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From One Stable Marriage to the Next: How Long Is the Way?
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.
... 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. ...
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 . ...
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]. ...
We provide a linear description of the unconstrained stable allocations problem by proving that the corresponding polytope is affinely congruent to the order polytope of a partially ordered set. The same holds for stable matchings hence simplifying the derivation of known polyhedral results. We also show that this congruence no longer holds for the constrained version of stable allocations. As side outcomes, we characterise the neighbouring vertices of the order polytope and the partially ordered set associated with stable allocations.
An upper bound on the diameter of the Stable Matching (Stable Marriage) polytope is known to be ⌊n2⌋ where n is the number of men (or women) involved in the matching. The current work complements that result by providing a lower bound and an algorithm computing it. It also presents a class of Stable Matching instances for which the lower bound coincides with the above-mentioned upper bound.
We show that it isD
P
-hard to determine the combinatorial diameter of a polytope specified by linear inequalities with integer data. Our result partially resolves a long-term open question.
We prove that the combinatorial diameter of the skeleton of the polytope of feasible solutions of any m £ n transportation problem is less than 12 (m + n) 2 . The transportation problem (TP) is a classic problem in operations research. The problem was posed for the first time by Hitchcock in 1941 (8) and independently by Koopmans in 1947 (11), and appears in any standard introductory course on operations research. The m £ n TP has m supply points and n demand points. Each supply point i holds a quantity ri > 0, and each demand point j wants a quantity cj > 0, with m P i=1 ri = n P j=1 cj. A solution to the problem can be written as a m£n matrix X with entries decision variables xij having value equal to the amount transported from supply point i to demand point j. The
The third author [Oper. Res. Lett. 8, No. 3, 147-153 (1989; Zbl 0675.90058)] described the polytope whose extreme points are the stable (core) matchings in the Marriage Problem. The second author [Math. Program., Ser. A 54, No. 1, 57-67 (1992; Zbl 0773.90059)] simplified and extended this result. This paper explores a corresponding linear program, its dual and consequences of the fact that the dual solutions have an unusually direct relation to the primal solutions. This close relationship allows us to provide simple proofs both of Vande Vate and Rothblum’s results and of other important results about the core of marriage markets. These proofs help explain the structure shared by the marriage problem (without sidepayments) and the assignment game (with sidepayments). The paper further explores “fractional” matchings, which may be interpreted as lotteries over possible matches or as time-sharing arrangements. We show that those fractional matchings in the Stable Marriage Polytope form a lattice with respect to a partial ordering that involves stochastic dominance. Thus, all expected utility functions corresponding to the same ordinal preferences will agree on the relevant comparisons. Finally, we provide linear programming proofs of slightly stronger versions of known incentive compatibility results.
In the setting of the stable matching (SM) problem, it has been observed that some of the man-woman pairs cannot be removed although they participate in no stable matching, since such a removal would alter the set of solutions. These pairs are yet to be identified. Likewise (and despite the sizeable literature), some of the fundamental characteristics of the SM polytope (e.g., its dimension, its facets, etc.) have not been established. In the current work, we show that these two seemingly distant open issues are closely related. More specifically, we identify the pairs with the above-mentioned property and present a polynomial algorithm for producing a set of minimal preference lists. We utilize this result in the context of two different representations of the SM structure (rotation-poset graph and algebraic formulation) and derive the dimension of the SM polytope to obtain all alternative minimal linear descriptions.
We study the classical stable marriage and stable roommates problems using a polyhedral approach. We propose a new LP formulation for the stable roommates problem, which has a feasible solution if and only if the underlying roommates problem has a stable matching. Furthermore, for certain special weight functions on the edges, we construct a a-approximation algorithm for the optimal stable roommates problem. Our technique exploits features of the geometry of fractional solutions of this formulation. For the stable marriage problem, we show that a related geometry allows us to express any fractional solution in the stable marriage polytope as a convex combination of stable marriage solutions. This also leads to a genuinely simple proof of the integrality of the stable marriage polytope.
This expository paper develops the principal known results (and some new ones) on the stable matchings of marriage games in the language of directed graphs. This both unifies and simplifies the presentation and renders it more symmetric. In addition, it yields a new algorithm and a new proof for the existence of stable matchings, new proofs for many known facts, and some new results (notably concerning players' strategies and the properties of the stable matching polytope).
Blair (J. Combin. Theory Ser. A37 (1984), 353–356) showed that every finite distributive lattice is the weak dominance relation for some instance of the stable marriage problem, but the only bound given on the size of the instance was 2k for a k element lattice. In this note we describe a method which, for any distributive lattice L of k elements, constructs an instance of size at most k2 − k + 4. Further, we note that if the smallest instance for lattice L has size 2n, then the construction in this paper has size at most .
This paper demonstrates that the celebrated Gale-Shapley algorithm for obtaining stable matchings in stable marriage problems is essentially an application of the dual simplex method.
A stable matching is an assignment of n men to n women so that no two people prefer each other to their respective spouses. This paper describes the convex hull of the incidence vectors of stable matchings. With this description, one may solve the optimal stable marriage problem as a linear program.
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.
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.
The purpose of this paper is to extend a modified version of a recent result of Vande Vate (1989) which characterizes stable matchings as the extreme points of a certain polytope. Our proofs are simpler and more transparent than those of Vande Vate.
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.
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.
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.
This paper discusses properties of the graphs of 2-way and 3-way transportation polytopes, in particular, their possible numbers of vertices and their diameters. Our main results include a quadratic bound on the diameter of axial 3-way transportation polytopes and a catalogue of non-degenerate transportation polytopes of small sizes. The catalogue disproves five conjectures about these polyhedra stated in the monograph by Yemelichev et al. (1984). It also allowed us to discover some new results. For example, we prove that the number of vertices of an $m\times n$ transportation polytope is a multiple of the greatest common divisor of $m$ and $n$. Comment: 29 pages, 7 figures. Final version. Improvements to the exposition of several lemmas and the upper bound in Theorem 1.1 is improved by a factor of two