a hybrid breakout local search and reinforcement learning approach to the vertex separator problem abstract the vertex separator problem vsp is an np hard problem which emerges from a variety of important domains and applications in this paper we present an improved breakout local search for vsp named bls rle the distinguishing feature of this approach is a new parameter control mechanism that draws upon ideas from reinforcement learning theory to reach an interdependent decision on the number and on the type of perturbation moves the mechanism complies with the principle of first carrying out intensification and then employing minimal diversification only if needed it uses a dedicated sampling strategy for a rapid convergence towards a limited set of parameter values that appear to be the most convenient for the given state of search extensive experimental evaluations and statistical comparisons on a wide range of benchmark instances show significant improvement in performance of the proposed algorithm over the existing bls algorithm for vsp indeed out of the 422 tested instances bls rle was able to attain the best known solution in 93 8 of the cases which is around 20 higher compared to the existing bls in addition we provide detailed analyses to evaluate the importance of the key elements of the proposed method and to justify the degree of diversification introduced during perturbation

approximate solutions to the turbine balancing problem abstract the turbine balancing problem tbp is an np hard combinatorial optimization problem arising in the manufacturing and maintenance of turbine engines exact solution methods for solving the tbp are not appropriate since the problem has to be solved in real time and the input data is itself inaccurate in this paper the tbp is formulated as a quadratic assignment problem qap and we propose a heuristic algorithm for solving the resulting problem computational results on a set of instances provided by pratt whitney p w and from the literature indicate that the proposed algorithm outperforms the current methods used for solving the tbp and has the best overall performance with respect to other heuristic algorithms in the literature

qaplib a quadratic assignment problem library abstract after a short introduction into quadratic assignment problems we give a library of problem instances for the quadratic assignment problem the examples are listed y authors in alphabetical order

solving lift and project relaxations of binary integer programs we propose a method for optimizing the lift and project relaxations of binary integer programs introduced by lovasz and schrijver in particular we study both linear and semidefinite relaxations the key idea is a restructuring of the relaxations which isolates the complicating constraints and allows for a lagrangian approach we detail an enhanced subgradient method and discuss its efficient implementation computational results illustrate that our algorithm produces tight bounds more quickly than state of the art linear and semidefinite solvers

bounds for the quadratic assignment problem using the bundle method semidefinite programming sdp has recently turned out to be a very powerful tool for approximating some np hard problems the nature of the quadratic assignment problem qap suggests sdp as a way to derive tractable relaxations we recall some sdp relaxations of qap and solve them approximately using a dynamic version of the bundle method the computational results demonstrate the efficiency of the approach our bounds are currently among the strongest ones available for qap we investigate their potential for branch and bound settings by looking also at the bounds in the first levels of the branching tree

a new rounding procedure for the assignment problem with applications to dense graph arrangement problems we present a randomized procedure for rounding fractional perfect matchings to integral matchings if the original fractional matching satisfies any linear inequality then with high probability the new matching satisfies that linear inequality in an approximate sense this extends the well known lp rounding procedure of raghavan and thompson which is usually used to round fractional solutions of linear programs we use our rounding procedure to design an additive approximation algorithm to the quadratic assignment problem the approximation error of the algorithm is en 2 and it runs in n o log n e2 time we also describe polynomial time approximation schemes ptass for dense subcases of many well known np hard arrangement problems including minimum linear arrangement minimum cut linear arrangement maximum acyclic subgraph and betweenness

some concave convex representations of the quadratic assignment problem abstract the quadratic assignment problem suffers from the non convex non concave form of its objective function in this paper five alternative representations of the problem are given whose convexity concavity properties can in principle be exploited to some advantage

an exact algorithm for the quadratic assignment problem on a tree the tree qap is a special case of the quadratic assignment problem qap where the nonzero flows form a tree no condition is required for the distance matrix this problem is np complete and is also a generalization of the traveling salesman problem in this paper we present a branch and bound algorithm for the exact solution of the tree qap based on an integer programming formulation of the problem the bounds are computed using a lagrangian relaxation of this formulation to solve the relaxed problem we present a dynamic programming algorithm which is polynomially bounded the obtained lower bound is very sharp and equals the optimum in many cases this fact allows us to employ a reduction method to decrease the number of variables and leads to search trees with a small number of nodes compared to those usually encountered in problems of this type computational results are given for problems with size up to 25

complexity of a 3 dimensional assignment problem abstract we show that a certain 3 dimensional assignment problem is np complete to do this we show that the following problem is np complete given bipartite graphs g 1 g 2 with the same sets of vertices do there exist perfect matchings m 1 m 2 of g 1 g 2 respectively such that m 1 m 2 o

an algorithm for the three index assignment problem we describe a branch and bound algorithm for solving the axial three index assignment problem the main features of the algorithm include a lagrangian relaxation that incorporates a class of facet inequalities and is solved by a modified subgradient procedure to find good lower bounds a primal heuristic based on the principle of minimizing maximum regret plus a variable depth interchange phase for finding good upper bounds and a novel branching strategy that exploits problem structure to fix several variables at each node and reduce the size of the total enumeration tree computational experience is reported on problems with up to 78 equations and 17 576 variables the primal heuristics were tested on problems with up to 210 equations and 343 000 variables

statistical model with a standard γ distribution we study a statistical model consisting of n basic units which interact with each other by exchanging a physical entity according to a given microscopic random law depending on a parameter ensuremath lambda we focus on the equilibrium or stationary distribution of the entity exchanged and verify through numerical fitting of the simulation data that the final form of the equilibrium distribution is that of a standard gamma distribution the model can be interpreted as a simple closed economy in which economic agents trade money and a saving criterion is fixed by the saving propensity ensuremath lambda alternatively from the nature of the equilibrium distribution we show that the model can also be interpreted as a perfect gas at an effective temperature t ensuremath lambda where particles exchange energy in a space with an effective dimension d ensuremath lambda

computing periodic orbits with arbitrary precision this paper deals with the computation of periodic orbits of dynamical systems up to any arbitrary precision these very high requirements are useful for example in the studies of complex pole location in many physical systems the algorithm is based on an optimized shooting method combined with a numerical ordinary differential equation ode solver tides that uses a taylor series method nowadays this methodology is the only one capable of reaching precision up to thousands of digits for odes the method is shown to be quadratically convergent some numerical tests for the paradigmatic lorenz model and the henon heiles hamiltonian are presented giving periodic orbits up to 1000 digits

positive definite completions of partial hermitian matrices abstract the question of which partial hermitian matrices some entries specified some free may be completed to positive definite matrices is addressed it is shown that if the diagonal entries are specified and principal minors composed of specified entries are positive then if the undirected graph of the specified entries is chordal a positive definite completion necessarily exists furthermore if this graph is not chordal then examples exist without positive definite completions in case a positive definite completion exists there is a unique matrix in the class of all positive definite completions whose determinant is maximal and this matrix is the unique one whose inverse has zeros in those positions corresponding to unspecified entries in the original partial hermitian matrix additional observations regarding positive definite completions are made

eigenvalue bounds versus semidefinite relaxations for the quadratic assignment problem it was recently demonstrated that a well known eigenvalue bound for the quadratic assignment problem qap actually corresponds to a semidefinite programming sdp relaxation however for this bound to be computationally useful the assignment constraints of the qap must first be eliminated and the bound then applied to a lower dimensional problem the resulting projected eigenvalue bound is one of the best available bounds for the qap especially when considering the quality of bounds relative to the complexity of obtaining them in this paper we show that the projected eigenvalue bound is also related to an sdp relaxation of the original qap this implicit sdp relaxation is similar to sdp relaxations of the qap proposed by lin and saigal on solving large scale semidefinite programming problems a case study of quadratic assignment problem department of industrial engineering and operations research university of michigan 1997 and zhao et al j combin optim 2 1998 pp 71 109

synchronization in weighted uncorrelated complex networks in a noisy environment optimization and connections with transport efficiency motivated by synchronization problems in noisy environments we study the edwards wilkinson process on weighted uncorrelated scale free networks we consider a specific form of the weights where the strength and the associated cost of a link is proportional to kikj beta with ki and kj being the degrees of the nodes connected by the link subject to the constraint that the total edge cost is fixed we find that in the mean field approximation on uncorrelated scale free graphs synchronization is optimal at beta 1 numerical results based on exact numerical diagonalization of the corresponding network laplacian confirm the mean field results with small corrections to the optimal value of beta employing our recent connections between the edwards wilkinson process and resistor networks and some well known connections between random walks and resistor networks we also pursue a naturally related problem of optimizing performance in queue limited communication networks utilizing local weighted routing schemes

some remarks on ab percolation in high dimensions in this paper we consider the ab percolation model on z d and zd let phalt zd be the critical probability for ab percolation on zd we show that phalt zd 1 2d2 if the probability of a site to be in state a is 2d2 for some fixed 1 then the probability that ab percolation occurs converges as d to the unique strictly positive solution y of the equation y 1 exp y we also find the limit for the analogous quantities for oriented ab percolation on z d in particular phalt z d 2 d2 we further obtain a small extension to the two parameter problem in which even vertices of zd have probability pa of being in state a and odd vertices have probability pb of being in state b but without relation between pa and pb the principal tools in the proofs are a method of penrose 1993 for asymptotics of percolation on graphs with vertices of high degree and the second moment method

breakout local search for the steiner tree problem with revenue budget and hop constraints the steiner tree problem stp is one of the most popular combinatorial optimization problems with various practical applications in this paper we propose a breakout local search bls algorithm for an important generalization of the stp the steiner tree problem with revenue budget and hop constraints stprbh which consists of determining a subtree of a given undirected graph which maximizes the collected revenues subject to both budget and hop constraints starting from a probabilistically constructed initial solution bls uses a neighborhood search ns procedure based on several specifically designed move operators for local optimization and employs an adaptive diversification strategy to escape from local optima the diversification mechanism is implemented by adaptive perturbations guided by dedicated information of discovered high quality solutions computational results based on 240 benchmarks show that bls produces competitive results with respect to several previous approaches for the 56 most challenging instances with unknown optimal results bls succeeds in improving 49 and matching one best known results within reasonable time for the 184 instances which have been solved to optimality bls can also match 167 optimal results

a new lower bound for the quadratic assignment problem we introduce a new lower bound for the quadratic assignment problem based on a sequence of equivalent formulations of the problem we present a procedure for obtaining tight bounds by sequentially applying our approach in conjunction with the a assad and w xu bound and the n christofides and m gerrard bound

a cooperative parallel tabu search algorithm for the quadratic assignment problem in this study we introduce a cooperative parallel tabu search algorithm cpts for the quadratic assignment problem qap the qap is an np hard combinatorial optimization problem that is widely acknowledged to be computationally demanding these characteristics make the qap an ideal candidate for parallel solution techniques cpts is a cooperative parallel algorithm in which the processors exchange information throughout the run of the algorithm as opposed to independent concurrent search strategies that aggregate data only at the end of execution cpts accomplishes this cooperation by maintaining a global reference set which uses the information exchange to promote both intensification and strategic diversification in a parallel environment this study demonstrates the benefits that may be obtained from parallel computing in terms of solution quality computational time and algorithmic flexibility a set of 41 test problems obtained from qaplib were used to analyze the quality of the cpts algorithm additionally we report results for 60 difficult new test instances the cpts algorithm is shown to provide good solution quality for all problems in acceptable computational times out of the 41 test instances obtained from qaplib cpts is shown to meet or exceed the average solution quality of many of the best sequential and parallel approaches from the literature on all but six problems whereas no other leading method exhibits a performance that is superior to this

on the behavior of the lorenz equation backward in time abstract the sets of solutions to the lorenz equations that exist backward in time and are bounded at an exponential rate determined by the eigenvalues of the linear part of the equation are examined the set associated with the middle eigenvalue is shown to project surjectively onto a plane thereby providing a lower estimate for its dimension specific bounds are also found for a cone containing this set

an extreme point algorithm for a local minimum solution to the quadratic assignment problem abstract in this paper the network structure of basic solutions to the quadratic assignment problem qap is revisited the concept of a relative local star minimum is introduced results characterizing a relative local star minimum are obtained then an extreme point algorithm for qap is proposed

an exact algorithm for the general quadratic assignment problem abstract we develop an algorithm that is based on the linearization and decomposition of a general quadratic assignment problem of size n into n2 linear assignment problems of size n 1 the solutions to these subproblems are used to calculate a lower bound for the original problem and this bound is then used in an exact branch and bound procedure these subproblems are similar to the minors defined by lawler 16 but permit us to calculate tighter bounds computational experience is given for solution to optimality of general quadratic assignment problems is sizes up to n 10

simulated annealing and tabu search in the long run a comparison on qap tasks simulated annealing sa and tabu search ts are compared on the quadratic assignment problem a recent work on the same benchmark suite argued that sa could achieve a reasonable solution quality with fewer function evaluations than ts the discussion is extended by showing that the conclusions must be changed if the task is hard or a very good approximation of the optimal solution is desired or if cpu time is the relevant parameter in addition a recently proposed version of ts the reactive tabu search solves the problem of finding the proper list size with an automatic memory based reaction mechanism

a new bound for the quadratic assignment problem based on convex quadratic programming we describe a new convex quadratic programming bound for the quadratic assignment problem qap the construction of the bound uses a semidefinite programming representation of a basic eigenvalue bound for qap the new bound dominates the well known projected eigenvalue bound and appears to be competitive with existing bounds in the trade off between bound quality and computational effort

the bipartite quadratic assignment problem and extensions abstract we introduce a new binary quadratic program that subsumes the well known quadratic assignment problem this problem is shown to be np hard when some associated parameters are restricted to simple values but solvable in polynomial time when some other parameter values are restricted three different neighborhood structures are introduced a best solution in all these neighborhoods can be identified in polynomial time even though two of these neighborhoods are of exponential size different local search algorithms and their enhancements using tabu search are developed using these neighborhoods experimental analysis show that two hybrid algorithms obtained by combining these neighborhoods in specific ways yield results that are superior to the algorithms that use these neighborhoods separately extensive computational results and statistical analysis of the resulting data are presented

the fractal property of the lorenz attractor in a 1963 paper lorenz inferred that the lorenz attractor must be an infinite complex of surfaces we investigate this fractal property of the lorenz attractor in two ways firstly we obtain explicit plots of the fractal structure of the lorenz attractor using symbolic dynamics and multiple precision computations of periodic orbits the method we derive for multiple precision computation is based on iterative refinement and can compute even highly unstable periodic orbits with long symbol sequences with as many as 100 digits of accuracy ordinary numerical integrations are much too crude to show even the coarsest splitting of surfaces and there appear to be no other explicit computations of the fractal structure in the extensive literature about the lorenz attractor secondly we apply a well known formula that gives the hausdorff dimension of the lorenz attractor in terms of the characteristic multipliers of its unstable periodic orbits the formula converges impressively and the hausdorff dimension of the lorenz attractor appears to be 2 0627160 we use comparison with explicit computations of the fractal structure and discuss the accuracy of this formula and its applicability to the lorenz equations additionally we apply periodic orbit theory to the lorenz attractor and exhibit its spectral determinant and compute its lyapunov exponent

lagrangian decomposition for integer nonlinear programming with linear constraints we present a lagrangean decomposition to study integer nonlinear programming problems solving the dual lagrangean relaxation we have to obtain at each iteration the solution of a nonlinear programming with continuous variables and an integer linear programming decreasing iteratively the primal dual gap we propose two algorithms to treat the integer nonlinear programming

a branch and bound algorithm for the quadratic assignment problem based on the hungarian method this paper presents a new branch and bound algorithm for solving the quadratic assignment problem qap the algorithm is based on a dual procedure dp similar to the hungarian method for solving the linear assignment problem our dp solves the qap in certain cases i e for some small problems n 7 and for numerous larger problems 7 n 16 that arise as sub problems of a larger qap such as the nugent 20 the dp however does not guarantee a solution it is used in our algorithm to calculate lower bounds on solutions to the qap as a result of a number of recently developed improvements the dp produces lower bounds that are as tight as any which might be useful in a branch and bound algorithm these are produced relatively cheaply especially on larger problems experimental results show that the computational complexity of our algorithm is lower than known methods and that its actual runtime is significantly shorter than the best known algorithms for qaplib test instances of size 16 through 22 our method has the potential for being improved and therefore can be expected to aid in solving even larger problems

strengthening gilmore s bound for the quadratic assignment problem abstract finding bounds for the quadratic assignment problem is important as part of some procedures for solving this problem in this paper a commonly used bound due to gilmore is extended to a set of possible bounds analytic properties of the functions underlying these bounds are given and computational approaches are suggested which make use of these properties

heuristics for biquadratic assignment problems and their computational comparison abstract the biquadratic assignment problem biqap is a generalization of the quadratic assignment problem qap as for any hard optimization problem also for biqap a reasonable effort to cope with the problem is trying to derive heuristics which solve it suboptimally and which possibly yield a good trade off between the solution quality and the time and memory requirements in this paper we describe several heuristics for biqaps in particular pair exchange algorithms improvement methods and variants of simulated annealing and taboo search we implement these heuristics as c codes and analyze their performances

critical properties of an ising model with dilute long range interactions abstract statistical mechanical models with local interactions in d 1 dimension can be regarded as d 1 dimensional models with regular long range interactions in this paper we study the critical properties of ising models having v sites each having z randomly chosen neighbors for z 2 the model reduces to the d 1 ising model for z we get a mean field model we find that for finite z 2 the system has a second order phase transition characterized by a length scale l ln v and mean field critical exponents that are independent of z

a novel chaotic search for quadratic assignment problems abstract we propose a novel method for solving the quadratic assignment problems first we realize the conventional tabu search on a neural network and modify it to a chaotic version our novel method includes both effects of chaotic dynamics and tabu search we compare the performance of the novel chaotic search with the conventional tabu search and an exponential tabu search whose memory effect for tabu forbidding previous moves decays exponentially we show that the exponential tabu search has higher performance than the conventional tabu search and further that the novel method with a chaotic neural network exhibits the best performance we also propose a controlling method of the chaotic neural network for realizing easy and robust applications of our method then better performance can be realized without manual parameter setting for various problems

an improved annealing scheme for the qap abstract recently there has been some interest in the use of simulated annealing to obtain good solutions to a number of combinatorial problems this paper reports on the use of this method applied to the quadratic assignment problem i e the assignment of inter communicating objects to locations to minimize the total cost of communication between them the result is a much improved annealing scheme for this problem which performs well on a range of examples finding improved solutions for several of the largest problems available in the literature and requiring only modest amounts of computational effort

on the quality of heuristic solutions to a 19 19 quadratic assignment problem abstract recently murtagh et al communicated the best known solution to a 19 19 quadratic assignment problem presented and solved earlier by elshafei their approaches are based on the application of quadratic programming and on the construction of a good initial solution respectively their solutions are confronted with one better solution obtained by lashkari and jaisingh by means of linear assignment approximations and many better solutions obtained by the present author through the application of exchange heuristics the results stress the nonconvex character of the problem and show the rather misleading nature of the concept good initial solution the quality of initial and final solution prove to be rather weakly or even negatively correlated a simple but broad and randomized search results in significantly better solutions while simple statistical aids provide an estimate of the quality of these solutions finally we make remarks concerning some practical issues

a study of exponential neighborhoods for the travelling salesman problem and for the quadratic assignment problem this paper deals with exponential neighborhoods for combinatorial optimization problems exponential neighborhoods are large sets of feasible solutions whose size grows exponentially with the input length we are especially interested in exponential neighborhoods over which the tsp respectively the qap can be solved in polynomial time and we investigate combinatorial and algorithmical questions related to such neighborhoods first we perform a careful study of exponential neighborhoods for the tsp we investigate neighborhoods that can be defined in a simple way via assignments matchings in bipartite graphs partial orders trees and other combinatorial structures we identify several properties of these combinatorial structures that lead to polynomial time optimization algorithms and we also provide variants that slightly violate these properties and lead to np complete optimization problems whereas it is relatively easy to find exponential neighborhoods over which the tsp can be solved in polynomial time the corresponding situation for the qap looks pretty hopeless every exponential neighborhood that is considered in this paper provably leads to an np complete optimization problem for the qap

integrating the nonintegrable analytic structure of the lorenz system revisited abstract a study of the complex time analytic structure of the lorenz system in nonintegrable parameter regimes reveals the special sets of parameter values for which one time dependent integral of motion exists furthermore the analysis yields the exact form of the part of the integral with the highest homogeneous weight and a method to construct the rest of the integral recursive clustering of singularities in the chaotic regimes of the system is observed in computer studies and explained by a simple analytic argument the analytic techniques used in these studies a systematic resummation of a logarithmic psi series appears to be quite general and can provide explicit representations of a solution even in the chaotic regimes in the neighborhood of a given movable singularity furthermore we suggest that this technique provides a type of renormalization program to study a wide class of nonintegrable systems

a classification of formulations for the time dependent traveling salesman problem abstract the time dependent traveling salesman problem tdtsp is a generalization of the classical traveling salesman problem where the cost of any given arc is dependent of its position in the tour the tdtsp can model several real world applications e g one machine sequencing in this paper we present a classification of formulations for the tdtsp this framework includes both new and old formulations the new formulation presented in this paper is derived from a quadratic assignment model for the tdtsp in a first step lawler s transformation procedure is used to derive an equivalent linearized version of the quadratic model in a second step a stronger formulation is obtained by tightening some constraints of the previous formulation it is shown that in terms of linear relaxations the latter formulation is either equivalent or better than other formulations already known from the literature finally we compare these formulations with other well known formulations for the classical traveling salesman problem

iterated local search for the quadratic assignment problem iterated local search ils is a simple and powerful stochastic local search method this article presents and analyzes the application of ils to the quadratic assignment problem qap we justify the potential usefulness of an ils approach to this problem by an analysis of the qap search space however an analysis of the run time behavior of a basic ils algorithm reveals a stagnation behavior which strongly compromises its performance to avoid this stagnation behavior we enhance the ils algorithm using acceptance criteria that allow moves to worse local optima and we propose population based ils extensions an experimental evaluation of the enhanced ils algorithms shows their excellent performance when compared to other state of the art algorithms for the qap

lower bounds for nonlinear assignment problems using many body interactions abstract this paper concerns lower bounding techniques for the general adic assignment problem the nonlinear objective function is linearized by the introduction of additional variables and constraints thus yielding a mixed integer linear programming formulation of the problem the concept of many body interactions is introduced to strengthen this formulation and incorporated in a modified formulation obtained by lifting the original representation to a higher dimensional space this process involves two steps i addition of new variables and constraints and ii incorporation of the new variables in the objective function if this lifting process is repeated times on an adic assignment problem along with the incorporation of higher order interactions it results in the mixed integer formulation of an equivalent adic assignment problem the incorporation of many body interactions in the higher dimensional formulation improves its degeneracy properties and is also critical to the derivation of decomposition methods for the solution of these large scale mathematical programs in the higher dimensional space it is shown that a lower bound to the optimal solution of the corresponding linear programming relaxation can be obtained by dualizing a subset of constraints in this formulation and solving o n 2 1 linear assignment problems whose coefficients depend on the dual values moreover it is proved that the optimal solution to the lp relaxation is obtained if we use the optimal duals for the solution of the linear assignment problems this concept of many body interactions could be applied in designing algorithms for the solution of formulations obtained by lifting general milp s we illustrate all these concepts on the quadratic assignment problems with these decomposition bounds we have found the provably optimal solutions of two unsolved qap s of size 32 and have also improved upon existing lower bounds for other qap s

ranking scalar products to improve bounds for the quadratic assignment problem abstract the eigenvalue bound for the quadratic assignment problem qap is successively improved by considering a set of k best scalar products related to the qap an efficient procedure is proposedto find such a set of k best scalar products a class of qaps is described for which this procedure in general improves existing lower bounds and at the same time generates good suboptimal solutions the method leaves the user with a large flexibility in controlling the quality of the bound however since the method is sensitive to input data it should only be used in combination with other bounding rules

complex singularities and the lorenz attractor the lorenz attractor is one of the best known examples of applied mathematics however much of what is known about it is a result of numerical calculations and not of mathemat ical analysis as a step toward mathematical analysis we allow the time variable in the three dimensional lorenz system to be complex hoping that solutions that have resisted analysis on the real line will give up their secrets in the complex plane knowledge of singularities being fundamental to any investigation in the complex plane we build upon earlier work and give a complete and consistent formal development of complex singular ities of the lorenz system using the psi series the psi series contain two undetermined constants in addition the location of the singularity is undetermined as a consequence of the autonomous nature of the lorenz system we prove that the psi series converge using a technique that is simpler and more powerful than that of hille thus implying a two parameter family of singular solutions of the lorenz system we pose three ques tions answers to which may bring us closer to understanding the connection of complex singularities to lorenz dynamics

a survey for the quadratic assignment problem the quadratic assignment problem qap one of the most difficult problems in the np hard class models many real life problems in several areas such as facilities location parallel and distributed computing and combinatorial data analysis combinatorial optimization problems such as the traveling salesman problem maximal clique and graph partitioning can be formulated as a qap in this paper we present some of the most important qap formulations and classify them according to their mathematical sources we also present a discussion on the theoretical resources used to define lower bounds for exact and heuristic algorithms we then give a detailed discussion of the progress made in both exact and heuristic solution methods including those formulated according to metaheuristic strategies finally we analyze the contributions brought about by the study of different approaches

the algebraic approximation of attractors the finite dimensional case abstract in this article we present a constructive method for approximating attractors the approximate sets are parts of algebraic sets i e the zeros of high degree polynomial functions and they can approximate the attractor at an arbitrary high level of accuracy

network based formulations of the quadratic assignment problem abstract we present two formulations of the quadratic assignment problem qap that result in network flow problems with integer variables and side constraints a linearization of the qap is obtained in both cases by considering each facility to consist of two parts a source for all outgoing flows from that facility and a sink for all incoming flows to the facility preliminary computational experience with both approaches is presented

a study of stochastic local search algorithms for the biobjective qap with correlated flow matrices this paper studies the performance of two stochastic local search algorithms for the biobjective quadratic assignment problem with different degrees of correlation between the flow matrices the two algorithms follow two fundamentally different ways of tackling multiobjective combinatorial optimization problems the first is based on the component wise ordering of the objective value vectors of neighboring solutions while the second is based on different scalarizations of the objective function vector our experimental results suggest that the performance of the algorithms with respect to solution quality and computation time depends strongly on the correlation between the flow matrices in addition some variants of these stochastic local search algorithms obtain very good solutions in short computation time

computing lower bounds for the quadratic assignment problem with an interior point algorithm for linear programming an example of the quadratic assignment problem qap is the facility location problem in which n facilities are assigned at minimum cost to n sites between each pair of facilities there is a given amount of flow contributing a cost equal to the product of the flow and the distance between sites to which the facilities are assigned proving optimality of qaps has been limited to instances having fewer than 20 facilities largely because known lower bounds are weak we compute lower bounds for a wide range of qaps using a linear programming based lower bound studied by z drezner 1995 on the majority of qaps tested a new best known lower bound is computed on 87 of the instances we produced the best known lower bound on several instances including some having more the 20 facilities the lower bound is tight the linear programs which can be large even for small qaps are solved with an interior point code that uses a preconditioned conjugate gradient algorithm to compute the interior point directions attempts to solve these instances using the cplex primal simplex algorithm as well as the cplex barrier primal dual interior point method were successful only for the smallest instances

application of the simulated annealing algorithm to the combinatorial optimisation problem with permutation property an investigation of generation mechanism focusing on the generation mechanism of random permutation solutions this paper investigates the application of the simulated annealing sa algorithm to the combinatorial optimisation problems with permutation property six types of perturbation scheme for generating random permutation solutions are introduced they are proved to satisfy the asymptotical convergence requirements the results of experimental evaluations on traveling salesman problem tsp flow shop scheduling problem fsp and quadratic assignment problem qap reveal that the efficiencies of the perturbation schemes are different in searching a solution space by adopting a proper perturbation scheme the sa algorithm has shown to produce very efficient solutions to different combinatorial optimisation problems with permutation property

box inequalities for quadratic assignment polytopes linear programming based lower bounds have been considered both for the general as well as for the symmetric quadratic assignment problem several times in the recent years their quality has turned out to be quite good in practice investigations of the polytopes underlying the corresponding integer linear programming formulations the non symmetric and the symmetric quadratic assignment polytope have been started during the last decade 34 31 21 22 they have lead to basic knowledge on these polytopes concerning questions like their dimensions affine hulls and trivial facets however no large class of facet defining inequalities that could be used in cutting plane procedures had been found we present in this paper the first such class of inequalities the box inequalities which have an interesting origin in some well known hypermetric inequalities for the cut polytope computational experiments with a cutting plane algorithm based on these inequalities show that they are very useful with respect to the goal of solving quadratic assignment problems to optimality or to compute tight lower bounds the most effective ones among the new inequalities turn out to be indeed facet defining for both the non symmetric as well as for the symmetric quadratic assignment polytope

random graphs with arbitrary degree distributions and their applications recent work on the structure of social networks and the internet has focused attention on graphs with distributions of vertex degree that are significantly different from the poisson degree distributions that have been widely studied in the past in this paper we develop in detail the theory of random graphs with arbitrary degree distributions in addition to simple undirected unipartite graphs we examine the properties of directed and bipartite graphs among other results we derive exact expressions for the position of the phase transition at which a giant component first forms the mean component size the size of the giant component if there is one the mean number of vertices a certain distance away from a randomly chosen vertex and the average vertex vertex distance within a graph we apply our theory to some real world graphs including the worldwide web and collaboration graphs of scientists and fortune 1000 company directors we demonstrate that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world while in others there is a measurable discrepancy between theory and reality perhaps indicating the presence of additional social structure in the network that is not captured by the random graph

breakout local search for the quadratic assignment problem the quadratic assignment problem qap is one of the most studied combinatorial optimization problems with various practical applications in this paper we present breakout local search bls for solving qap bls explores the search space by a joint use of local search and adaptive perturbation strategies experimental evaluations on the set of qaplib benchmark instances show that the proposed approach is able to attain current best known results for all but two instances with an average computing time of less than 4 5hours comparisons are also provided to show the competitiveness of the proposed approach with respect to the best performing qap algorithms from the literature

limits on relief through constrained exchange on random graphs abstract agents are represented by nodes on a random graph e g small world each agent is endowed with a zero mean random value that may be either positive or negative all agents attempt to find relief i e to reduce the magnitude of that initial value to zero if possible through exchanges the exchange occurs only between the agents that are linked a constraint that turns out to dominate the results the exchange process continues until pareto equilibrium is achieved only 40 90 of the agents achieved relief on small world graphs with mean degree between 2 and 40 even fewer agents achieved relief on scale free like graphs with a truncated power law degree distribution the rate at which relief grew with increasing degree was slow only at most logarithmic for all of the graphs considered viewed in reverse the fraction of nodes that achieve relief is resilient to the removal of links

scaling and percolation in the small world network model in this paper we study the small world network model of watts and strogatz which mimics some aspects of the structure of networks of social interactions we argue that there is one nontrivial length scale in the model analogous to the correlation length in other systems which is well defined in the limit of infinite system size and which diverges continuously as the randomness in the network tends to zero giving a normal critical point in this limit this length scale governs the crossover from large to small world behavior in the model as well as the number of vertices in a neighborhood of given radius on the network we derive the value of the single critical exponent controlling behavior in the critical region and the finite size scaling form for the average vertex vertex distance on the network and using series expansion and pad e approximants find an approximate analytic form for the scaling function we calculate the effective dimension of small world graphs and show that this dimension varies as a function of the length scale on which it is measured in a manner reminiscent of multifractals we also study the problem of site percolation on small world networks as a simple model of disease propagation and derive an approximate expression for the percolation probability at which a giant component of connected vertices first forms in epidemiological terms the point at which an epidemic occurs the typical cluster radius satisfies the expected finite size scaling form with a cluster size exponent close to that for a random graph all our analytic results are confirmed by extensive numerical simulations of the model

recent advances in the solution of quadratic assignment problems the quadratic assignment problem qap is notoriously difficult for exact solution methods in the past few years a number of long open qaps including those posed by steinberg 1961 nugent et al 1968 and krarup 1972 were solved to optimality for the first time the solution of these problems has utilized both new algorithms and novel computing structures we describe these developments as well as recent work which is likely to result in the solution of even more difficult instances

centennial history of hilbert s 16th problem the second part of hilbert s 16th problem deals with polynomial differential equations in the plane it remains unsolved even for quadratic polynomials there were several attempts to solve it that failed yet the problem inspired significant progress in the geometric theory of planar differential equations as well as bifurcation theory normal forms foliations and some topics in algebraic geometry the dramatic history of the problem as well as related developments are presented below 1 the problem and its counterparts what may be said about the number and location of limit cycles of a planar polynomial vector field of degree n the limit cycle is an isolated closed orbit of a vector field this second part of hilbert s 16th problem appears to be one of the most persistent in the famous hilbert list h second only to the riemann function conjecture traditionally hilbert s question is split into three each one requiring a stronger answer problem 1 is it true that a planar polynomial vector field has but a finite number of limit cycles problem 2 is it true that the number of limit cycles of a planar polynomial vector field is bounded by a constant depending on the degree of the polynomials only the bound on the number of limit cycles in problem 2 is denoted by h n and known as the hilbert number linear vector fields have no limit cycles hence h 1 0 it is still unknown whether or not h 2 exists problem 3 give an upper bound for h n a solution to any of these problems implies a solution for the previous ones only the first problem is solved now the positive answer was established in e92 i91 there are analytic counterparts of problems 1 and 2 received by the editors december 2001 2000 mathematics subject classification primary 34cxx 34mxx 37f75

static pairwise annihilation in complex networks we study static annihilation on complex networks in which pairs of connected particles annihilate at a constant rate during time through a mean field formalism we compute the temporal evolution of the distribution of surviving sites with an arbitrary number of connections this general formalism which is exact for disordered networks is applied to kronecker erd os r enyi i e poisson and scale free networks we compare our theoretical results with extensive numerical simulations obtaining excellent agreement although the mean field approach applies in an exact way neither to ordered lattices nor to small world networks it qualitatively describes the annihilation dynamics in such structures our results indicate that the higher the connectivity of a given network element the faster it annihilates this fact has dramatic consequences in scale free networks for which once the hubs have been annihilated the network disintegrates and only isolated sites are left

computational results of an interior point algorithm for large scale linear programming this paper gives computational results for an efficient implementation of a variant of dual projective algorithm for linear programming the implementation uses the preconditioned conjugate gradient method for computing projections our computational experience reported in this paper indicates that this algorithm has potential as an alternative for solving very large lps in which the direct methods fail due to memory and cpu time requirements the conjugate gradient algorithm was able to find very accurate directions even when the system was ill conditioned the paper also discusses a new mathematical technique called the reciprocal estimates for estimating the primal variables we have conducted extensive computational experiments on problems representative of large classes of applications of current interest we have also chosen instances of the problems of future potential interest which could not be solved in the past due to the weakness of the prior solution methods but which represent a large class of new applications the hypergraph model is such an example comparison of our implementation with minos 5 1 shows that our implementation is orders of magnitude faster than minos 5 1 for these problems

a borel transform method for locating singularities of taylor and fourier series given a taylor series with a finite radius of convergence its borel transform defines an entire function a theorem of polya relates the large distance behavior of the borel transform in different directions to singularities of the original function with the help of the new asymptotic interpolation method of van der hoeven we show that from the knowledge of a large number of taylor coefficients we can identify precisely the location of such singularities as well as their type when they are isolated there is no risk of getting artefacts with this method which also gives us access to some of the singularities beyond the convergence disk the method can also be applied to fourier series of analytic periodic functions and is here tested on various instances constructed from solutions to the burgers equation large precision on scaling exponents up to twenty accurate digits can be achieved

quadratic assignment problems abstract this paper surveys quadratic assignment problems qap at first several applications of this problem class are described and mathematical formulations of qaps are given then some exact solution methods and good heuristics are outlined their computational behaviour is illustrated by numerical results further recent results on the asymptotic probabilistic behaviour of qaps are outlined