approximation algorithms for the capacitated multi item lot sizing problem via flow cover inequalities we study the classical capacitated multi item lot sizing problem with hard capacities there are n items each of which has a specified sequence of demands over a finite planning horizon of t discrete periods the demands are known in advance but can vary from period to period all demands must be satisfied on time each order incurs a time dependent fixed ordering cost regardless of the combination of items or the number of units ordered but the total number of units ordered cannot exceed a given capacity c on the other hand carrying inventory from period to period incurs holding costs the goal is to find a feasible solution with minimum overall ordering and holding costs we show that the problem is strongly np hard and then propose a novel facility location type lp relaxation that is based on an exponentially large subset of the well known flow cover inequalities the proposed lp can be solved to optimality in polynomial time via an efficient separation procedure for this subset of inequalities moreover the optimal solution of the lp can be rounded to a feasible integer solution with cost that is at most twice the optimal cost this provides a 2 approximation algorithm which is the first constant approximation algorithm for the problem we also describe an interesting on the fly variant of the algorithm that does not require solving the lp a priori with all the flow cover inequalities as a by product we obtain the first theoretical proof regarding the strength of flow cover inequalities in capacitated inventory models

tight formulations for some simple mixed integer programs and convex objective integer programs we study the polyhedral structure of simple mixed integer sets that generalize the two variable set s z r 1 z 1 s z b these sets form basic building blocks that can be used to derive tight formulations for more complicated mixed integer programs for four such sets we give a complete description by valid inequalities and or an integral extended formulation and we also indicate what constraints can be added without destroying integrality we apply these results to provide tight formulations for certain piecewise linear convex objective integer programs and in a companion paper we exploit them to provide polyhedral descriptions and computationally effective mixed integer programming formulations for discrete lot sizing problems

aggregation and mixed integer rounding to solve mips in this paper we discuss the use of mixed integer rounding mir inequalities to solve mixed integer programs mir inequalities are essentially gomory mixed integer cuts however as we wish to use problem structure we insist that mir inequalities be generated from constraints or simple aggregations of constraints of the original problem this idea is motivated by the observation that several strong valid inequalities based on specific problem structure can be derived as mir inequalities here we present and test a separation routine for such mir inequalities that includes a heuristic row aggregation procedure to generate a single knapsack plus continuous variables constraint complementation of variables and finally the generation of an mir inequality inserted in a branch and cut system the results suggest that such a routine is a useful additional tool for tackling a variety of mixed integer programming problems

a branch and cut algorithm for capacitated network design problems we present a branch and cut algorithm to solve capacitated network design problems given a capacitated network and point to point traffic demands the objective is to install more capacity on the edges of the network and route traffic simultaneously so that the overall cost is minimized we study a mixed integer programming formulation of the problem and identify some new facet defining inequalities these inequalities together with other known combinatorial and mixed integer rounding inequalities are used as cutting planes to choose the branching variable we use a new rule called knapsack branching we also report on our computational experience using real life data

level repulsion for band 3 3 random matrices in view of the great interest in band random matrices the authors investigate the level repulsion for the simplest case of 3 3 band matrices a non linear repulsion is found

determination of effective brain connectivity from activity correlations effective connectivity embodied in transfer functions is derived from symmetric network activity correlations under task free conditions via a recent causal spectral factorization method this generalizes previous covariance based analyses to include frequency dependencies and time delays results are verified against analytic solutions of equations that have reproduced many aspects of experimental brain dynamics and against cases of more complex connectivity robustness to noise is also demonstrated

on explicit wiener hopf factorization of 2 2 matrices in a vicinity of a given matrix as it is known the existence of the wienerhopf factorization for a given matrix is a well studied problem severe difficulties arise however when one needs to compute the factors approximately a

semiclassical level spacings when regular and chaotic orbits coexist the authors calculate semiclassical limiting level spacing distributions p s for systems whose classical energy surface is divided into a number of separate region in which motion is regular or chaotic in the calculation it is assumed that the spectrum is the superposition of statistically independent sequences of levels from each of the classical phase space regions sequences from regular regions having poisson distributions and those from irregular regions having wigner distributions the formulae for p s depend on the sum of the liouville measures of all the classical regular regions and on the separate liouville measures of the significant chaotic regions

diversification strategies in local search for a nonbifurcated network loading problem abstract this paper examines a variant of the network loading problem a network design problem found in the telecommunications industry in this problem facilities of fixed capacity must be installed on the edges of an undirected network to carry the flow from a central vertex to a set of demand vertices the objective is to minimize the total installation costs in this work the nonbifurcated version of the problem is considered where the demand at any given vertex must be satisfied through a single path the proposed heuristics alternate between a construction phase and a local search phase each new construction phase except the first one is part of a diversification strategy aimed at providing a new starting point for the following local search phase different diversification strategies are tested and compared on large scale instances with up to 500 vertices

continuous knapsack sets with divisible capacities we study two continuous knapsack sets mathcal y ge y and mathcal y le y with n n integer one unbounded continuous and m m bounded continuous variables in either ge or le form when the coefficients of the integer variables are integer and divisible we show in both cases that the convex hull is the intersection of the bound constraints and 2 m 2m polyhedra arising as the convex hulls of continuous knapsack sets with a single unbounded continuous variable the latter convex hulls are completely described by an exponential family of partition inequalities and a polynomial size extended formulation is known in the ge case we also provide an extended formulation for the le case it follows that given a specific objective function optimization over both mathcal y ge y and mathcal y le y can be carried out by solving m m polynomial size linear programs a further consequence of these results is that the coefficients of the continuous variables all take the values 0 or 1 after scaling in any non trivial facet defining inequality of the convex hull of such sets

generalized wigner surmise for 2 2 random matrices we present new analytical results concerning the spectral distributions for 2 2 random real symmetric matrices which generalize the wigner surmise

poisson to wigner crossover transition in the nearest neighbor statistics of random points on fractals we show that the nearest neighbor spacing distribution for a model that consists of random points uniformly distributed on a self similar fractal is the brody distribution of random matrix theory in the usual context of hamiltonian systems the brody parameter does not have a definite physical meaning but in the model considered here the brody parameter is actually the fractal dimension exploiting this result we introduce a new model for a crossover transition between poisson and wigner statistics random points on a continuous family of self similar curves with fractal dimensions between 1 and 2 the implications to quantum chaos are discussed and a connection to conservative classical chaos is introduced

a capacity scaling heuristic for the multicommodity capacitated network design problem in this paper we propose a capacity scaling heuristic using a column generation and row generation technique to address the multicommodity capacitated network design problem the capacity scaling heuristic is an approximate iterative solution method for capacitated network problems based on changing arc capacities which depend on flow volumes on the arcs by combining a column and row generation technique and a strong formulation including forcing constraints this heuristic derives high quality results and computational effort can be reduced considerably the capacity scaling heuristic offers one of the best current results among approximate solution algorithms designed to address the multicommodity capacitated network design problem

projections of the capacitated network loading problem abstract consider an undirected network and a set of commodities with specified demands between various pairs of nodes of the network given two types of capacitated facilities that can be installed loaded for arc dependent costs we have to determine the integer number of facilities to load on each arc in order to send the required flow of all commodities at minimum total cost we present a natural mixed integer programming formulation of the problem and then consider its single commodity and multicommodity versions we develop equivalent formulations in a lower dimensional space by projecting out the flow variables and study the polyhedral properties of the corresponding projection cones our results strengthen an existing result for multicommodity flow problems we also characterize several classes of facet defining inequalities for this lower dimensional polyhedron and conclude by identifying some open problems and future research directions

on splittable and unsplittable flow capacitated network design arc set polyhedra we study the polyhedra of splittable and unsplittable single arc set relaxations of multicommodity flow capacitated network design problems we investigate the optimization problems over these sets and the separation and lifting problems of valid inequalities for them in particular we give a linear time separation algorithm for the residual capacity inequalities 19 and show that the separation problem of c strong in equalities 7 is np hard but can be solved over the subspace of fractional variables only we introduce two classes of inequalities for the unsplittable flow problems we present a summary of computational experiments with a branch and cut algorithm for multicommodity flow capacitated network design problems to test the effectiveness of the results presented here empirically

random matrix ensembles with column row constraints ii we analyze statistical properties of a complex system subjected to conditions which manifests through specific constraints on the column row sum of the matrix elements of its hermitian operators the presence of additional constraints besides real symmetric nature leads to new correlations among their eigenfunctions hinders a complete delocalization of dynamics and affects the eigenvalues too the statistical analysis of the latter indicates the presence of a new universality class analogous to that of a special type of brownian ensemble appearing between poisson and gaussian orthogonal ensemble

valid inequalities based on simple mixed integer sets in this paper we use facets of simple mixed integer sets with three variables to derive a parametric family of valid inequalities for general mixed integer sets we call these inequalities two step mir inequalities as they can be derived by applying the simple mixed integer rounding mir principle of wolsey 1998 twice the two step mir inequalities define facets of the master cyclic group polyhedron of gomory 1969 in addition they dominate the strong fractional cuts of letchford and lodi 2002

valid inequalities and facets of the capacitated plant location problem recently several successful applications of strong cutting plane methods to combinatorial optimization problems have renewed interest in cutting plane methods and polyhedral characterizations of integer programming problems in this paper we investigate the polyhedral structure of the capacitated plant location problem our purpose is to identify facets and valid inequalities for a wide range of capacitated fixed charge problems that contain this prototype problem as a substructure

design of capacitated multicommodity networks with multiple facilities this paper addresses the problem of designing a multicommodity network using several facilities with different costs and capacities the problem is addressed in a special context of designing private telecommunications networks using fractional t1 services the algorithm starts from any given solution of the problem and gradually improves it by solving a series of subproblems arriving at a local minimum the subproblem is defined over a subset of links called the subnetwork by using one of the links as the base link it is shown that the subproblem can be formulated as a multiple choice knapsack problem that is solved by dynamic programming computational results and lower bounds are reported on problems of up to 20 nodes and up to 3 facilities on most problems the algorithm produces solutions within about 5 of lower bound on the average although it was not possible to compute lower bounds for larger problems heuristic solutions and running times are reported for problems of up to 99 nodes and four facilities

manufacturer s mixed pallet design problem abstract we study a problem faced by a major beverage producer the company produces and distributes several brands to various customers from its regional distributors for some of these brands most customers do not have enough demand to justify full pallet shipments therefore the company decided to design a number of mixed or rainbow pallets so that its customers can order these unpopular brands without deviating too much from what they initially need we formally state the company s problem as determining the contents of a pre determined number of mixed pallets so as to minimize the total inventory holding and backlogging costs of its customers over a finite horizon we first show that the problem is np hard we then formulate the problem as a mixed integer linear program and incorporate valid inequalities to strengthen the formulation finally we use company data to conduct a computational study to investigate the efficiency of the formulation and the impact of mixed pallets on customers total costs

on the algorithmization of janashia lagvilava matrix spectral factorization method we consider three different ways of algorithmization of the janashia lagvilava spectral factorization method the first algorithm is faster than the second one however it is only suitable for matrices of low dimension the second algorithm on the other hand can be applied to matrices of substantially larger dimension the third algorithm is a superfast implementation of the method but only works in the polynomial case under the additional restriction that the zeros of the determinant are not too close to the boundary all three algorithms fully utilize the advantage of the method which carries out spectral factorization of leading principal submatrices step by step the corresponding results of numerical simulations are reported in order to describe the characteristic features of each algorithm and compare them to other existing algorithms

strong formulations for mixed integer programs valid inequalities and extended formulations we examine progress over the last fifteen years in finding strong valid inequalities and tight extended formulations for simple mixed integer sets lying both on the easy and hard sides of the complexity frontier most progress has been made in studying sets arising from knapsack and single node flow sets and a variety of sets motivated by different lot sizing models we conclude by citing briefly some of the more intriguing new avenues of research

on invariant 2 2 β ensembles of random matrices abstract we introduce and solve exactly a family of invariant 2 2 random matrices depending on one parameter and we show that rotational invariance and real dyson index are not incompatible properties the probability density for the entries contains a weight function and a multiple trace trace interaction term which corresponds to the representation of the vandermonde squared coupling on the basis of power sums as a result the effective dyson index eff of the ensemble can take any real value in an interval two weight functions gaussian and non gaussian are explored in detail and the connections with ensembles of dumitriu edelman and the so called poisson wigner crossover for the level spacing are respectively highlighted a curious spectral twinning between ensembles of different symmetry classes is unveiled as a consequence the identification between symmetry group orthogonal unitary or symplectic and the exponent of the vandermonde 1 2 4 is shown to be potentially deceptive the proposed technical tool more generically allows for designing actual matrix models which i are rotationally invariant ii have a real dyson index eff iii have a pre assigned confining potential or alternatively level spacing profile the analytical results have been checked through numerical simulations with an excellent agreement eventually we discuss possible generalizations and further directions of research

a note on the level spacings distribution of the hamiltonians in the transition region between integrability and chaos it is well known that there is no strict universality of the spectral fluctuations of quantum hamiltonians whose classical counterparts undergo the transition from integrability to complete chaos the author discusses the level spacings distribution p s and explains why the semiclassical formulae of berry and robnik cannot be correct for small s there is no global universality of p s for nearly integrable systems but the approach to the integrability as the perturbation parameter in goes to zero can be universal this is reflected in the fact that the slope of dp ds at s 0 for small in is universally inversely proportional to in the author gives two models in terms of two dimensional random matrices one of them being based on maximum entropy considerations the author also points out the connection to the statistics of zeros of random functions and discusses the numerical evidence

a polyhedral study of the semi continuous knapsack problem we study the convex hull of the feasible set of the semi continuous knapsack problem in which the variables belong to the union of two intervals such structure arises in a wide variety of important problems e g blending and portfolio selection we show how strong inequalities valid for the semi continuous knapsack polyhedron can be derived and used in a branch and cut scheme for problems with semi continuous variables to demonstrate the effectiveness of these inequalities which we call collectively semi continuous cuts we present computational results on real instances of the unit commitment problem as well as on a number of randomly generated instances of linear programming with semi continuous variables

eigenvalue spacing statistics of a four matrix model of some four by four random matrices a simple four matrix model consisting of some 4 4 real and imaginary random matrices and three parameters is introduced it is shown that the eigenvalue spacing statistics of an ensemble of such matrices can be used to describe transitions between all the wigner surmises of random matrix theory formulae for the nearest neighbour spacing distributions of the ensemble are given for various parameter values

on capacitated network design cut set polyhedra this paper provides an analysis of capacitated network design cut set polyhedra we give a complete linear description of the cut set polyhedron of the single commodity single facility capacitated network design problem then we extend the analysis to single commodity multifacility and multicommodity multifacility capacitated network design problems valid inequalities described here are applicable to directed network design problems with any number of facility types and any level of capacities we report results from a computational study done for testing the effectiveness of the new inequalities

on the facets of the mixed integer knapsack polyhedron we study the mixed integer knapsack polyhedron that is the convex hull of the mixed integer set defined by an arbitrary linear inequality and the bounds on the variables we describe facet defining inequalities of this polyhedron that can be obtained through sequential lifting of inequalities containing a single integer variable these inequalities strengthen and or generalize known inequalities for several special cases we report computational results on using the inequalities as cutting planes for mixed integer programming

spectral fluctuation properties of hamiltonian systems the transition region between order and chaos the authors study numerically the classical dynamical behaviour and the spectral fluctuation properties of a class of hamiltonian systems with two degrees of freedom the quantum mechanical properties of these systems are monotonic but non universal functions of the fraction of classical phase space filled by chaotic trajectories it is found that the observed spectral fluctuation measures can be reproduced by a random matrix model which depends on one parameter only

valid linear inequalities for fixed charge problems many problems in the operations research management science literature can be formulated with both zero one and continuous variables however the exact optimization of such mixed zero one models remains a computational challenge in this paper we propose to study mixed problems from a mathematical point of view that is similar in spirit to recent research on purely combinatorial problems that has investigated systems of defining linear inequalities or facets of the underlying polytope at least two numerical studies have validated this line of research computationally and the advances in the problem solving capabilities are considerable we expect that similar gains are possible for the mixed zero one problem more precisely we consider the mixed integer programs whose feasible region x is composed of i a simple additive constraint in the continuous variables xj for j 1 2 n and ii constraints 0 xj mjyj defined by binary variables yj for j 1 2 n this type of feasible region arises in a variety of mixed integer problems and particularly in network problems with fixed charges on the arcs we derive two classes of facet defining linear inequalities of the convex hull of x and show that the second of these classes gives a complete description of the convex hull when mj m for all j we also develop methods to detect violated inequalities from these classes so that these facets can be used as cutting planes to strengthen the formulations of certain mixed integer problems

level repulsion near integrability a random matrix analogy using the analogy between the statistics of the levels of quantum hamiltonians and the eigenvalues of random matrices the authors use an appropriate choice of the latter in order to study the transition region near integrability they show that the nearest neighbour spacing distribution is linear for small spacings while the inverse of its slope is proportional to the amplitude of the integrability destroying perturbation

logarithmic terms in the spectral statistics of band random matrices the authors present a detailed analysis of the nearest neighbour spacing distribution nnsd for random matrices on which the non zero elements are confined to a band around the diagonal the vanishing of the remaining elements induces non analytic logarithmic behaviour for the nnsd at very small spacings a conjective for the precise behaviour of these logarithmic terms is presented and supported by extensive numerical computations

single commodity robust network design with finite and hose demand sets we study a single commodity robust network design problem srnd defined on an undirected graph our goal is to determine minimum cost capacities such that any traffic demand from a given uncertainty set can be satisfied by a feasible single commodity flow we consider two ways of representing the uncertainty set either as a finite list of scenarios or as a polytope we propose a branch and cut algorithm to derive optimal solutions to srnd built on a capacity based integer linear programming formulation it is strengthened with valid inequalities derived as 0 frac 1 2 0 12 chvatal gomory cuts since the formulation contains exponentially many constraints we provide practical separation algorithms extensive computational experiments show that our approach is effective in comparison to existing approaches from the literature as well as to solving a flow based formulation by a general purpose solver

spacing distributions for real symmetric 2 2 generalized gaussian ensembles for ensembles of 2 2 real symmetric matrices the normalized spacing distributions p s form a family whose parameter space is the unit cube with coordinates determined by the means and variances of the diagonal and off diagonal elements the cube contains a variety of spacing distributions that are calculated analytically they include the wigner poisson transition distributions with singularities and gaussians unfolding is superfluous for 2 2 matrices but it can be implemented giving rise to a further variety of spacing distributions some surprising

mixing sets linked by bidirected paths recently there has been considerable research on simple mixed integer sets called mixing sets and closely related sets arising in uncapacitated and constant capacity lot sizing this in turn has led to study of more general sets called network dual sets for which it is possible to derive extended formulations whose projection gives the convex hull of the network dual set unfortunately this formulation cannot be used in general to optimize in polynomial time furthermore the inequalities defining the convex hull of a network dual set in the original space of variables are known only for some special cases here we study two new cases in which the continuous variables of the network dual set are linked by a bidirected path in the first case which is motivated by lot sizing problems with lost sales we provide a description of the convex hull as the intersection of the convex hulls of 2 n mixing sets where n is the number of continuous variables of the set however optimization is polynomial as only n 1 of the sets are required for any given objective function in the second case generalizing single arc flow sets we describe again the convex hull as the intersection of an exponential number of mixing sets and also give a combinatorial polynomial time separation algorithm

design of survivable networks using three and four partition facets this paper considers the problem of designing a multicommodity network with single facility type subject to the requirement that under failure of any single edge the network should permit a feasible flow of all traffic we study the polyhedral structure of the problem by considering the multigraph obtained by shrinking the nodes but not the edges in a k partition of the original graph a key theorem is proved according to which a facet of the k node problem defined on the multigraph resulting from a k partition is also facet defining for the larger problem under a mild condition after reviewing the prior work on two partition inequalities we develop two classes of three partition inequalities and a large number of inequality classes based on four partitions proofs of facet defining status for some of these are provided while the rest are stated without proof computational results show that the addition of three and four partition inequalities results in substantial increase in the bound values compared to those possible with two partition inequalities alone problems of 35 nodes and 80 edges with fully dense traffic matrices have been solved optimally within a few minutes of computer time

modeling and solving the two facility capacitated network loading problem this paper studies a topical and economically significant capacitated network design problem that arises in the telecommunications industry in this problem given point to point communication demand in a network must be met by installing loading capacitated facilities on the arcs loading a facility incurs an arc specific and facility dependent cost this paper develops modeling and solution approaches for loading facilities to satisfy the given demand at minimum cost we consider two approaches for solving the underlying mixed integer program a lagrangian relaxation strategy and a cutting plane approach that uses three classes of valid inequalities that we identify for the problem we show that a linear programming formulation that includes these inequalities always approximates the value of the mixed integer program at least as well as the lagrangian relaxation bound our computational results on a set of prototypical telecommunication data show that including these inequalities considerably improves the gap between the integer programming formulation and its linear programming relaxation from an average of 25 to an average of 8 these results show that strong cutting planes can be an effective modeling and algorithmic tool for solving problems of the size that arise in the telecommunications industry

algorithmic complexity of the eigenvalue sequence of a nonintegrable hamiltonian system irregularity of the eigenvalue sequence in a quantum nonintegrable system is investigated from the viewpoint of algorithmic complexity theory an example of a real symmetric matrix whose eigenvalue sequence has null complexity but whose nearest neighbor level spacing shows the wigner type distribution is presented

semiclassical theory of spectral rigidity the spectral rigidity l of a set of quantal energy levels is the mean square deviation of the spectral staircase from the straight line that best fits it over a range of l mean level spacings in the semiclassical limit 0 formulae are obtained giving l as a sum over classical periodic orbits when l l max where l max n 1 for a system of n freedoms l is shown to display the following universal behaviour as a result of properties of very long classical orbits if the system is classically integrable all periodic orbits filling tori l 1 5 l as in an uncorrelated poisson eigenvalue sequence if the system is classically chaotic all periodic orbits isolated and unstable and has no symmetry l in l 2 2 d if 1 l l max as in the gaussian unitary ensemble of random matrix theory if the system is chaotic and has time reversal symmetry l in l 2 e if 1 l l max as in the gaussian orthogonal ensemble when l l max l saturates non universally at a value determined by short classical orbits of order n 1 for integrable systems and in 1 for chaotic systems these results are obtained by using the periodic orbit expansion for the spectral density together with classical sum rules for the intensities of long orbits and a semiclassical sum rule restricting the manner in which their contributions interfere for two examples l is studied in detail the rectangular billiard integrable and the riemann zeta function assuming its zeros to be the eigenvalues of an unknown quantum system whose unknown classical limit is chaotic

spacing distributions for point processes on a regular fractal the homogeneous poisson point process in rd denoted by pd is a basic model of stochastic geometry and modern statistical physics using ideas from fractal geometry geometrical statistics and random matrix theory we introduce the model of random points on a self similar fractal as a model of intermediate statistics in the sense that the interpoint spacing statistics of the model are intermediate between those of p1 and p2 when the fractal dimension is in between 1 and 2 and intermediate between those of p2 and p3 when the fractal dimension is in between 2 and 3 and so on we also introduce the idea of using a continuous family of such models to interpolate between p1 and p2 and thereby effectuate crossover transitions between p1 statistics and p2 statistics we first derive the kth nearest neighbor spacing distribution for the general model and then study the interpoint spacing statistics of several realizations of the model involving sierpinski fractals in r2 and r3 we also study a realization of a continuous interpolation between p1 and p2 in particular a continuous interpolation between a point process on a line and a point process on a plane filling curve using the continuous family of self similar koch curves in r2 in the latter study we specifically analyze the second nearest neighbor interpoint spacing statistics which undergo a crossover transition between semi poisson and ginibre statistics

exact conditions for preservation of the partial indices of a perturbed triangular 2 2 matrix function the possible instability of partial indices is one of the important constraints in the creation of approximate methods for the factorization of matrix functions this paper is devoted to a study of a specific class of triangular matrix functions given on the unit circle with a stable and unstable set of partial indices exact conditions are derived that guarantee a preservation of the unstable set of partial indices during a perturbation of a matrix within the class thus even in this probably simplest of cases when the factorization technique is well developed the structure of the parametric space guiding the types of matrix perturbations is non trivial

a stabilized structured dantzig wolfe decomposition method we discuss an algorithmic scheme which we call the stabilized structured dantzig wolfe decomposition method for solving large scale structured linear programs it can be applied when the subproblem of the standard dantzig wolfe approach admits an alternative master model amenable to column generation other than the standard one in which there is a variable for each of the extreme points and extreme rays of the corresponding polyhedron stabilization is achieved by the same techniques developed for the standard dantzig wolfe approach and it is equally useful to improve the performance as shown by computational results obtained on an application to the multicommodity capacitated network design problem

the convex hull of two core capacitated network design problems the network loading problem nlp is a specialized capacitated network design problem in which prescribed point to point demand between various pairs of nodes of a network must be met by installing loading a capacitated facility we can load any number of units of the facility on each of the arcs at a specified arc dependent cost the problem is to determine the number of facilities to be loaded on the arcs that will satisfy the given demand at minimum cost

discretized formulations for capacitated location problems with modular distribution costs abstract in this paper we study a discretization reformulation technique in the context of a facility location problem with modular link costs we present a so called traditional model and a straightforward discretized model with a general objective function whose variable coefficients are computed by solving a simple knapsack problem we show that the linear programming relaxation of the discretized model dominates the linear programming relaxation of the original model the discretized model suggests quite intuitive valid inequalities that considerably improve the linear programming relaxation of the original model computational results based on randomly generated data show that in the context of problems with modular costs the proposed discretized models perform significantly better than the traditional models an important outcome of our research is the result whose proof is also presented in this paper that a restricted version of the discretized model gives an extended description of the convex hull of the integer solutions of a subproblem that usually arises in network design problems with modular costs

strong inequalities for capacitated survivable network design problems we present several classes of facet defining inequalities to strengthen polyhedra arising as subsystems of network design problems with survivability constraints these problems typically involve assigning capacities to a network with multicommodity demands such that after a vertex or edge deletion at least some prescribed fraction of each demand can be routed

level spacing for band random matrices the authors examine the spacing distribution of the eigenvalues of tridiagonal real symmetric random matrices the elements of which are distributed according to a gaussian law they show explicitly that for 4 4 matrices the distribution at small spacings behaves as s log2 s they surmise that for n n matrices the behaviour is s logn 2s and they present numerical results which support this conjecture

polyhedral results for the edge capacity polytope network loading problems occur in the design of telecommunication networks in many different settings for instance bifurcated or non bifurcated routing also called splittable and unsplittable can be considered in most settings the same polyhedral structures return a better understanding of these structures therefore can have a major impact on the tractability of polyhedral guided solution methods in this paper we investigate the polytopes of the problem restricted to one arc edge of the network the undirected directed edge capacity problem for the non bifurcated routing case as an example one of the basic variants of network loading is described including an integer linear programming formulation as the edge capacity problems are relaxations of this network loading problem their polytopes are intimately related we give conditions under which the inequalities of the edge capacity polytopes define facets of the network loading polytope we describe classes of strong valid inequalities for the edge capacity polytopes and we derive conditions under which these constraints define facets for the diverse classes the complexity of lifting projected variables is stated the derived inequalities are tested on i the edge capacity problem itself and ii the described variant of the network loading problem the results show that the inequalities substantially reduce the number of nodes needed in a branch and cut approach moreover they show the importance of the edge subproblem for solving network loading problems

a new method of matrix spectral factorization a new algorithm of matrix spectral factorization is proposed which can be applied to compute an approximate spectral factor of any positive definite matrix function which satisfies the paley wiener condition

the continuous knapsack set we study the convex hull of the continuous knapsack set which consists of a single inequality constraint with n n non negative integer and m m non negative bounded continuous variables when n 1 n 1 this set is a generalization of the single arc flow set studied by magnanti et al math program 60 233 250 1993 we first show that in any facet defining inequality the number of distinct non zero coefficients of the continuous variables is bounded by 2 n n 2n n our next result is to show that when n 2 n 2 this upper bound is actually 1 this implies that when n 2 n 2 the coefficients of the continuous variables in any facet defining inequality are either 0 or 1 after scaling and that all the facets can be obtained from facets of continuous knapsack sets with m 1 m 1 the convex hull of the sets with n 2 n 2 and m 1 m 1 is then shown to be given by facets of either two variable pure integer knapsack sets or continuous knapsack sets with n 2 n 2 and m 1 m 1 in which the continuous variable is unbounded the convex hull of these two sets has been completely described by agra and constantino discrete optim 3 95 110 2006 finally we show via an example that when n 3 n 3 the non zero coefficients of the continuous variables can take different values

survivable network design with shared protection routing abstract in this paper we study the problem of designing a survivable telecommunication network with shared protection routing we develop a heuristic algorithm to solve this problem recent results in the area of global re routing have been used to obtain very tight lower bounds for the problem our results indicate that in a majority of problem instances the average gap between the heuristic solutions and the lower bounds is within 5 computational experience is reported on randomly generated problem instances with up to 35 nodes 80 edges and 595 demand pairs and also on the instances available in sndlib database

fixed charge multicommodity network design using p partition facets abstract we are given an undirected network g v e and a set of traffic demands to install a potential edge e e we incur a cost fe to provide a positive capacity ae the objective is to select edges at minimum cost so as to permit a feasible multicommodity flow of all traffic we study structure of the projection polytope of this problem in the space of binary variables associated with fixed charges by relating facets of a p node problem p 2 3 or 4 defined over a multi graph obtained by a p partition of v to the facets of the original problem inspired from the well known cover inequalities of the knapsack problem we develop the notion of p partition cover inequalities we present necessary and sufficient conditions for such inequalities to be facet defining for p 3 and 4 a simple heuristic approach for separating and adding such violated inequalities is presented and implemented for p values up to 10 we report optimal solutions for problems with 30 nodes 60 edges and fully dense demand matrices within a few minutes of cpu time for most instances some results for dense graph problems are also reported

a comparison of heuristics and relaxations for the capacitated plant location problem abstract approaches proposed in the literature for the capacitated plant location problem are compared the comparison is based on new theoretical and computational results the main emphasis is on relaxations in particular dominance relations among the various relaxations found in the literature are identified in the computational study the relaxations are compared as a function of various characteristics of the test problems several of these relaxations can be used to generate heuristic feasible solutions that are better than the classical greedy or interchange heuristics both in computing time and in the quality of the solutions found