (1962). Vande Vate (1989) and Rothblum (1992) identified a set of linear inequalities which define a polytope whose extreme points correspond to the stable matchings. Points in the polytope are called fractional stable matchings. Here we identify a unique representation of fractional stable matchings as a convex combination of stable matchings that are arrangeable in a man-decreasing order. We refer to this representation and to a dual one, in terms of woman-decreasing order, as the canonical monotone representations. These representations can be interpreted as time-sharing stable matchings where particular stable matchings are used at each time-instance but the scheduled stable matchings are (occasionally) switched over time. The new representations allow us to extend, in a natural way, the lattice structure of the set of stable matchings to the set of all fractional stable matchings.
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