polynomial time algorithm for an optimal stable assignment with multiple partners this paper considers the many to many version of the stable marriage problem where each man and woman has a strict preference ordering on the members of the opposite sex that he or she considers acceptable further each man and woman wishes to be matched to as many acceptable partners as possible up to his or her specified quota in this setup a polynomial time algorithm for finding a stable matching that minimizes the sum of partner ranks across all men and women is provided it is argued that this sum can be used as an optimality criterion for minimizing total dissatisfaction if the preferences over partner combinations satisfy a no complementarities condition the results in this paper extend those already known for the one to one version of the problem

hierarchical latin hypercube sampling abstractlatin hypercube sampling lhs is a robust scalable monte carlo method that is used in many areas of science and engineering we present a new algorithm for generating hierarchic latin hypercube sets hlhs that are recursively divisible into lhs subsets based on this new construction we introduce a hierarchical incremental lhs hilhs method that allows the user to employ lhs in a flexibly incremental setting this overcomes a drawback of many lhs schemes that require the entire sample set to be selected a priori or only allow very large increments we derive the sampling properties for hlhs designs and hilhs estimators we also present numerical studies that showcase the flexible incrementation offered by hilhs

popular edges and dominant matchings given a bipartite graph g a cup b e with strict preference lists and given an edge e in e we ask if there exists a popular matching in g that contains e we call this the popular edge problem a matching m is popular if there is no matching m such that the vertices that prefer m to m outnumber those that prefer m to m it is known that every stable matching is popular however g may have no stable matching with the edge e in this paper we identify another natural subclass of popular matchings called dominant matchings and show that if there is a popular matching that contains the edge e then there is either a stable matching that contains e or a dominant matching that contains e this allows us to design a linear time algorithm for identifying the set of popular edges when preference lists are complete we show an o n 3 algorithm to find a popular matching containing a given set of edges or report that none exists where n a b

complexity study for the robust stable marriage problem abstract the robust stable marriage problem rsm is a variant of the classic stable marriage problem in which the robustness of a given stable matching is measured by the number of modifications required to find an alternative stable matching should some pairings break due to an unforeseen event we focus on the complexity of finding an a b supermatch an a b supermatch is defined as a stable matching in which if any a non fixed men women break up it is possible to find another stable matching by changing the partners of those a men women and the partners of at most b others we first discuss a model based on independent sets for finding 1 1 supermatches secondly in order to show that deciding whether or not there exists a 1 b supermatch is np complete we first introduce a sat formulation for which the decision problem is np complete by using schaefer s dichotomy theorem we then show the equivalence between this sat formulation and finding a 1 1 supermatch on a specific family of instances we also focus on studying the threshold between the cases in p and np complete for this problem

polyhedral aspects of stable marriage 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

approximability results for stable marriage problems with ties we consider instances of the classical stable marriage problem in which persons may include ties in their preference lists we show that in such a setting strong lower bounds hold for the approximability of each of the problems of finding an egalitarian minimum regret and sex equal stable matching we also consider stable marriage instances in which persons may express unacceptable partners in addition to ties in this setting we prove that there are constants such that each of the problems of approximating a maximum and minimum cardinality stable matching within factors of respectively is np hard under strong restrictions we also give an approximation algorithm for both problems that has a performance guarantee expressible in terms of the number of lists with ties this significantly improves on the best known previous performance guarantee for the case that the ties are sparse our results have applications to large scale centralized matching schemes

the invariant polynomials degrees of the kronecker sum of two linear operators and additive theory let g be an abelian group let a and b be finite non empty subsets of g by a b we denote the set of all elements a b with a a and b b for c a b c a b is the cardinality of the set of pairs a b such that a b c we call c a b the multiplicity of c in a b let i be a positive integer we denote by i a b or briefly by i the cardinality of the set of the elements of a b that have multiplicity greater than or equal to i let f be a field let p be the characteristic of f in case of finite characteristic and if f has characteristic 0 let a and b be finite non empty subsets of f we will prove that for every l 1 min a b one has a 1 l lmin p a b l this statement on the multiplicities of the elements of a b generalizes cauchy davenport theorem in fact cauchy davenport is exactly inequality a for l 1 when f zp inequality a was proved in j m pollard j london math soc 8 1974 460 462 see also m b nathanson additive number theory inverse problems and the geometry of sumsets springer new york 1996

the complexity of counting stable marriages in an instance of size n of the stable marriage problem each of n men and n women ranks the members of the opposite sex in order of preference a stable matching is a complete matching of men and women such that no man and woman who are not partners both prefer each other to their actual partners under the matching it is well known that at least one stable matching exists for every stable marriage instance so that the decision version of the problem always has a yes answer furthermore efficient algorithms are known for the determination of such a stable matching so that the search version of the problem is polynomially solvable however by exploring the structure of the set of stable matchings for any particular instance of the problem and exploiting its relationship with the set of antichains of an associated partially ordered set we prove that the enumeration version of the problem determining the number of stable matchings is p complete and therefore cannot be solved in polynomial time if

large sample properties of simulations using latin hypercube sampling latin hypercube sampling mckay conover and beckman 1979 is a method of sampling that can be used to produce input values for estimation of expectations of functions of output variables the asymptotic variance of such an estimate is obtained the estimate is also shown to be asymptotically normal asymptotically the variance is less than that obtained using simple random sampling with the degree of variance reduction depending on the degree of additivity in the function being integrated a method for producing latin hypercube samples when the components of the input variables are statistically dependent is also described these techniques are applied to a simulation of the performance of a printer actuator

a central limit theorem for latin hypercube sampling summary latin hypercube sampling lhs is a technique for monte carlo integration due to mckay conover and beckman m stein proved that lhs integrals have smaller variance than independent and identically distributed monte carlo integration the extent of the variance reduction depending on the extent to which the integrand is additive we extend stein s work to prove a central limit theorem variance estimation methods for nonparametric regression can be adapted to provide n 12 consistent estimates of the asymptotic variance in lhs moreover the skewness can be estimated at this rate the variance reduction may be explained in terms of certain control variates that cannot be directly measured we also show how to combine control variates with lhs finally we show how these results lead to a frequentist approach to computer experimentation

the stable marriage problem with restricted pairs a stable matching is a complete matching of men and women such that no man and woman who are not partners both prefer each other to their actual partners under the matching in an instance of the stable marriage problem each of the n men and n women ranks the members of the opposite sex in order of preference it is well known that at least one stable matching exists for every stable marriage problem instance we consider extensions of the stable marriage problem obtained by forcing and by forbidding sets of pairs we present a characterization for the existence of a solution for the stable marriage with forced and forbidden pairs problem in addition we describe a reduction of the stable marriage with forced and forbidden pairs problem to the stable marriage with forbidden pairs problem finally we also present algorithms for finding a stable matching all stable pairs and all stable matchings for this extension the complexities of the proposed algorithms are the same as the best known algorithms for the unrestricted version of the problem

an application of group theory to matrices and to ordinary differential equations abstract a result concerned with groups is proved from which several applications can be derived we estimate e g the number of distinct eigenvalues of the kronecker product and sum of two given matrices a b when a as well as b has distinct eigenvalues we also discuss the order of the linear ode whose solutions are the products of solutions of two given linear odes when such odes are in certain classes

blockers and antiblockers of stable matchings an implicit linear description of the stable matching polytope is provided in terms of the blocker and antiblocker sets of constraints of the matroid kernel polytope the explicit identification of both these sets is based on a partition of the stable pairs in which each agent participates here we expose the relation of such a partition to rotations we provide a time optimal algorithm for obtaining such a partition and establish some new related results most importantly that this partition is unique

matching games with partial information abstract we analyze different ways of pairing agents in a bipartite matching problem with regard to its scaling properties and to the distribution of individual satisfactions then we explore the role of partial information and bounded rationality in a generalized marriage problem comparing the benefits obtained by self searching and by a matchmaker finally we propose a modified matching game intended to mimic the way consumers information makes firms to enhance the quality of their products in a competitive market

what matchings can be stable the testable implications of matching theory this paper studies the falsifiability of two sided matching theory when agents preferences are unknown a collection of matchings is rationalizable if there are preferences for the agents involved so that the matchings are stable we show that there are nonrationalizable collections of matchings hence the theory is falsifiable we also characterize the rationalizable collections of matchings which leads to a test of matching theory in the spirit of revealed preference tests of individual optimizing behavior

a new approach to stable matching problems it is shown that stable matching problems are the same as problems about stable configurations of x networks consequences include easy proofs of old theorems a new simple algorithm for finding a stable matching an understanding of the difference between stable marriage and stable roommates np completeness of three party stable marriage cc completeness of several stable matching problems and a fast parallel reduction from the stable marriage problem to the assignment problem

testing for inequality constraints in singular models by trimming or winsorizing the variance matrix there are many applications in which a statistic follows at least asymptotically a normal distribution with a singular or nearly singular variance matrix a classic example occurs in linear regre

a unifying approach to the structures of the stable matching problems abstract it is well known that the structure of the set of stable marriages of a stable marriage instance can be represented as a finite distributive lattice and conversely every finite distributive lattice is a set of stable marriages for some stable marriage instance recently irving 12 and gusfield 9 propose some representations of the set of all stable assignments for a given solvable instance of the stable roommates problem in this paper we will give a unifying approach to the structures of the stable marriage problem and the stable roommates problem to achieve this purpose we first study the duality in the structure of a stable marriage instance then transform every stable roommates instance into a corresponding stable marriage instance and obtain the structure of the stable roommates instance directly from that of the corresponding stable marriage instance the main results of this paper are 1 there is a one one correspondence between the set of stable marriages for a stable marriage instance and the set of feasible words of some faigle geometry 2 there is a one one correspondence between the set of stable assignments for a stable roommates instance and the set of basic words of some faigle geometry

an algorithm for a super stable roommates problem in this paper we describe an efficient algorithm that decides if a stable matching exists for a generalized stable roommates problem where instead of linear preferences agents have partial preference orders on potential partners furthermore we may forbid certain partnerships that is we are looking for a matching such that none of the matched pairs is forbidden and yet no blocking pair forbidden or not exists to solve the above problem we generalize the first algorithm for the ordinary stable roommates problem

stable fractional matchings abstract we study a generalization of the classical stable matching problem that allows for cardinal preferences as opposed to ordinal and fractional matchings as opposed to integral in this cardinal setting stable fractional matchings can have much larger social welfare than stable integral ones our goal is to understand the computational complexity of finding an optimal i e welfare maximizing stable fractional matching we consider both exact and approximate stability notions and provide simple approximation algorithms with weak welfare guarantees our main result is that somewhat surprisingly achieving better approximations is computationally hard to the best of our knowledge these are the first computational complexity results for stable fractional matchings in the cardinal model en route to these results we provide a number of structural observations that could be of independent interest

an efficient algorithm for the optimal stable marriage in an instance of size n of the stable marriage problem each of n men and n women ranks the members of the opposite sex in order of preference a stable matching is a complete matching of men and women such that no man and woman who are not partners both prefer each other to their actual partners under the matching it is well known 2 that at least one stable matching exists for every stable marriage instance however the classical gale shapley algorithm produces a marriage that greatly favors the men at the expense of the women or vice versa the problem arises of finding a stable matching that is optimal under some more equitable or egalitarian criterion of optimality this problem was posed by knuth 6 and has remained unsolved for some time here the objective of maximizing the average or equivalently the total satisfaction of all people is used this objective is achieved when a person s satisfaction is measured by the position of his her partner in his her preference list by exploiting the structure of the set of all stable matchings and using graph theoretic methods an o n 4 algorithm for this problem is derived

on randomized approximation for finding a level ideal of a poset and the generalized median stable matchings this study is concerned with finding a level ideal li of a partially ordered set poset given a finite poset p the level of each element p p is defined as the number of ideals that do not include p then the problem is to find the ith li the ideal consisting of elements whose levels are less than a given integer i the concept of a level ideal is naturally derived from the generalized median stable matchings introduced by teo and sethuraman teo c p j sethuraman 1998 the geometry of fractional stable matchings and its applications math oper res 23 4 874 891 in the context of fairness of matchings in a stable marriage problem cheng cheng c t 2010 understanding the generalized median stable matchings algorithmica58 1 34 51 showed that finding the ith li is p hard when i n where n is the total number of ideals of p this paper shows that finding the ith li is p hard even if i n1 c where c is an arbitrary constant at least one meanwhile we present a polynomial time exact algorithm when i o log n c where c is an arbitrary positive constant we also devise two randomized approximation schemes for the ideals of a poset by using an oracle of an almost uniform sampler

a unified approach to finding good stable matchings in the hospitals residents setting the hospitals residents hr problem is a many to one generalization of the stable marriage sm problem researchers have been interested in variants of stable matchings that either satisfy a set of additional contraints or are optimal with respect to some cost function in this paper we show that broad classes of feasibility and optimization stable matching problems in the hr setting can be solved efficiently provided certain tasks such as checking the feasibility of a stable matching or computing the cost of a stable matching can also be done efficiently to prove our results we make use of an hr instance s meta rotation poset to explore its stable matchings an algorithm that can discover all the meta rotations of the instance serves as a starting point for all our algorithms

efficient algorithms for generalized stable marriage and roommates problems we consider a generalization of the stable roommates problem sr in which preference lists may be partially ordered and forbidden pairs may be present denoted by srpf this includes as a special case a corresponding generalization of the classical stable marriage problem sm denoted by smpf by extending previous work of feder we give a two step reduction from srpf to 2 sat this has many consequences including fast algorithms for a range of problems associated with finding optimal stable matchings and listing all solutions given variants of sr and sm for example given an smpf instance i we show that there exists an o m succinct certificate for the unsolvability of i an o m algorithm for finding all the super stable pairs in i an o m kn algorithm for listing all the super stable matchings in i an o m 1 5 algorithm for finding an egalitarian super stable matching in i and an o m algorithm for finding a minimum regret super stable matching in i where n is the number of men m is the total length of the preference lists and k is the number of super stable matchings in i analogous results apply in the case of srpf

generalized derivations and additive theory we return to the theme of generalized derivations related to symmetric functions to correct the hypothesis of one of the main theorems of our first paper so that all cases are now properly covered

nested latin hypercube designs we propose an approach to constructing nested latin hypercube designs such designs are useful for conducting multiple computer experiments with different levels of accuracy a nested latin hypercube design with two layers is defined to be a special latin hypercube design that contains a smaller latin hypercube design as a subset our method is easy to implement and can accommodate any number of factors we also extend this method to construct nested latin hypercube designs with more than two layers illustrative examples are given some statistical properties of the constructed designs are derived

three fast algorithms for four problems in stable marriage the stable marriage problem 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 gs 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 n s time and o n 2 space bounded algorithm which is time and space optimal to enumerate all stable marriages where s is the set of them and we give an o n 2 time algorithm to find the mini

a note on the minimal polynomial of the kronecker sum of two linear operators abstract we give a lower bound for the degree of the minimal polynomial of the kronecker sum of two linear operators f and g in terms of the degrees of the minimal polynomials of f and g our result generalizes the known cauchy davenport theorem