a note on the selection of benders cuts a new cut selection criterion for benders cuts is proposed and computationally analyzed the results show that the new criterion is more robust and often considerably faster than the standard ones

the traveling salesman problem an overview of exact and approximate algorithms in this paper some of the main known algorithms for the traveling salesman problem are surveyed the paper is organized as follows 1 definition 2 applications 3 complexity analysis 4 exact algorithms 5 heuristic algorithms 6 conclusion

lp bounds in an interval graph algorithm for orthogonal packing feasibility we consider the feasibility problem opp orthogonal packing problem in higher dimensional orthogonal packing given a set of d dimensional d 2 rectangular items decide whether all of them can be orthogonally packed in the given rectangular container without rotation the one dimensional 1d bar lp relaxation of opp reduces the latter to a 1d cutting stock problem where the packing of each stock bar represents a possible 1d stitch through an opp layout the dual multipliers of the lp provide us with another kind of powerful bounding information conservative scales we investigate how the set of possible 1d packings can be tightened using the overlapping information of item projections on the axes with the goal to tighten the relaxation we integrate the bar relaxation into an interval graph algorithm for opp which operates on such overlapping relations numerical results on 2d and 3d instances demonstrate the efficiency of tightening leading to a speedup and stabilization of the algorithm

formulations and benders decomposition algorithms for multidepot salesmen problems with load balancing this paper describes new models and exact solution algorithms for the fixed destination multidepot salesmen problem defined on a graph with n nodes where the number of nodes each salesman is to visit is restricted to be in a predefined range such problems arise when the time to visit a node takes considerably longer as compared to the time of travel between nodes in which case the number of nodes visited in a salesman s tour is the determinant of their load the new models are novel multicommodity flow formulations with o n 2 binary variables which is contrary to the existing formulations for the same and similar problems that typically include o n 3 binary variables the paper also describes benders decomposition algorithms based on the new formulations for solving the problem exactly results of the computational experiments on instances derived from tsplib show that some of the proposed algorithms perform remarkably well in cases where formulations solved by a state of the art optimization code fail to yield optimal solutions within reasonable computation time

an exact algorithm for the two dimensional strip packing problem this paper considers the two dimensional strip packing problem 2sp in which a set of rectangular items have to be orthogonally packed without overlapping into a strip of a given width and infinite height by minimizing the overall height of the packing the 2sp is np hard in the strong sense and finds many practical applications we propose reduction procedures lower and upper bounds and an exact algorithm for the 2sp the new lower bounds are both combinatorial bounds and bounds derived from different relaxations of mathematical formulations of the 2sp the new upper bounds are obtained by constructive heuristics based on different strategies to place the items into the strip the new exact method is based on a branch and bound approach computational results on different sets of test problems derived from the literature show the effectiveness of the new lower and upper bounds and of the new exact algorithm

a near optimal solution to a two dimensional cutting stock problem we present an asymptotic fully polynomial approximation scheme for strip packing or packing rectangles into a rectangle of fixed width and minimum height a classicalnp hard cutting stock problem the algorithm based on a new linear programming relaxation finds a packing ofn rectangles whose total height is within a factor of 1 e of optimal up to an additive term and has running time polynomial both in n and in 1 e

elliptic equations and maps of bounded length distortion on considere l equation elliptique d ordre 2 l u i f 1 n 1 a ij ju 0 ou les coefficients a ij sont des fonctions c 1 dans un domaine d de r n

a two stage intelligent search algorithm for the two dimensional strip packing problem this paper presents a two stage intelligent search algorithm for a two dimensional strip packing problem without guillotine constraint in the first stage a heuristic algorithm is proposed which is based on a simple scoring rule that selects one rectangle from all rectangles to be packed for a given space in the second stage a local search and a simulated annealing algorithm are combined to improve solutions of the problem in particular a multi start strategy is designed to enhance the search capability of the simulated annealing algorithm extensive computational experiments on a wide range of benchmark problems from zero waste to non zero waste instances are implemented computational results obtained in less than 60 seconds of computation time show that the proposed algorithm outperforms the supposedly excellent algorithms reported recently on average it performs particularly better for large instances

weak bld mappings and hausdorff measure abstract we prove that if x y is a mapping of weak bounded length distortion from a quasiconvex and complete metric space x to any metric space y then for any lipschitz mapping f r k e x we have that hausdorff measure h k f e 0 equals zero in x if and only if h k f e 0 in y this generalizes an earlier result of hajlasz and malekzadeh where the target space y was a euclidean space y r n

wolfe s theorem for weakly differentiable cochains abstract a fundamental theorem of wolfe isometrically identifies the space of flat differential forms of dimension m in r n with the space of flat m cochains that is the dual space of flat chains of dimension m in r n the main purpose of the present paper is to generalize wolfe s theorem to the setting of sobolev differential forms and sobolev cochains in r n a suitable theory of sobolev cochains has recently been initiated by the second and third author it is based on the concept of upper norm and upper gradient of a cochain introduced in analogy with heinonen koskela s concept of upper gradient of a function

uniformization of two dimensional metric surfaces we establish uniformization results for metric spaces that are homeomorphic to the euclidean plane or sphere and have locally finite hausdorff 2 measure applying the geometric definition of quasiconformality we give a necessary and sufficient condition for such spaces to be qc equivalent to the euclidean plane disk or sphere moreover we show that if such a qc parametrization exists then the dilatation can be bounded by 2 as an application we show that the euclidean upper bound for measures of balls is a sufficient condition for the existence of a 2 qc parametrization this result gives a new approach to the bonk kleiner theorem on parametrizations of ahlfors 2 regular spheres by quasisymmetric maps

a node current based 2 index formulation for the fixed destination multi depot travelling salesman problem abstract the travelling salesman problem forms a basis for many optimisation problems in logistics finance and engineering several variants exist to accommodate for different problem types in this paper we discuss the fixed destination multi depot travelling salesman problem where several salesmen will start from different depots and they are required to return to the depot they originated from we propose a novel formulation for this problem using 2 index binary variables and node currents and compare it to other 2 index formulations from the literature this novel formulation requires less binary variables and continuous variables to formulate a problem resulting in lower computation times using a large benchmark the effectiveness of the new formulation is demonstrated

bidimensional packing by bilinear programming we consider geometric problems in which rectangles have to be packed into identical squares these problems turn out to be very hard in practice and ilp formulations in which variables specify the coordinates in the packing perform very poorly while most methods developed so far are based on simple geometric considerations a recent landmark result of fekete and schepers suggests to put these geometric aspects aside and use the most advanced tools for the one dimensional case in this paper we make additional progress in this direction especially on the basic question does a given set of rectangles fit into a square which turns out to be the bottleneck of all the approaches known given a set of rectangles and the associated convex hull of rectangle subsets that fit into a square we derive a wide class of valid inequalities for this convex hull from a complete description of the two knapsack polytopes associated with the widths and the heights of the rectangles respectively in addition we illustrate how to solve the associated separation problem as a bilinear program for which we develop a solution method that turns out to be fast in practice and show that the integer solutions that satisfy all these constraints generally correspond to vertices of the original convex hull for the benchmark instances in the literature the same tools are used to derive lower bounds for the two dimensional bin packing problem corresponding to the determination of an optimal pair of so called dual feasible functions these lower bounds in many cases equal those obtained by the customary set covering formulation for which column generation is very hard but are computable within a time that is smaller by some orders of magnitude this allows us to close a few of the benchmark instances in the literature

quasisymmetric structures on surfaces we show that a locally ahlfors 2 regular and locally linearly locally contractible metric surtace is locally quasisymmetrically equivalent to tne disk we also discuss an application of this result to the problem of characterizing surfaces embedded in some euclidean spaces that are locally bi lipschitz equivalent to a ball in the plane

on symmetry detection a straight line is an axis ofsymmetry of a planar figure if the figure is invariant to reflection with respect to that line the purpose of this correspondence is to describe an o n log n time algorithm for enumerating all the axes of symmetry of a planar figure which is made up of possibly intersecting segments circles points etc the solution involves a reduction of the problem to a combinatorial question on words our algorithm is optimal since we can establish an n log n time lower bound for this problem

integer linear programming formulations of multiple salesman problems and its variations in this paper we extend the classical multiple traveling salesman problem mtsp by imposing a minimal number of nodes that a traveler must visit as a side condition we consider single and multidepot cases and propose integer linear programming formulations for both with new bounding and subtour elimination constraints we show that several variations of the multiple salesman problem can be modeled in a similar manner computational analysis shows that the solution of the multidepot mtsp with the proposed formulation is significantly superior to previous approaches

scheduling inspired models for two dimensional packing problems abstract we propose two exact algorithms for two dimensional orthogonal packing problems whose main components are simple mixed integer linear programming models based on the different forms of time representation in scheduling formulations we extend the concept of multiple time grids into a second dimension and propose a hybrid discrete continuous space formulation by relying on events to continuously locate the rectangles along the strip height we aim to reduce the size of the resulting mathematical problem when compared to a pure discrete space model with hopes of achieving a better computational performance through the solution of a set of 29 test instances from the literature we show that this was mostly accomplished primarily because the associated search strategy can quickly find good feasible solutions prior to the optimum which may be very important in real industrial environments we also provide a comprehensive comparison to seven other conceptually different approaches that have solved the same strip packing problems

skewed rotation symmetry group detection we present a novel and effective algorithm for affinely skewed rotation symmetry group detection from real world images we define a complete skewed rotation symmetry detection problem as discovering five independent properties of a skewed rotation symmetry group 1 the center of rotation 2 the affine deformation 3 the type of the symmetry group 4 the cardinality of the symmetry group and 5 the supporting region of the symmetry group in the image we propose a frieze expansion fe method that transforms rotation symmetry group detection into a simple 1d translation symmetry detection problem we define and construct a pair of rotational symmetry saliency maps complemented by a local feature method frequency analysis using discrete fourier transform dft is applied to the frieze expansion patterns feps to uncover the types cyclic dihedral and o 2 the cardinalities and the corresponding supporting regions concentric or otherwise of multiple rotation symmetry groups in an image the phase information of the fep is used to rectify affinely skewed rotation symmetry groups our result advances the state of the art in symmetry detection by offering a unique combination of region based feature based and frequency based approaches experimental results on 170 synthetic and natural images demonstrate superior performance of our rotation symmetry detection algorithm over existing methods

exact solution of the two dimensional finite bon packing problem given a set of rectangular pieces to be cut from an unlimited number of standardized stock pieces bins the two dimensional finite bin packing problem is to determine the minimum number of stock pieces that provide all the pieces the problem is np hard in the strong sense and finds many practical applications in the cutting and packing area we analyze a well known lower bound and determine its worst case performance we propose not lower bounds which are used within a branch and bound algorithm for the exact solution of the problem extensive computational testing on problem instances from the literature involving up to 120 pieces shows the effectiveness of the proposed approach

logic based benders decomposition benders decomposition uses a strategy of learning from one s mistakes the aim of this paper is to extend this strategy to a much larger class of problems the key is to generalize the linear programming dual used in the classical method to an inference dual solution of the inference dual takes the form of a logical deduction that yields benders cuts the dual is therefore very different from other generalized duals that have been proposed the approach is illustrated by working out the details for propositional satisfiability and 0 1 programming problems computational tests are carried out for the latter but the most promising contribution of logic based benders may be to provide a framework for combining optimization and constraint programming methods

mathematical formulations for a 1 full truckload pickup and delivery problem abstract we address a generalization of the asymmetric traveling salesman problem where routes have to be constructed to satisfy customer requests which either involve the pickup or delivery of a single commodity a vehicle is to be routed such that the demand and the supply of the customers is satisfied under the objective to minimize the total distance traveled the commodities which are collected from the pickup customers can be used to accommodate the demand of the delivery customers in this paper we present three mathematical formulations for this problem class and apply branch and cut algorithms to optimally solve the model formulations for two of the models we derive benders cuts based on the classical and the generalized benders decomposition finally we analyze the different mathematical formulations and associated solution approaches on well known data sets from the literature

exact algorithms for the two dimensional strip packing problem with and without rotations we propose exact algorithms for the two dimensional strip packing problem 2sp with and without 90 rotations we first focus on the perfect packing problem pp which is a special case of 2sp wherein all given rectangles are required to be packed without wasted space and design branch and bound algorithms introducing several branching rules and bounding operations a combination of these rules yields an algorithm that is especially efficient for feasible instances of pp we then propose several methods of applying the pp algorithms to 2sp our algorithms succeed in efficiently solving benchmark instances of pp with up to 500 rectangles and those of 2sp with up to 200 rectangles they are often faster than existing exact algorithms specially tailored for problems without rotations

robust midsagittal plane extraction from normal and pathological 3 d neuroradiology images this paper focuses on extracting the ideal midsagittal plane imsp from three dimensional 3 d normal and pathological neuroimages the main challenges in this work are the structural asymmetry that may exist in pathological brains and the anisotropic unevenly sampled image data that is common in clinical practice we present an edge based cross correlation approach that decomposes the plane fitting problem into discovery of two dimensional symmetry axes on each slice followed by a robust estimation of plane parameters the algorithm s tolerance to brain asymmetries input image offsets and image noise is quantitatively evaluated we find that the algorithm can extract the imsp from input 3 d images with 1 large asymmetrical lesions 2 arbitrary initial rotation offsets 3 low signal to noise ratio or high bias field the imsp algorithm is compared with an approach based on maximization of mutual information registration and is found to exhibit superior performance under adverse conditions finally no statistically significant difference is found between the midsagittal plane computed by the imsp algorithm and that estimated by two trained neuroradiologists

bld mappings in w 2 2 are locally invertible we prove that mappings of bounded length distortion are local homeomorphisms if they have l 2 integrable weak second derivatives

combinatorial benders cuts for mixed integer linear programming mixed integer programs mips involving logical implications modeled through big m coefficients are notoriously among the hardest to solve in this paper we propose and analyze computationally an automatic problem reformulation of quite general applicability aimed at removing the model dependency on the big m coefficients our solution scheme defines a master integer linear problem ilp with no continuous variables which contains combinatorial information on the feasible integer variable combinations that can be distilled from the original mip model the master solutions are sent to a slave linear program lp which validates them and possibly returns combinatorial inequalities to be added to the current master ilp the inequalities are associated to minimal or irreducible infeasible subsystems of a certain linear system and can be separated efficiently in case the master solution is integer the overall solution mechanism closely resembles the benders one but the cuts we produce are purely co

combinatorial benders cuts for the strip packing problem we study the strip packing problem in which a set of two dimensional rectangular items has to be packed in a rectangular strip of fixed width and infinite height with the aim of minimizing the height used the problem is important because it models a large number of real world applications including cutting operations where stocks of materials such as paper or wood come in large rolls and have to be cut with minimum waste scheduling problems in which tasks require a contiguous subset of identical resources and container loading problems arising in the transportation of items that cannot be stacked one over the other the strip packing problem has been attacked in the literature with several heuristic and exact algorithms nevertheless benchmark instances of small size remain unsolved to proven optimality in this paper we propose a new exact method that solves a large number of the open benchmark instances within a limited computational effort our method is based on a benders decomposition in whi

strip packing with precedence constraints and strip packing with release times the strip packing problem seeks to tightly pack a set of n rectangles into a strip of fixed width and arbitrary height the rectangles model tasks and the height models time this paper examines two variants of strip packing when the rectangles to be placed have precedence constraints and when the rectangles have release times strip packing is used to model scheduling problems in which tasks require a contiguous subset of identical resources that are arranged in a linear topology the variants studied here are motivated by scheduling tasks for dynamically reconfigurable field programmable gate arrays fpgas comprised of a linear arrangement of k homogeneous computing resources where k is a fixed positive integer and each task occupies a contiguous subset of these resources for the case in which tasks have precedence constraints we give an o logn approximation algorithm we then consider the special case in which all the rectangles have uniform height and reduce it to the resource constrained scheduling studied by garey graham johnson and yao thereby extending their asymptotic results to our special case problem we also give an absolute 3 approximation for this special case problem for strip packing with release times we provide an asymptotic polynomial time approximation scheme we make the standard assumption that the rectangles have height at most 1 in addition we also require widths to be in 1k 1 for the fpga application this would imply that the rectangles are at least as wide as a column our running time is polynomial in n and 1 e but exponential in k

3 d object recognition with symmetric models symmetry extraction and encoding object recognition systems which employ solid models and range data have been a topic of interest for several years model databases have the potential to become large in some environments this paper proposes a pair of techniques for incorporating knowledge of the symmetries of object models into the recognition process the effects of symmetric models on the speed of an object recognition system is examined in the context of an implemented system employing invariant feature indexing as a correspondence building mechanism groups of model surfaces are enumerated and examined to yield a list of segment label permutations which summarize the model s symmetry this symmetry extraction process is followed by a symmetry encoding procedure which replaces groups of features which are indistinguishable because of symmetry with a single prototype feature group experiments with a large model database demonstrate the utility of these symmetry extraction and encoding techniques

a new exact method for the two dimensional orthogonal packing problem abstract the two dimensional orthogonal packing problem 2 opp consists in determining if a set of rectangles items can be packed into one rectangle of fixed size bin in this paper we propose two exact algorithms for solving this problem the first algorithm is an improvement on a classical branch bound method whereas the second algorithm is based on a new relaxation of the problem we also describe reduction procedures and lower bounds which can be used within enumerative methods we report computational experiments for randomly generated benchmarks which demonstrate the efficiency of both methods the second method is competitive compared to the best previous methods it can be seen that our new relaxation allows an efficient detection of non feasible instances

dynamic programming and strong bounds for the 0 1 knapsack problem two new algorithms recently proved to outperform all previous methods for the exact solution of the 0 1 knapsack problem this paper presents a combination of such approaches where in additi on valid inequalities are generated and surrogate relaxed and a new initial core problem is adopted the algorithm is able to solve all classical test instances with up to 10 000 variables in less than 0 2 seconds on a hp9000 735 99 computer the c language implementation of the algorithm is available on the internet

quasiconformal maps in metric spaces with controlled geometry this paper develops the foundations of the theory of quasiconformal maps in metric spaces that satisfy certain bounds on their mass and geometry the principal message is that such a theory is both relevant and viable the first main issue is the problem of definition which we next describe quasiconformal maps are commonly understood as homeomorphisms that distort the shape of infinitesimal balls by a uniformly bounded amount this requirement makes sense in every metric space given a homeomorphism f from a metric space x to a metric space y then for x x and r 0 set

integer programming formulations of vehicle routing problems abstract various mathematical formulations are available for situations represented by vehicle routing problems the assignment based integer programming formulations of these problems are more common and easy to understand such formulations are discussed in this paper and a much simpler formulation for the vechicle routing problem is presented for the case when all the vehicles have the same load capacity and maximum allowable cost per route

accelerating benders decomposition algorithmic enhancement and model selection criteria this paper proposes methodology for improving the performance of benders decomposition when applied to mixed integer programs it introduces a new technique for accelerating the convergence of the algorithm and theory for distinguishing good model formulations of a problem that has distinct but equivalent mixed integer programming representations the acceleration technique is based upon selecting judiciously from the alternate optima of the benders subproblem to generate strong or pareto optimal cuts this methodology also applies to a much broader class of optimization algorithms that includes dantzig wolfe decomposition for linear and nonlinear programs and related cutting plane type algorithms that arise in resource directive and price decomposition when specialized to network location problems this cut generation technique leads to very efficient algorithms that exploit the underlying structure of these models in discussing the proper formulation of mixed integer programs we suggest criteri

combinatorial benders cuts for assembly line balancing problems with setups abstract the classical assembly line balancing problem consists of assigning assembly work to workstations in the presence of setup times that depend on the sequence of tasks assigned to each workstation the problem becomes more complicated given that two interdependent problems namely assignment and sequencing must be solved simultaneously the hierarchical nature of these two problems also suggest a natural decomposition of the problem this paper adopts such an approach and describes an exact algorithm based on benders decomposition to solve both simple and mixed model assembly line balancing problems with setups the algorithm is tested on a set of benchmark instances and numerically compared against a mixed integer linear programming formulation of the problem solved using a commercial optimizer

3 d object recognition using bipartite matching embedded in discrete relaxation the authors show how large efficiencies can be achieved in model based 3 d vision by combining the notions of discrete relaxation and bipartite matching the computational approach presented is capable of pruning large segments of search space an indispensable step when the number of objects in the model library is large and when recognition of complex objects with a large number of surfaces is called for bipartite matching is used for quick wholesale rejection of inapplicable models and for the determination of compatibility of a scene surface with a potential model surface taking into account relational considerations the time complexity function associated with those aspects of the procedure that are implemented via bipartite matching is provided the algorithms do not take more than a couple of iterations even for objects with more than 30 surfaces

multi objective vehicle routing problems routing problems such as the traveling salesman problem and the vehicle routing problem are widely studied both because of their classic academic appeal and their numerous real life applications similarly the field of multi objective optimization is attracting more and more attention notably because it offers new opportunities for defining problems this article surveys the existing research related to multi objective optimization in routing problems it examines routing problems in terms of their definitions their objectives and the multi objective algorithms proposed for solving them