statistics of stable marriages in the stable marriage problem n men and n women have to be matched by pairs under the constraint that the resulting matching is stable we study the statistical properties of stable matchings in the large n limit using both numerical and analytical methods generalizations of the model including singles and unequal numbers of men and women are also investigated

analysis of one dimensional helmholtz equation with pml boundary in this paper the linear conforming finite element method for the one dimensional berenger s pml boundary is investigated and well posedness of the given equation is discussed furthermore optimal error estimates and stability in the l 2 or h 1 norm are derived under the assumption that h h email protected 2 and h email protected 3 are sufficiently small where h is the mesh size and w denotes a fixed frequency numerical examples are presented to validate the theoretical error bounds

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

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

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

hard variants of stable marriage the stable marriage problem and its many variants have been widely studied in the literature gusfield and irving the stable marriage problem structure and algorithms mit press cambridge ma 1989 roth and sotomayor two sided matching a study in game theoretic modeling and analysis econometric society monographs vol 18 cambridge university press cambridge 1990 knuth stable marriage and its relation to other combinatorial problems crm proceedings and lecture notes vol 10 american mathematical society providence ri 1997 partly because of the inherent appeal of the problem partly because of the elegance of the associated structures and algorithms and partly because of important practical applications such as the national resident matching program roth j political economy 92 6 1984 991 and similar large scale matching schemes here we present the first comprehensive study of variants of the problem in which the preference lists of the participants are not necessarily complete and not necessarily totally ordered we show that under surprisingly restrictive assumptions a number of these variants are hard and hard to approximate the key observation is that in contrast to the case where preference lists are complete or strictly ordered or both a given problem instance may admit stable matchings of different sizes in this setting examples of problems that are hard are finding a stable matching of maximum or minimum size determining whether a given pair is stableeven if the indifference takes the form of ties on one side only the ties are at the tails of lists there is at most one tie per list and each tie is of length 2 and finding or approximating both an egalitarian and a minimum regret stable matching however we give a 2 approximation algorithm for the problems of finding a stable matching of maximum or minimum size we also discuss the significant implications of our results for practical matching schemes

absorbing boundary conditions for difference approximations to the multi dimensional wave equation construction des conditions aux limites absorbantes pour l equation des ondes multidimensionnelle a l aide d approximations aux differences formulation sous forme canonique des conditions aux limites absorbantes analytiques

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

leibniz algebroid associated with a nambu poisson structure the notion of a leibniz algebroid is introduced and it is shown that each nambu poisson manifold has associated a canonical leibniz algebroid this fact permits one to define the modular class of a nambu poisson manifold as an appropriate cohomology class extending the well known modular class of poisson manifolds

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

on the minority game analytical and numerical studies abstract we investigate further several properties of the minority game we have recently introduced we explain the origin of the phase transition and give an analytical expression of 2 n in the n 2 m region the ability of the players to learn a given payoff is also analyzed and we show that the darwinian evolution process tends to a self organized state in particular the lifetime distribution is a power law with exponent 2 furthermore we study the influence of identical players on their gain and on the systems performance finally we show that large brains always take advantage of small brains

dynamics of generalized poisson and nambu poisson brackets a unified setting for generalized poisson and nambu poisson brackets is discussed it is proved that a nambu poisson bracket of even order is a generalized poisson bracket characterizations of poisson morphisms and generalized infinitesimal automorphisms are obtained as coisotropic and lagrangian submanifolds of product and tangent manifolds respectively

the marriage problem from the bar of appointments to the agency abstract we study the stable marriage problem from different points of view we proposed a microscopic dynamic that led the system to a stationary state that we are able to characterize analytically then we derive a thermodynamical description of the nash equilibrium states of the system that agree very well with the results of monte carlo simulations finally through large scale numerical simulations we compare the global optimum of the society with the stable marriage of lower energy we showed that both states are strongly correlated and that the selfish attitude results in a benefit for most of the practitioners belonging to blocking pairs in the global optimum of the society

statistical mechanics of a simplified bipartite matching problem an analytical treatment we perform an analytical study of a simplified bipartite matching problem in which there exists a constant matching energy and both heterosexual and homosexual pairings are allowed we obtain the partition function in a closed analytical form and we calculate the corresponding thermodynamic functions of this model we conclude that the model is favored at high temperatures for which the probabilities of heterosexual and homosexual pairs tend to become equal in the limits of low and high temperatures the system is extensive however this property is lost in the general case there exists a relation between the matching energies for which the system becomes more stable under external thermal perturbations as the difference of energies between the two possible matches increases the system becomes more ordered while the maximum of entropy is achieved when these energies are equal in this limit there is a first order phase transition between two phases with constant entropy

stability of poisson equilibria and hamiltonian relative equilibria by energy methods we develop a general stability theory for equilibrium points of poisson dynamical systems and relative equilibria of hamiltonian systems with symmetries including several generalisations of the energy casimir and energy momentum methods using a topological generalisation of lyapunov s result that an extremal critical point of a conserved quantity is stable we show that a poisson equilibrium is stable if it is an isolated point in the intersection of a level set of a conserved function with a subset of the phase space that is related to the topology of the symplectic leaf space at that point this criterion is applied to generalise the energy momentum method to hamiltonian systems which are invariant under non compact symmetry groups for which the coadjoint orbit space is not hausdorff we also show that a g stable relative equilibrium satisfies the stronger condition of being a stable where a is a specific group theoretically defined subset of g which contains the momentum isotropy subgroup of the relative equilibrium the results are illustrated by an application to the stability of a rigid body in an ideal irrotational fluid

on a cutting plane heuristic for the stable roommates problem and its applications abstract we propose a new cutting plane heuristic for the classical stable roommates problem our approach utilises a new linear programming formulation for the problem and the underlying geometric properties of the fractional solution to construct the violated inequality we test the approach on moderate size problems with encouraging computational performance to further illustrate the versatility of this approach we also show how it can be suitably extended to handle variants of the basic stable roommates model

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

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

beauty and distance in the stable marriage problem the stable marriage problem has been introduced in order to describe a complex system where individuals attempt to optimise their own satisfaction subject to mutually conflicting constraints due to the potential large applicability of such model to describe all the situation where different objects has to be matched pairwise the statistical properties of this model have been extensively studied in this paper we present a generalisation of this model introduced in order to take into account the presence of correlations in the lists and the effects of distance when the players are supposed to be represented by a position in space

a frequency domain parallel method for the numerical approximation of parabolic problems a naturally parallelizable numerical method is introduced and analyzed for parabolic partial differential equations instead of solving the problem in the space time formulation we propose to solve it in the space frequency formulation existence and uniqueness are given error estimates are given for each single frequency also given is a full estimate for errors coming from truncation numerical quadrature rule in fourier inversion and finite element discretization in the space frequency domain a numerical experiment on a parallel mimd machine is also given

the marriage problem and the fate of bachelors abstract in the marriage problem a variant of the bi parted matching problem each member has a wish list expressing his her preference for all possible partners this list consists of random positive real numbers drawn from a certain distribution one searches the lowest cost for the society at the risk of breaking up pairs in the course of time minimization of a global cost function hamiltonian is performed with statistical mechanics techniques at a finite fictitious temperature the problem is generalized to include bachelors needed in particular when the groups have different size and polygamy exact solutions are found for the optimal solution t 0 the entropy is found to vanish quadratically in t also other evidence is found that the replica symmetric solution is exact implying at most a polynomial degeneracy of the optimal solution whether bachelors occur or not depends not only on their intrinsic qualities or lack thereof but also on global aspects of the chance for pair formation in society

a normal form theorem around symplectic leaves we prove the poisson geometric version of the local reeb stability from foliation theory and of the slice theorem from equivariant geometry which is also a generalization of conn s linearization theorem

chinese family name distributions in multiple scales abstract the distribution of chinese family names is investigated based on data of the fifth national census of china in 2000 including 7329 chinese family names and 1 28 billion people the cumulative distribution function cdf of family name in multiple scales is presented and the correlation between distribution and scale is discussed the fitting results show that cdfs at the scales of country province city and county all follow stretched exponential truncated power law while the fitting parameters vary greatly a characteristic index is proposed to measure the extent the distribution is stretched exponential like the index increases as the scale goes down indicating that the distribution at lower scale is more stretched exponential like and the distribution at higher scale is more power law like such correlation between distribution and scale is verified by investigating how the distribution changes as the scale is expanded county by county furthermore the correlation is qualitatively explained by artificially constructing multiple scales with a partition process

crowd anticrowd theory of the minority game abstract the minority game is a simple yet highly non trivial agent based model for a complex adaptive system here we provide an explanation of the game s fluctuations which is both intuitive and quantitative and which applies over the entire parameter range of interest the physical idea behind our theory is to describe the interplay between crowds of like minded agents and their anticorrelated partners anticrowds

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

absorbing boundary conditions for wave propagation in viscoelastic media we derive absorbing boundary conditions for viscoelastic waves error estimates are given for such absorbing boundary conditions we treat the problem in the space frequency domain formulation after taking the fourier transformation in time well posedness is shown for the elliptic problems thus created and fourier invertibility is also proved we also present some results of numerical simulations

a parallel method for the numerical solution of integro differential equation with positive memory abstract an efficient parallel numerical method is proposed for an integro differential equation with positive memory instead of solving the equation in classical time marching methods which require massive storage of solutions of previous time steps in order to advance to a next time step the fourier laplace transformation in time is applied to obtain a set of complex valued elliptic problems parameterized by points on a contour in the complex plane using the independence of an elliptic problem corresponding to one contour point is independent of those elliptic problems corresponding to other contour points all elliptic problems can be solved in parallel essentially without data communications then the time domain solution can be obtained by the fourier laplace inversion formula an error analysis and the numerical implementation of this parallel method is presented

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

vertical transmission of culture and the distribution of family names abstract a stochastic model for the evolution of a growing population is proposed in order to explain empirical power law distributions in the frequency of family names as a function of the family size preliminary results show that the predicted exponents are in good agreement with real data the evolution of family name distributions is discussed in the frame of vertical transmission of cultural features

the u n free rigid body integrability and stability analysis of the equilibria a natural extension of the free rigid body dynamics to the unitary group u n is considered the dynamics is described by the euler equation on the lie algebra u n which has a bi hamiltonian structure and it can be reduced onto the adjoint orbits as in the case of the so n the complete integrability and the stability of the isolated equilibria on the generic orbits are considered by using the method of bolsinov and oshemkov in particular it is shown that all the isolated equilibria on generic orbits are lyapunov stable c 2015 elsevier inc all rights reserved

statistical properties and network structure of the marriage problem abstract we study the statistical properties of the distribution of total energies or costs over the set of n configurations for the marriage problem with n men and n women analytical results and exhaustive numerical calculations for the whole configuration space are presented taking into account several definitions of the individual energies the distribution of the minimal energy is evaluated approximately moreover we point out a natural association between the configuration space of the marriage problem and a network we study the topological and metric properties of this network as well as the distribution of local energy minima on it

rank maximal matchings structure and algorithms abstract let g a p e be a bipartite graph where a denotes a set of applicants p denotes a set of posts and ranks on the edges denote preferences of the agents over posts a matching m in g is rank maximal if it matches the maximum number of applicants to their top rank post subject to this the maximum number of applicants to their second rank post and so on in this paper we develop a switching graph characterization of rank maximal matchings which is a useful tool that encodes all rank maximal matchings in an instance the characterization leads to simple and efficient algorithms for several interesting problems in particular we give an efficient algorithm to compute the set of rank maximal pairs in an instance we show that the problem of counting the number of rank maximal matchings is p complete and also give an fpras for the problem finally we consider the problem of deciding whether a rank maximal matching is popular among all the rank maximal matchings in a given instance and give an efficient algorithm for the 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

quantum dating market abstract we consider the dating market decision problem under the quantum mechanics point of view quantum states whose associated amplitudes are modified by men strategies are used to represent women the grover quantum search algorithm is used as a playing strategy success is more frequently obtained by playing quantum than playing classic

frequency domain formulation of linearized navier stokes equations abstract a naturally parallelizable formulation is considered for solving linearized time dependent navier stokes equations the evolution problem is first converted into a complex valued elliptic system by fourier transformation existence and uniqueness are then given for the resulting problem for each frequency stability and regularity depending on frequency are analyzed next standard finite element methods are used to approximate solutions for the transformed elliptic systems finally time dependent solutions are constructed by fourier inversion with a full estimate of errors generated in the truncation in the fourier transformation quadrature rules and finite element approximations

on the geometric quantization of jacobi manifolds the geometric quantization of jacobi manifolds is discussed a natural cohomology termed lichnerowicz jacobi on a jacobi manifold is introduced and using it the existence of prequantization bundles is characterized to do this a notion of contravariant derivatives is used in such a way that the procedure developed by vaisman for poisson manifolds is naturally extended a notion of polarization is discussed and the quantization problem is studied the existence of prequantization representations is also considered

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

asymptotic and lyapunov stability of constrained and poisson equilibria reference cag article 2005 003doi 10 1016 j jde 2004 09 016view record in web of science record created on 2008 11 20 modified on 2017 09 27

arbitrated matching formulation and protocol abstract this paper presents an arbitration approach to bipartite matching in which a set of actors are matched up with a set of mates in a one to one manner the arbitration is composed of two stages viz stage i in which ordinal preference is elicited on a need to basis until all stable matchings are obtained stage ii is sequential bargaining in which risk preference information is extracted by eliciting certainty equivalents of default randomizations declared by the arbitrator stage ii comes out with a particular stable subset randomization of which is preferable for all to that of the entire stable set stage ii is particularly effective when actors and mates are risk averse whence reducing uncertainty with a smaller randomization set proves to be advantageous to all we also prove that it is superior to the traditional one off approach in terms of the elicitation effort required as well as strategyproofness

an accurate and efficient scheme for wave propagation in linear viscoelastic media the problem of wave propagation in a linear viscoelastic medium can be described mathematically as an exponential evolution operator of the form em t acting on a vector representing the initial conditions where m is a spatial operator matrix and t is the time variable techniques like finite difference for instance are based on a taylor expansion of this evolution operator we propose an optimal polynomial approximation of em t based on the powerful method of interpolation in the complex plane in a domain which includes the eigenvalues of the matrix m the new time integration technique is implemented to solve the isotropic viscoacoustic equation of motion the algorithm is tested for the problem of wave propagation in a homogeneous medium and compared with second order temporal differencing and the spectral chebychev method the algorithm solves efficiently and with machine accuracy the problem of seismic wave propagation with a dissipation mechanism in the sonic band a characteristic of sedimen

incorporating inertia into multiagent systems we consider a model that demonstrates the crucial role of inertia and stickiness in multiagent systems based on the minority game the inertia of an agent is introduced into the game model by allowing agents to apply hypothesis testing when choosing their best strategies thereby reducing their reactivity toward changes in the environment we find by extensive numerical simulations that our game shows a remarkable improvement of global cooperation throughout the whole phase space in other words the maladaptation behavior due to over reaction of agents is removed these agents are also shown to be advantageous over the standard ones which are sometimes too sensitive to attain a fair success rate we also calculate analytically the minimum amount of inertia needed to achieve the above improvement our calculation is consistent with the numerical simulation results finally we review some related works in the field that show similar behaviors and compare them to our work

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

the geometry of fractional stable matchings and its applications 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 2 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

pair formation within multi agent populations abstract we present a simple model for the formation of pairs in multi agent populations of type a and b which move freely on a spatial network each agent of population a and b is labeled as a i and b j with i 1 n a and j 1 n b and carries its own individual list of characteristics or phenotype when agents from opposite populations encounter one another on the network they can form a relationship if not already engaged in one the length of time for which any given pair stays together depends on the compatibility of the two constituent agents possible applications include the human dating scenario and the commercial domain where two types of businesses a and b have members of each type looking for a business partner i e a i b j r ij the pair r ij then survives for some finite time before dissociating r ij a i b j there are many possible generalizations of this basic setup here we content ourselves with some initial numerical results for the simplest of network topologies together with some accompanying analytic analysis

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

statistical mechanics of systems with heterogeneous agents minority games we study analytically a simple game theoretical model of heterogeneous interacting agents we show that the stationary state of the system is described by the ground state of a disordered spin model which is exactly solvable within the simple replica symmetric ansatz such a stationary state differs from the nash equilibrium where each agent maximizes her own utility the latter turns out to be characterized by a replica symmetry broken structure numerical results fully agree with our analytical findings

recursion operators and bi hamiltonian formulations of the landau lifshitz equation hierarchies using a polynomial pencil of lax pairs we give a new bi hamiltonian formulation for the landau lifshitz equation hierarchy and find a new recursion operator related to it

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

the structure of the stable roomate problem efficient representation and enumeration of all stable assignments the stable roommates problem is a well known problem of matching 2n people into disjoint pairs to achieve a certain type of stability the problem strictly generalizes the better known stable marriage problem it has been previously shown irving and leather siam j comput 15 1986 pp 655 667 that the set of stable marriages for a given instance can be compactly represented and this representation has been exploited to yield a number of very efficient algorithms concerned with stable marriage irving leather and gusfield j assoc comput mach 34 1987 pp 532 543 gusfield siam j comput 16 1987 pp 111 128 gusfield et al j combin theory ser a 1987 pp 304 309 in this paper we generalize the structure of the stable marriages to obtain two efficiently computed small implicit representations of the set of all stable roommate assignments for any given instance one representation is a partial order pi on o n 2 elements such that the stable assignments ar

sex oriented stable matchings of the marriage problem with correlated and incomplete information in the stable marriage problem two sets of agents must be paired according to mutual preferences which may happen to conflict we present two generalizations of its sex oriented version aiming to take into account correlations between the preferences of agents and costly information their effects are investigated both numerically and analytically