Marriage Theorem Research Articles

On Weighted Kernels of Two Posets

A recent result of Aharoni Berger and Gorelik (Order 31(1), 35–43, 2014) is a weighted generalization of the well-known theorem of Sands Sauer and Woodrow (Theory Ser. B 33(3), 271–275, 1982) on monochromatic paths. The authors prove the existence of a so called weighted kernel for any pair of weighted posets on the same ground set. In this work, we point out that this result is closely related to the stable marriage theorem of Gale and Shapley (Amer. Math. Monthly 69(1), 9–15, 1962), and we generalize Blair’s theorem by showing that weighted kernels form a lattice under a certain natural order. To illustrate the applicability of our approach, we prove further weighted generalizations of the Sands Sauer Woodrow result.

On Simple Multiple Access Networks

We investigate a simple multiple access network (SMAN) where k independent sources of unit rates multicast their information to a set of sinks, via n commonly shared relays. All links are assumed to have unit capacity. Given such a SMAN, a coding scheme for the relays is called optimal if each sink can retrieve all information from the sources under at most ⌊n-k+1/2⌋ node/link errors. We study the problem of designing the sparsest SMAN, i.e., the SMAN that has the least number of edges, that supports an optimal coding scheme for the relays. Additionally, the SMAN must satisfy either of the following constraints: 1) Connection Constraint: Each relay can be connected only to a given subset of sources or 2) Balance Constraint: Each relay must be connected to approximately the same number of sources. We provide two polynomial time algorithms to identify the cases where such a SMAN exists together with its optimal coding scheme designed over sufficiently large fields. One algorithm is based on a nontrivial modification of the well-known Gale-Ryser algorithm, whereas the other is based on a novel generalization of the famous Hall's marriage theorem. We also propose a combinatorial approach to construct optimal coding schemes over small fields and settle the problem for a special case.

Finite structure for Friedman ideals of finite trees

By extending Hall's classical marriage theorem, we find a finite structure theorem for every Friedman ideal of finite trees.

On the strength of marriage theorems and uniformity

Kierstead showed that every computable marriage problem has a computable matching under the assumption of computable expanding Hall condition and computable local finiteness for boys and girls. The strength of the marriage theorem reaches or if computable expanding Hall condition or computable local finiteness for girls is weakened. In contrast, the provability of the marriage theorem is maintained in even if local finiteness for boys is completely removed. Using these conditions, we classify the strength of variants of marriage theorems in the context of reverse mathematics. Furthermore, we introduce another condition that also makes the marriage theorem provable in , and investigate the sequential and Weihrauch strength of marriage theorems under that condition.

Geometric variants of Hallʼs Theorem through Spernerʼs Lemma

Hallʼs marriage theorem can be regarded as a condition for the existence of a rainbow set of distinct points. Motivated by this interpretation, we introduce the notion of geometric Hall-type problems. We prove that a linear Hall-type condition guarantees the existence of a rainbow set of pairwise disjoint unit balls in Rn. We also prove that a quadratic Hall-type condition guarantees the existence of a rainbow set of points in general position on the plane. To prove these results, we present and extend a topological technique by Aharoni and Haxell that uses Spernerʼs lemma in tight triangulations of the k-dimensional simplex.

Multichannel Opportunistic Carrier Sensing for Stable Channel Access Control in Cognitive Radio Systems

In this paper we propose to use the well known game theoretic Gale-Shapley stable marriage theorem from game theory as a basis for spectrum allocation in cognitive radio networks. We analyze the performance of the proposed solution and provide tight lower and upper bounds on both the stable allocation and the optimal allocation performance. Then we present a novel opportunistic multichannel medium access control technique that achieves stable allocation within a single CSMA contention window. We discuss practical implementation issues and put forward two other varieties of the algorithm which have lower implementation complexity. Finally, we provide simulated examples.

A generalization of Hungarian method and Hall’s theorem with applications in wireless sensor networks

In this paper, we consider various problems concerning quasi-matchings and semi-matchings in bipartite graphs, which generalize the classical problem of determining a perfect matching in bipartite graphs. We prove a generalization of Hall’s marriage theorem, and present an algorithm that solves the problem of determining a lexicographically minimum g-quasi-matching (that is a set F of edges in a bipartite graph such that in one set of the bipartition every vertex v has at least g(v) incident edges from F, where g is a so-called need mapping, while on the other side of the bipartition the distribution of degrees with respect to F is lexicographically minimum). We obtain that finding a lexicographically minimum quasi-matching is equivalent to minimizing any strictly convex function on the degrees of the A-side of a quasi-matching and use this fact to prove a more general statement: the optima of any component-based strictly convex cost function on any subset of L1-sphere in Nn are precisely the lexicographically minimal elements of this subset. We also present an application in designing optimal CDMA-based wireless sensor networks.

The group marriage problem

Let G be a permutation group acting on [ n ] = { 1 , … , n } and V = { V i : i = 1 , … , n } be a system of n subsets of [ n ] . When is there an element g ∈ G so that g ( i ) ∈ V i for each i ∈ [ n ] ? If such a g exists, we say that G has a G-marriage subject to V . An obvious necessary condition is the orbit condition: for any nonempty subset Y of [ n ] , there is an element g ∈ G such that the image of Y under g is contained in ⋃ y ∈ Y V y . Keevash observed that the orbit condition is sufficient when G is the symmetric group S n ; this is in fact equivalent to the celebrated Hall's Marriage Theorem. We prove that the orbit condition is sufficient if and only if G is a direct product of symmetric groups. We extend the notion of orbit condition to that of k-orbit condition and prove that if G is the cyclic group C n where n ⩾ 4 or G acts 2-transitively on [ n ] , then G satisfies the ( n − 1 ) -orbit condition subject to V if and only if G has a G-marriage subject to V .

Balanced list edge-colourings of bipartite graphs

Galvin solved the Dinitz conjecture by proving that bipartite graphs are edge-choosable. We improve Galvin’s method and deduce from any colouring of the edges of bipartite graph G some further list edge-colouring properties of G. In particular, for bipartite graphs, it follows from the existence of balanced bipartite edge-colourings that balanced list edge-colourings exist as well. While the key to Galvin’s proof is the stable marriage theorem of Gale and Shapley, our result is based on the well-known “many-to-many” version of the stable matching theorem.

The Stable Roommates Problem with Choice Functions

The stable marriage theorem of Gale and Shapley states that for n men and n women there always exists a stable marriage scheme, that is, a set of marriages such that no man and woman mutually prefer one another to their partners. The stable marriage theorem was generalized in two directions: the stable roommates problem is the “one-sided” version, where any two agents on the market can form a partnership. The generalization by Kelso and Crawford is in the “two-sided” model, but on one side of the market agents have a so-called substitutable choice function, and stability is interpreted in a natural way. It turned out that even if both sides of the market have substitutable choice functions, there still exists a stable assignment. The latter version contains the “many-to-many” model where up to a personal quota, polygamy is allowed for both men and women in the two-sided market. The goal of this work is to solve the stable partnership problem, a generalization of the one-sided model with substitutable choice functions. We do not quite reach that: besides substitutability, we also need for the result the so called increasing property (or, in other terminology, the law of aggregate demand). Luckily, choice functions in well-known stable matching theorems comply with this property. The main result is a generalization of Irving’s algorithm, the first efficient method to solve the stable roommates problem. We deduce from the algorithm a generalization of Tan’s result on characterizing the existence of a stable matching and prove a generalization of the rural hospital theorem and the so-called splitting property of many-to-many stable matchings. We show that our algorithm is linear-time in some sense and indicate that for general (i.e. not necessarily increasing) substitutable choice functions the stable partnership problem is NP-complete.

Hall number for list colorings of graphs: Extremal results

Given a graph G = ( V , E ) and sets L ( v ) of allowed colors for each v ∈ V , a list coloring of G is an assignment of colors φ ( v ) to the vertices, such that φ ( v ) ∈ L ( v ) for all v ∈ V and φ ( u ) ≠ φ ( v ) for all u v ∈ E . The choice number of G is the smallest natural number k admitting a list coloring for G whenever | L ( v ) | ≥ k holds for every vertex v . This concept has an interesting variant, called Hall number, where an obvious necessary condition for colorability is put as a restriction on the lists L ( v ) . (On complete graphs, this condition is equivalent to the well-known one in Hall’s Marriage Theorem.) We prove that vertex deletion or edge insertion in a graph of order n > 3 may make the Hall number decrease by as much as n − 3 . This estimate is tight for all n . Tightness is deduced from the upper bound that every graph of order n has Hall number at most n − 2 . We also characterize the cases of equality; for n ≥ 6 these are precisely the graphs whose complements are K 2 ∪ ( n − 2 ) K 1 , P 4 ∪ ( n − 4 ) K 1 , and C 5 ∪ ( n − 5 ) K 1 . Our results completely solve a problem raised by Hilton, Johnson and Wantland [A.J.W. Hilton, P.D. Johnson, Jr., E. B. Wantland, The Hall number of a simple graph, Congr. Numer. 121 (1996), 161–182, Problem 7] in terms of the number of vertices, and strongly improve some estimates due to Hilton and Johnson [A.J.W. Hilton, P.D. Johnson, Jr., The Hall number, the Hall index, and the total Hall number of a graph, Discrete Appl. Math. 94 (1999), 227–245] as a function of maximum degree.

Commuting conjugacy classes: an application of Hall's marriage theorem to group theory

Let G be a finite group. Given two conjugacy classes C and D of G, we shall say that C commutes withD, and write C ∼ D, if there exist elements c ∈ C and d ∈ D such that c and d commute. In this paper we are particularly concerned with the case where G has a normal subgroup H such that G/H is cyclic; in this case, since G/H is abelian, each conjugacy class of G is entirely contained within a particular coset of H. We establish some results concerning the distribution between the cosets of H of pairs of conjugacy classes of G related by ∼. In order to state our theorems, we make the following definition: a conjugacy class gG of G is non-split if gG = gH , and split otherwise. Observe that gG is non-split if and only if the centre CentG(g) of G meets every coset of H in G, or equivalently, if and only if g commutes with an element in a generating coset of G/H. In particular, if a coset Ht generates the quotient group G/H, then all of the conjugacy classes in Ht are non-split.

Regular bipartite graphs are antimagic

AbstractA labeling of a graph G is a bijection from E(G) to the set {1, 2,… |E(G)|}. A labeling is antimagic if for any distinct vertices u and v, the sum of the labels on edges incident to u is different from the sum of the labels on edges incident to v. We say a graph is antimagic if it has an antimagic labeling. In 1990, Hartsfield and Ringel conjectured that every connected graph other than K2 is antimagic. In this article, we show that every regular bipartite graph (with degree at least 2) is antimagic. Our technique relies heavily on the Marriage Theorem. © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 173–182, 2009

A Solution for the Reward Allocation Problem in Linear Production Games with Flexible Resources

In this paper we study an extension of the linear production game (LPG) introduced by Owen (1975). We discuss LPG with players who have flexible resources. Here, flexibility means that a player has a universal resource (e.g. working time) which he can split up (e.g. to execute different tasks). To model resource flexibility we extend the original LPG with constraints similar to the condition in Hall’s famous marriage theorem. The resources of the players can be used in several linear production processes, alone or together with other players. The resulting profit has to be divided among the players according to an allocation rule. We will show that this new variant of cooperative games – like the original linear LPG – is balanced, hence, has a nonempty core, and that we can obtain an allocation in the core by using the shadow prices of the flexible resources.

List multicolorings of graphs with measurable sets

AbstractIn an ordinary list multicoloring of a graph, the vertices are “colored” with subsets of pre‐assigned finite sets (called “lists”) in such a way that adjacent vertices are colored with disjoint sets. Here we consider the analog of such colorings in which the lists are measurable sets from an arbitrary atomless, semifinite measure space, and the color sets are to have prescribed measures rather than prescribed cardinalities. We adapt a proof technique of Bollobás and Varopoulos to prove an analog of one of the major theorems about ordinary list multicolorings, a generalization and extension of Hall's marriage theorem, due to Cropper, Gyárfás, and Lehel. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 179–193, 2007

A Fixed-Point Approach to Stable Matchings and Some Applications

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.

Combinatorial properties of the family of maximum stable sets of a graph

The stability number α ( G ) of a graph G is the size of a maximum stable set of G , core (G)=⋂{S : S is a maximum stable set in G }, and ξ ( G )=| core ( G )|. In this paper we prove that for a graph G the following assertions are true: (i) if G has no isolated vertices, and ξ ( G )⩽1, then G is quasi-regularizable; (ii) if the order of G is n , and α ( G )>( n + k −min{1,| N ( core ( G ))|})/2, for some k ⩾1, then ξ ( G )⩾ k +1; moreover, if n + k −min{1,| N ( core ( G ))|} is even, then ξ ( G )⩾ k +2. The last finding is a strengthening of a result of Hammer, Hansen, and Simeone, which states that ξ ( G )⩾1 is true whenever α ( G )> n /2. In the case of König–Egerváry graphs, i.e., for graphs enjoying the equality α ( G )+ μ ( G )= n , where μ ( G ) is the maximum size of a matching of G , we prove that | core ( G )|>| N ( core ( G ))| is a necessary and sufficient condition for α ( G )> n /2. Furthermore, for bipartite graphs without isolated vertices, ξ ( G )⩾2 is equivalent to α ( G )> n /2. We also show that Hall's Marriage Theorem is true for König–Egerváry graphs, and, it is sufficient to check Hall's condition only for one specific stable set, namely for core ( G ).

Maximal nests of subspaces, the matrix Bruhat decomposition, and the marriage theorem - with an application to graph coloring

Using the celebrated Marriage Theorem of P. Hall, we give an elementary combinato- rial proof of the theorem that asserts that given two maximal nests N1 and N2 in a finite dimensional vector space V , there is an ordered basis of V that generates N1 and a permutation of that ordered basis that generates N2. From this theorem one easily obtains the Matrix Bruhat Decomposition. A generalization to matroids is discussed, and an application to graph coloring is given.

The Theorem of the k -1 Happy Divorces

Given a binary relation R between the elements of two sets X and Y and a natural number k, it is shown that there exist k injective maps f1, f2,...,fk: \( X \hookrightarrow Y \) with \( \# \{f_1(x), f_2(x),...,f_k(x)\}=k \quad{\rm and}\quad (x,f_1(x)), (x, f_2(x)),...,(x, f_k(x)) \in R \) for all \( x \in X \) if and only if the inequality \( k \cdot \# A \leq \sum_{y \in Y} min(k, \#\{a \in A \mid (a,y) \in R\}) \) holds for every finite subset A of X, provided \( \{y \in Y \mid (x,y) \in R\} \) is finite for all \( x \in X \).¶Clearly, as suggested by this paper's title, this implies that, in the context of the celebrated Marriage Theorem, the elements x in X can (simultaneously) marry, get divorced, and remarry again a partner from their favourite list as recorded by R, for altogether k times whenever (a) the list of favoured partners is finite for every \( x \in X \)and (b) the above inequalities all hold.¶In the course of the argument, a straightforward common generalization of Bernstein's Theorem and the Marriage Theorem will also be presented while applications regarding (i) bases in infinite dimensional vector spaces and (ii) incidence relations in finite geometry (inspired by Conway's double sum proof of the de Bruijn-Erdos Theorem) will conclude the paper.

On non-injectivity of Borel functions selecting points from compact sets

We show that no Borel function on the hyperspace of a compact metric space X, selecting points from nonempty closed sets in X, can be injective on a residual set in the hyperspace. As a consequence, we show that a measure-theoretic analogue of the marriage theorem for finite sets, obtained by R. D. Mauldin [7], [3], fails in the Baire category setting. This answers a question from [1].