heuristics for the container loading problem abstract the knapsack container loading problem is the problem of loading a subset of rectangular boxes into a rectangular container of fixed dimensions such that the volume of the packed boxes is maximized a new heuristic based on the wall building approach is proposed which decomposes the problem into a number of layers which again are split into a number of strips the packing of a strip may be formulated and solved optimally as a knapsack problem with capacity equal to the width or height of the container the depth of a layer as well as the thickness of each strip is decided through a branch and bound approach where at each node only a subset of branches is explored several ranking rules for the selection of the most promising layer depths and strip widths are presented and the performance of the corresponding algorithms is experimentally compared for homogeneous and heterogeneous instances the best ranking rule is then used in a comprehensive computational study involving large sized instances these computational results show that instances with a total box volume up to 90 easily may be solved to optimality and that average fillings of the container volume exceeding 95 may be obtained for large sized instances

sur le theoreme d atiyah singer on donne une version simplifiee de la demonstration probabiliste du theoreme de l indice pour l operateur de dirac donnee par j m bismut au lieu d utiliser la construction du mouvement brownien au moyen du fibre des reperes on utilise la construction de schwartz en plongeant la variete ambiante dans un espace vectoriel de dimension plus grande on evite aussi l utilisation de la decomposition de l espace de wiener en deux utilisee par j m bismut le calcul des variations stochastiques etant notre outil principal de ce fait on perd la relation avec la cohomologie de l espace des lacets

dynamic programming based algorithms for the discounted 0 1 knapsack problem abstract the discounted 0 1 knapsack problem dkp is an extension of the classical 0 1 knapsack problem kp that consists of selecting a set of item groups where each group includes three items and at most one of the three items can be selected the dkp is more challenging than the kp because four choices of items in an item group diversify the selection of the items consequently it is not possible to solve the dkp based on a classical definition of a core consisting of a small number of relevant variables this paper partitions the dkp into several easier sub problems to achieve problem reductions by imitating the core concept of the kp to derive an alternative core for the dkp numerical experiments with dp based algorithms are conducted to evaluate the effectiveness of the problem partition by solving the partitioned problem and the original problem based on different types of dkp instances

a family of composite discrete bivariate distributions with uniform marginals for simulating realistic and challenging optimization problem instances abstract we consider a family of composite bivariate distributions or probability mass functions pmfs with uniform marginals for simulating optimization problem instances for every possible population correlation except the extreme values there are an infinite number of valid joint distributions in this family we quantify the entropy for all member distributions including the special cases under independence and both extreme correlations greater variety is expected across optimization problem instances simulated based on a high entropy pmf we present a closed form solution to the problem of finding the joint pmf that maximizes entropy for a specified population correlation and we show that this entropy maximizing pmf belongs to our family of pmfs we introduce the entropy range as a secondary indicator of the variety of instances that may be generated for a particular correlation finally we discuss how to systematically control entropy and correlation to simulate a set of synthetic problem instances that includes challenging examples and examples with realistic characteristics

an exact approach for the bilevel knapsack problem with interdiction constraints and extensions we consider the bilevel knapsack problem with interdiction constraints an extension of the classic 0 1 knapsack problem formulated as a stackelberg game with two agents a leader and a follower that choose items from a common set and hold their own private knapsacks first the leader selects some items to be interdicted for the follower while satisfying a capacity constraint then the follower packs a set of the remaining items according to his knapsack constraint in order to maximize the profits the goal of the leader is to minimize the follower s total profit we derive effective lower bounds for the bilevel knapsack problem and present an exact method that exploits the structure of the induced follower s problem the approach strongly outperforms the current state of the art algorithms designed for the problem we extend the same algorithmic framework to the interval min max regret knapsack problem after providing a novel bilevel programming reformulation also for this problem the proposed approach outperforms the exact algorithms available in the literature

new analogs of split algorithms for arbitrary toeplitz plus hankel matrices fast algorithms for solving arbitrary toeplitz plus hankel systems of equations are presented the algorithms are analogs of the split levinson and schur algorithms although the more general toeplitz plus hankel structure requires that the algorithms be based on a four term recurrence relations with the previous split algorithms are considered the algorithms require roughly half as many multiplications as previous fast algorithms for toeplitz plus hankel systems

an integrated cutting stock and sequencing problem in this paper an integrated problem formulated as an integer linear programming problem is presented to find an optimal solution to the cutting stock problem under particular pattern sequencing constraints the solution uses a lagrangian approach the dual problem is solved using a modified subgradient method a heuristic for the integrated problem is also presented the computational results obtained from a set of unidimensional instances that use these procedures are reported

a fast algorithm for linear estimation of two dimensional isotropic random fields the problem considered involves estimating a two dimensional isotropic random field given noisy observations of this field over a disk of finite radius by expanding the field and observations in fourier series and exploiting the covariance structure of the resulting fourier coefficient processes recursions are obtained for efficiently constructing the linear least squares estimate of the field as the radius of the observation disk increases these recursions are similar to the levinson equations of one dimensional linear prediction in the spectral domain they take the form of schrodinger equations which are used to give an inverse spectral interpretation of our estimation procedure

an exact algorithm for constrained two dimensional two staged cutting problems theconstrained two dimensional cutting c tdc problem consists of determining acutting pattern of a set ofn small rectangular piece types on a rectangular stock plates with lengthl and widthw to maximize the sum of the profits of the pieces to be cut each piece typei i 1 n is characterized by a lengthl i a widthw i a profit or weight c i and an upper demand valueb i the upper demand value is the maximum number of pieces of typei that can be cut ons in this paper we study the two staged c tdc problem noted c 2tdc it is a classical variant of the c tdc where each piece is produced in the final cutting pattern by at most two cuts we solve the c 2tdc problem using an exact algorithm that is mainly based on a bottom up strategy we introduce new lower and upper bounds and propose new strategies that eliminate several duplicate patterns we evaluate the performance of the proposed exact algorithm on problem instances extracted from the literature and compare it to the performance of an existing exact algorithm

an expanding core algorithm for the exact 0 1 knapsack problem abstract a new branch and bound algorithm for the exact solution of the 0 1 knapsack problem is presented the algorithm is based on solving an expanding core which intially only contains the break item but which is expanded each time the branch and bound algorithm reaches the border of the core computational experiments show that most data instances are optimally solved without sorting or preprocessing a great majority of the items detailed program sketches are provided and computational experiments are reported indicating that the algorithm presented not only is shorter but also faster and more stable than any other algorithm hitherto proposed

a minimal algorithm for the 0 1 knapsack problem several types of large sized 0 1 knapsack problems kp may be easily solved but in such cases most of the computational effort is used for sorting and reduction in order to avoid this problem it has been proposed to solve the so called core of the problem a knapsack problem defined on a small subset of the variables the exact core cannot however be identified before kp is solved to optimality thus previous algorithms had to rely on approximate core sizes in this paper we present an algorithm for kp where the enumerated core size is minimal and the computational effort for sorting and reduction also is limited according to a hierarchy the algorithm is based on a dynamic programming approach where the core size is extended by need and the sorting and reduction is performed in a similar lazy way computational experiments are presented for several commonly occurring types of data instances experience from these tests indicate that the presented approach outperforms any known algorithm for kp

efficient enumeration of maximal k degenerate induced subgraphs of a chordal graph abstract in this paper we consider the problem of listing the maximal k degenerate induced subgraphs of a chordal graph and propose an output sensitive algorithm using delay o m g for any n vertex chordal graph with m edges where g n is the maximum size of a clique in g degeneracy is a well known sparsity measure and k degenerate subgraphs are a notion of sparse subgraphs which generalizes other problems such as independent sets 0 degenerate subgraphs and forests 1 degenerate subgraphs many efficient enumeration algorithms are designed by solving the so called extension problem which asks whether there exists a maximal solution containing a given set of nodes but no node from a forbidden set we show that solving this problem is np complete for maximal k degenerate induced subgraphs motivating the need for additional techniques

index theorem and equivariant cohomology on the loop space in this paper extending ideas of witten and atiyah we describe some relations of equivariant cohomology on the loop space of a manifold to the path integral representation of the index of the dirac operator on a twisted spin complex in particular the natural extension of the chern character to the whole loop space is described also it is shown that the non zero free homotopy classes of the loop space have 0 measure for the index measures associated to the index problem

performance prediction and preselection for optimization and heuristic solution procedures the operations research literature contains numerous studies on the design and application of optimization and heuristic solution procedures these studies identify a particular optimization problem suggest a general solution procedure and then customize that procedure to improve its efficiency and or accuracy in contrast this paper shows how to use existing solution procedures more effectively we develop a methodology for predicting the relative performance of alternative procedures using easily computed problem characteristics this methodology enables us for any given data set to preselect a solution procedure we apply this preselection methodology to the 0 1 knapsack problem for which two successful optimization procedures dynamic programming and branch and search are available extensive computational testing indicates that substantial savings in average computation time are achieved the benefits of our work include faster and cheaper identification of effective solution procedures as well as an improved understanding of the relationship between problem characteristics and the performance of various procedures our methodology can be applied to many optimization problems to develop easily implemented guidelines for selecting appropriate solution procedures

the witten complex and the degenerate morse inequalities on utilise le complexe introduit par witten pour demontrer des inegalites de morse dans le cas ou les points critiques de la fonction de morse sont isoles et aussi les inegalites de morse degenerees de bott

an exact recursion for the composite nearest neighbor degeneracy for a 2 n lattice space a set theoretic argument is utilized to develop a recursion relation that yields exactly the composite nearest neighbor degeneracy for simple indistinguishable particles distributed on a 2 n lattice space the associated generating functions as well as the expectation of the resulting statistics are also treated

analogues of split levinson schur and lattice algorithms for three dimensional random field estimation problems fast algorithms for computing the linear least squares estimate of a three dimensional random field from noisy observations inside a sphere are proposed the algorithms can be viewed as three dimensional analogues of the split levinson schur and lattice algorithms of linear prediction since they exploit an assumed toeplitz plus hankel structure of the double radon transform of the random field covariance therefore these algorithms require fewer computations than would the solution of the three dimensional wiener hopf integral equation unlike previous generalized levinson algorithms no quarter plane or asymmetric half plane support assumptions for the filter are necessary nor is the three dimensional filtering problem treated as a multichannel vector filtering problem the algorithms work in three stages first the three dimensional split schur algorithm computes a potential from the covariance of the random field this potential is a three dimensional analogue of the parameter appearing in the

an exact algorithm for the subset sum problem abstract the subset sum problem ssp is defined as given n positive integers w1 wn find a combination amongst them such that their sum is the closest to but not exceeding a positive integer c we suggest an exact algorithm by introducing a new type of core problem and also by using an improved version of bellman s recursion we show that the resulting algorithm is bounded in time and space resource utilizations respectively by o max n log 2 c 2 c c log 2 c and o n c in addition to the sharp memory requirement decrease in comparison with any dynamic programming based algorithm the search space for a vast range of instances is restricted only to feasible states

a minimal algorithm for the multiple choice knapsack problem abstract the multiple choice knapsack problem is defined as a 0 1 knapsack problem with the addition of disjoined multiple choice constraints as for other knapsack problems most of the computational effort in the solution of these problems is used for sorting and reduction but although o n algorithms which solve the linear multiple choice knapsack problem without sorting have been known for more than a decade such techniques have not been used in enumerative algorithms in this paper we present a simple o n partitioning algorithm for deriving the optimal linear solution and show how it may be incorporated in a dynamic programming algorithm such that a minimal number of classes are enumerated sorted and reduced computational experiments indicate that this approach leads to a very efficient algorithm which outperforms any known algorithm for the problem

a characterization of binary decision diagrams binary decision diagrams bdds are a representation of boolean functions its use in the synthesis simulation and testing of boolean circuits has been proposed by various researchers in all these applications of bdds solutions to some fundamental computational problems are needed a characterization of bdds in terms of the complexity of these computational problems is presented a tighter bound on the size of an ordered bdd that can be computed from a given boolean circuit is presented on the basis of the results a case is made for exploring the use of repeated bdds with a small number of repeated variables and free bdds for some applications for which only ordered bdds have been used so far

the dirac ramond operator on loops in flat space in this paper a rigorous construction of the s1 equivariant dirac operator i e dirac ramond operator on the space of mean zero loops in rd is given and its equivariant l2 index computed essential use is made of infinite tensor product representations of the canonical anticommutation relations algebra

a new algorithm for the 0 1 knapsack problem we present a new algorithm for the optimal solution of the 0 1 knapsack problem which is particularly effective for large size problems the algorithm is based on determination of an appropriate small subset of items and the solution of the corresponding core problem from this we derive a heuristic solution for the original problem which with high probability can be proved to be optimal the algorithm incorporates a new method of computation of upper bounds and efficient implementations of reduction procedures the corresponding fortran code is available we report computational experiments on small size and large size random problems comparing the proposed code with all those available in the literature

efficient solution of a toeplitz plus hankel coefficient matrix system of equations there are well known fast algorithms such as the levinson recursion for solving linear equations with a toeplitz or hankel coefficient matrix this paper extends the saving obtained by the levinson recursion for inversion of a toeplitz matrix to problems involving the inversion of matrices which are the sum of toeplitz and hankel matrices

self adjusting grid networks to minimize expected path length given a network infrastructure e g data center or on chip network and a distribution on the source destination requests the expected path route length is an important measure for the performance efficiency and power consumption of the network in this work we initiate a study on self adjusting networks networks that use local distributed mechanisms to adjust the position of the nodes e g virtual machines in the network to best fit the route requests distribution finding the optimal placement of nodes is defined as the minimum expected path length mepl problem this is a generalization of the minimum linear arrangement mla problem where the network infrastructure is a line and the computation is done centrally in contrast to previous work we study the distributed version and give efficient and simple approximation algorithms for interesting and practically relevant special cases of the problem in particular we consider grid networks in which the distribution of requests is a symmetric product distribution in this setting we show that a simple greedy policy of position switching between neighboring nodes to locally minimize an objective function achieves good approximation ratios we are able to prove this result using the useful notions of expected rank of the distribution and the expected distance to the center of the graph

discrete fast algorithms for two dimensional linear prediction on a polar raster new generalized split levinson and schur algorithms for the two dimensional linear least squares prediction problem on a polar raster are derived the algorithms compute the prediction filter for estimating a random field at the edge of a disk from noisy observations inside the disk the covariance function of the random field is assumed to have a toeplitz plus hankel structure for both its radial part and its transverse angular part this assumption is valid for some types of random fields such as isotropic random fields the algorithms generalize the split levinson and schur algorithms in two ways 1 to two dimensions and 2 to toeplitz plus hankel covariances

sparse solutions of sparse linear systems fixed parameter tractability and an application of complex group testing a vector with at most k nonzeros is called k sparse we show that enumerating the support vectors of k sparse solutions to a system ax b of r sparse linear equations i e where the rows of a are r sparse is fixed parameter tractable fpt in the combined parameter r k we give different branching algorithms based on the close relationship to the hitting set problem in fixed rank hypergraphs for r 2 the problem is simple for 0 1 matrices a we can also compute an o rk r kernel for systems of linear inequalities we get an fpt result in the combined parameter d k where d is the total number of minimal solutions this is achieved by interpreting the problem as a case of group testing in the complex model the problems stem from the reconstruction of chemical mixtures by observable reaction products

robust efficiency measures for linear knapsack problem variants abstract numerous combinatorial optimization applications such as the mobile retailer product mix problem utilize the multidemand multidimensional knapsack problem variant in which there are multiple knapsack constraints requiring a weighted summation of the variables to be less than or equal to a nonnegative value and multiple demand constraints requiring a weighted summation of the variables to be greater than or equal to a nonnegative value the purpose of this paper is to demonstrate that core variables and efficiency measures concepts used in the most efficient solvers to date for binary knapsack problems and some of its variants can be extended to the multidemand multidimensional knapsack problem variant specifically new efficiency measure calculations are provided and their properties are mathematically proven and experimentally demonstrated the contribution of such measures to knapsack problem research is that these measures are applicable to all knapsack problem variants with a single linear objective function and linear constraints of any quantity the applicability of these new measures is demonstrated through the development of three heuristic procedures a fixed core heuristic a genetic algorithm and a kernel search heuristic the results from these tests are compared with the results from a commercial solver and an existing heuristic the findings from these tests demonstrate that the fixed core and kernel search heuristics developed for this paper are the most efficient solvers to date for hard multidemand multidimensional knapsack problems

fast algorithms for linear least squares smoothing problems in one and two dimensions using generalized discrete bellman siegert krein resolvent identities new fast algorithms for linear least square smoothing problems in one and two dimensions are derived these are discrete and multidimensional generalizations of the bellman siegert krein resolvent identity which has been applied to the continuous one dimensional stationary smoothing problem by t kailath 1969 the new equations relate the linear least squares prediction filters associated with discrete random fields to the smoothing filters for those fields this results in new fast algorithms for deriving the latter from the former in particular used in conjunction with recently developed generalized one two dimensional split levinson and schur algorithms for covariances with block toeplitz plus hankel structure these algorithms can be used to compute smoothing filters for random fields defined on a polar raster using fewer computations than those required by previous algorithms

an improved balanced algorithm for the subset sum problem abstract we present balsublast which is an improved balanced algorithm for the subset sum problem that was designed to solve benchmarks that require the exhaustion of the search space and where there are many subsets with the same sum this new algorithm which spends time o n 2 w max and space o n w max where w max is the largest of n items has obtained excellent performance in comparison to other known algorithms

the atiyah singer index theorem for families of dirac operators two heat equation proofs the purpose of this paper is to give two heat equation proofs of the index theorem of atiyah singer for a family of dirac operators

algorithms for interval structures with applications we present new algorithms for two problems on interval structures that arise in computer aided manufacturing and in other areas we give an o kn time algorithm for the single source k link shortest path problem on an interval graph with n weighted vertices and two o n time algorithms for a generalized version of the optimal color spanning problem for n points on a real line where each point is assigned one of m colors email protected n a standard approach for solving the k link shortest path problem would take o kn 2 time and thus our result offers a linear time improvement the previously best known algorithm for the optimal color spanning problem in r 1 takes o n time and space we provide two algorithms for a generalized version of this problem in which each color must appear a specified minimum number of times one of these two solutions is suitable for an online processing of the streaming input points it uses o m working space for the ordinary one dimensional optimal color spanning problem we also show several applications of our algorithms in computer aided manufacturing and in other areas

a physicist s approach to number partitioning the statistical physics approach to the number partioning problem a classical np hard problem is both simple and rewarding very basic notions and methods from statistical mechanics are enough to obtain analytical results for the phase boundary that separates the easy to solve from the hard to solve phase of the npp as well as for the probability distributions of the optimal and sub optimal solutions in addition it can be shown that solving a number portioning problem of size n to some extent corresponds to locating the minimum in an unsorted list of o 2n numbers considering this correspondence it is not surprising that known heuristics for the partitioning problem are not signi4cantly better than simple random search

the atiyah singer theorems a probabilistic approach i the index theorem abstract the atiyah singer index theorem for classical elliptic complexes is proved by using probabilistic methods the general idea is to use a probabilistic construction of the heat equation kernel which permits the direct derivation of the index formulas without using the theory of invariants of gilkey stochastic calculus on the exterior algebra is then used to find the classical local formula for the index theorem the lefschetz fixed point formulas of atiyah singer will be proved in part ii

upper bounds and algorithms for hard 0 1 knapsack problems it is well known that many instances of the 0 1 knapsack problem can be effectively solved to optimality also for very large values of n the number of binary variables while other instances cannot be solved for n equal to only a few hundreds we propose upper bounds obtained from the mathematical model of the problem by adding valid inequalities on the cardinality of an optimal solution and relaxing it in a lagrangian fashion we then introduce a specialized iterative technique for determining the optimal lagrangian multipliers in polynomial time a branch and bound algorithm is finally developed computational experiments prove that several classes of hard instances are effectively solved even for large values of n

a mixture of dynamic programming and branch and bound for the subset sum problem given n items each having a weight wi and a container of capacity w the subset sum problemssp is to select a subset of the items whose total weight is closest to without exceeding w the paper presents a mixed approach depth first search dynamic programming to the exact solution of the problem an extensive computational experience is presented comparing the proposed algorithm with that of ahrens finke as well as with the balas zemel algorithm for large problems both easy and hard problems with values of n up to 10 000 are considered

parallel time and space upper bounds for the subset sum problem three new parallel scalable algorithms for solving the subset sum problem in o np c w m i n time and o n c space in the pram model are presented where n is the number of objects c is the capacity w m i n is the smallest weight and p is the number of processors these time and space bounds are better than the direct parallelization of bellman s algorithm which was the most efficient known result

approximate and exact algorithms for the fixed charge knapsack problem abstract the subject of this paper is the formulation and solution of a variation of the classical binary knapsack problem the variation that is addressed is termed the fixed charge knapsack problem in which sub sets of variables activities are associated with fixed costs these costs may represent certain set ups and or preparations required for the associated sub set of activities to be scheduled several potential real world applications as well as problem extensions generalizations are discussed the efficient solution of the problem is facilitated by a standard branch and bound algorithm based on 1 a non iterative polynomial algorithm to solve the lp relaxation 2 various heuristic procedures to obtain good candidate solutions by adjusting the lp solution and 3 powerful rules to peg the variables computational experience shows that the suggested branch and bound algorithm shows excellent potential in the solution of a wide variety of large fixed charge knapsack problems

tolerance analysis for 0 1 knapsack problems abstract post optimal analysis is the task of understanding the behavior of the solution of a problem due to changes in the data frequently post optimal analysis is as important as obtaining the optimal solution itself post optimal analysis for linear programming problems is well established and widely used however for integer programming problems the task is much more computationally demanding and various approaches based on branch and bound or cutting planes have been presented in the present paper we study how much coefficients in the original problem can vary without changing the optimal solution vector the so called tolerance analysis we show how to perform exact tolerance analysis for the 0 1 knapsack problem with integer coefficients in amortized time o clog n for each item where n is the number of items and c is the capacity of the knapsack amortized running times report the time used for each item when calculating tolerance limits of all items exact tolerance limits are the widest possible intervals while approximate tolerance limits may be suboptimal we show how various upper bounds can be used to determine approximate tolerance limits in time o log n or o 1 per item using the dantzig bound and dembo hammer bound respectively the running times and quality of the tolerance limits of all exact and approximate algorithms are experimentally compared showing that all tolerance limits can be found in less than a second the approximate bounds are of good quality for large sized instances while it is worth using the exact approach for smaller instances

brownian chen series and atiyah singer theorem the purpose of this work is to give a new and short proof of the atiyah singer local index theorem for the dirac operator on the spin bundle this proof is obtained by using heat semigroups approximations based on the truncation of brownian chen series

localization formulas superconnections and the index theorem for families in this paper we give a new proof of the localization formulas of berline and vergne 9 and duistermaat and heckman 18 when interpreted in the framework of atiyah 2 the probabilistic heat equation proof of the index theorem given in our paper 12 appears as the rigorous infinite dimensional version of this new proof of the localization formulas in finite dimensions the results of quillen 25 on superconnections are briefly presented the heat equation proofs 15 of the index theorem for families are described it is shown that in this framework the superconnections formalism is the operator theoretic description of integration along the fiber in the loop space

a curiosum concerning discrete time convolution it is shown that the discrete time convolution of two absolutely summable nowhere zero sequences may be identically equal to zero