lagrangian hamiltonian unified formalism for field theory the rusk skinner formalism was developed in order to give a geometrical unified formalism for describing mechanical systems it incorporates all the characteristics of lagrangian and hamiltonian descriptions of these systems including dynamical equations and solutions constraints legendre map evolution operators equivalence etc in this work we extend this unified framework to first order classical field theories and show how this description comprises the main features of the lagrangian and hamiltonian formalisms both for the regular and singular cases this formulation is a first step toward further applications in optimal control theory for partial differential equations

multivector fields and connections setting lagrangian equations in field theories the integrability of multivector fields in a differentiable manifold is studied then given a jet bundle j1e e m it is shown that integrable multivector fields in e are equivalent to integrable connections in the bundle e m that is integrable jet fields in j1e this result is applied to the particular case of multivector fields in the manifold j1e and connections in the bundle j1e m that is jet fields in the repeated jet bundle j1j1e in order to characterize integrable multivector fields and connections whose integral manifolds are canonical lifting of sections these results allow us to set the lagrangian evolution equations for first order classical field theories in three equivalent geometrical ways in a form similar to that in which the lagrangian dynamical equations of nonautonomous mechanical systems are usually given then using multivector fields we discuss several aspects of these evolution equations both for the regular and singular cases namely the existence and nonuniqueness o

the product matrix traveling salesman problem an application and solution heuristic in this paper we analyze an optimization problem that arises in the overhaul of a gas turbine engine this problem involves the placement of nozzle guide vanes in the nozzle of the engine the objective of the vane placement is to attain uniform flow equalized distribution of hot fuel gases about the circumference of the engine nozzle we show that this placement problem is equivalent to a traveling salesman problem tsp whose cost matrix is a product matrix exploiting properties of the special form of the cost matrix we give a heuristic algorithm for solving the problem and derive a posteriori lower bounds for the heuristic we show that the heuristic performs extremely well on both real and simulated data finally we present and develop the theoretical results that are the foundation of the proposed heuristic

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

halin graphs and the travelling salesman problem a halin graphh t c is obtained by embedding a treet having no nodes of degree 2 in the plane and then adding a cyclec to join the leaves oft in such a way that the resulting graph is planar these graphs are edge minimal 3 connected hamiltonian and in general have large numbers of hamilton cycles we show that for arbitrary real edge costs the travelling salesman problem can be polynomially solved for such a graph and we give an explicit linear description of the travelling salesman polytope the convex hull of the incidence vectors of the hamilton cycles for such a graph

computer construction of matrix representations of the covering group of the higman sims group in this paper we will describe how to use the meat axe of r parker 9 to construct two 28 dimensional matrix representations of the covering group of the higman sims group denoted by 2 hs over the finite field gf of order 5 these techniques have been developed in cambridge over a number of years the first use of them was the construction by norton and parker see s of the 112 dimensional representation of the sporadic simple group j4 over the field of order 2 these matrices have been used for example in determining the maximal subgroups of j4 see s there are also other examples involving sporadic groups such as the 11 l dimensional representation of the lyons group ly over the field of order 5 see 11 i the 45 dimensional representation of the triple cover 3 o n of the o nan group over the field of order 7 see 12 and the 248 dimensional representation of the thompson group th over the field of order 3 see is at the end of this paper we will write down two 28 x 28 matrices over gf which generate 2 hs

a new class of cutting planes for the symmetric travelling salesman problem a comprehensive class of cutting planes for the symmetric travelling salesman problem tsp is proposed which contains the known comb inequalities the path inequalities and the 3 star constraints as special cases its relation to the clique tree inequalities is discussed the cutting planes are shown to be valid for a relaxed version of the tsp the travelling salesman problem on a road network and under certain conditions to define facets of the polyhedron associated with this problem

broue s abelian defect group conjecture holds for the harada norton sporadic simple group hn abstract in representation theory of finite groups there is a well known and important conjecture due to m broue he conjectures that for any prime p if a p block a of a finite group g has an abelian defect group p then a and its brauer corresponding block b of the normaliser n g p of p in g are derived equivalent rickard equivalent this conjecture is called broue s abelian defect group conjecture we prove in this paper that broue s abelian defect group conjecture is true for a non principal 3 block a with an elementary abelian defect group p of order 9 of the harada norton simple group hn it then turns out that broue s abelian defect group conjecture holds for all primes p and for all p blocks of the harada norton simple group hn

the symmetric traveling salesman polytope and its graphical relaxation composition of valid inequalities the graphical relaxation of the traveling salesman problem is the relaxation obtained by requiring that the salesman visit each city at least once instead of exactly once this relaxation has already led to a better understanding of the traveling salesman polytope in cornuejols fonlupt and naddef 1985 we show here how one can compose facet inducing inequalities for the graphical traveling salesman polyhedron and obtain other facet inducing inequalities this leads to new valid inequalities for the symmetric traveling salesman polytope this paper is the first of a series of three papers on the symmetric traveling salesman polytope the next one studies the strong relationship between that polytope and its graphical relaxation and the last one applies all the theoretical developments of the two first papers to prove some new facet inducing results

multivector field formulation of hamiltonian field theories equations and symmetries we state the intrinsic form of the hamiltonian equations of first order classical field theories in three equivalent geometrical ways using multivector fields jet fields and connections thus these equations are given in a form similar to that in which the hamiltonian equations of mechanics are usually given then using multivector fields we study several aspects of these equations such as the existence and non uniqueness of solutions and the integrability problem in particular these problems are analysed for the case of hamiltonian systems defined in a submanifold of the multimomentum bundle furthermore the existence of first integrals of these hamiltonian equations is considered and the relation between cartan noether symmetries and general symmetries of the system is discussed noether s theorem is also stated in this context both the classical version and its generalization to include higher order cartan noether symmetries finally the equivalence between the lagrangian and hamiltonian formalisms is also discussed

the traveling salesman problem in graphs with some excluded minors given a graph and a length function defined on its edge set the traveling salesman problem can be described as the problem of finding a family of edges an edge may be chosen several times which forms a spanning eulerian subgraph of minimum length in this paper we characterize those graphs for which the convex hull of all solutions is given by the nonnegativity constraints and the classical cut constraints this characterization is given in terms of excluded minors a constructive characterization is also given which uses a small number of basic graphs

a multi start dynasearch algorithm for the time dependent single machine total weighted tardiness scheduling problem we extend the dynasearch technique recently proposed by congram et al in the context of time dependent combinatorial optimization problems as an application we consider a general time dependent idleness version of the well known single machine total weighted tardiness scheduling problem in which the processing time of a job depends on its starting time of execution we develop a multi start local search algorithm and present experimental results on several types of instances showing the superiority of the dynasearch neighborhood over the traditional one

gradient vector fields on cosymplectic manifolds the authors study some geometrical properties of gradient vector fields on cosymplectic manifolds thereby emphasizing the close analogy with hamiltonian systems on symplectic manifolds it is shown that gradient vector fields and more generally local gradient vector fields can be characterized in terms of lagrangian submanifolds of the tangent bundle with respect to an induced symplectic structure in addition the symmetry and reduction properties of gradient vector fields are investigated

testing the necklace condition for shortest tours and optimal factors in the plane a tour of a finite set p of points is a necklace tour if there are disks with the points in p as centers such that two disks intersect if and only if their centers are adjacent in it has been observed by sanders that a necklace tour is an optimal traveling salesman tour

the binested inequalities for the symmetric traveling salesman polytope in this paper we define a family of valid inequalities for the symmetric travelling salesman polytope which are defined on two nested sets of vertices of the graph these inequalities generalize the comb inequalities of chvital grotschel and padberg the clique tree inequalities of grotschel and pulleyblank the path inequalities of cornuejols fonlupt and naddef and the hyperstar inequalities of fleischman this is the largest known family of valid inequalities known so far facet inducing inequalities for the symmetric travelling salesman polytope contained in this class and in no other one are given proving that this is a proper generalization of the previously mentioned families

well solvable special cases of the traveling salesman problem a survey the traveling salesman problem tsp belongs to the most basic most important and most investigated problems in combinatorial optimization although it is an cal np hard problem many of its special cases can be solved efficiently in polynomial time we survey these special cases with emphasis on the results that have been obtained during the decade 1985 1995 this survey complements an earlier survey from 1985 compiled by gilmore lawler and shmoys the traveling salesman problem a guided tour of combinatorial optimization wiley chichester pp 87 143

efficiently solvable special cases of hard combinatorial optimization problems we survey some recent advances in the field of polynomially solvable special cases of hard combinatorial optimization problems like the travelling salesman problem quadratic assignment problems and steiner tree problems such special cases can be found by considering special cost structures the geometry of the problem the special topology of the underlying graph structure or by analyzing special algorithms in particular we stress the importance of recognition algorithms we comment on open problems in this area and outline some lines for future research in this field

generation of the exact pareto set in multi objective traveling salesman and set covering problems abstract the calculation of the exact set in multi objective combinatorial optimization moco problems is one of the most computationally demanding tasks as most of the problems are np hard in the present work we use augmecon2 a multi objective mathematical programming momp method which is capable of generating the exact pareto set in multi objective integer programming moip problems for producing all the pareto optimal solutions in two popular moco problems the multi objective traveling salesman problem motsp and the multi objective set covering problem moscp the computational experiment is confined to two objective problems that are found in the literature the performance of the algorithm is slightly better to what is already found from previous works and it goes one step further generating the exact pareto set to till now unsolved problems the results are provided in a dedicated site and can be useful for benchmarking with other momp methods or even multi objective meta heuristics momh that can check the performance of their approximate solution against the exact solution in motsp and moscp problems

a lower bound on the hamiltonian path completion number of a line graph given a line graph l g of a graph g v e the problem of finding the minimum number of edges to add to l g to have a hamiltonian path on l g is considered this problem related to different applications is known to be np hard this paper presents an o v e algorithm to determine a lower bound for the hamiltonian path completion number of a line graph the algorithm is based on finding a collection of edge disjoint trails dominating all the edges of the root graph g the algorithm is tested by an extensive experimental study showing good performance suggesting its use as a building block of exact as well as heuristic solution approaches for the problem

the crown inequalities for the symmetric traveling salesman polytope we define a new family of valid inequalities for the symmetric traveling salesman polytope stspn that is the convex hull of the incidence vectors of all hamiltonian cycles of a complete graph with n nodes the smallest value of n for which the inequalities are defined is 8 we call these inequalities crown inequalities and we prove that they are facet inducing for stspn for n 8 we describe some extensions of the crown inequalities and we prove that they induce facets of stspn as well finally we discuss some issues on the possible applications of these inequalities in a polyhedral cutting plane algorithm

special cases of the traveling salesman problem the paper surveys special cases of the traveling salesman problem that are in the literature the tables include polynomial time recognition algorithms and relationships among special cases as a result of research the most general special cases are identified the paper also contains a table of subclasses of the most general special cases applications and open questions are indicated

the three machine no wait flow shop is np complete l atelier d ecoulement de permutation de la machine 3 n ayant pas d attente dans le processus est montre etre np dur au sens fort ceci etablit une question ouverte bien connue en theorie de l ordonnancement

the traveling salesman problem on a graph and some related integer polyhedra given a graphg n e and a length functionl e the graphical traveling salesman problem is that of finding a minimum length cycle goingat least once through each node ofg this formulation has advantages over the traditional formulation where each node must be visited exactly once we give some facet inducing inequalities of the convex hull of the solutions to that problem in particular the so called comb inequalities of grotschel and padberg are generalized some related integer polyhedra are also investigated finally an efficient algorithm is given wheng is a series parallel graph

geometry of multisymplectic hamiltonian first order field theories in the jet bundle description of field theories multisymplectic models in particular there are several choices for the multimomentum bundle where the covariant hamiltonian formalism takes place as a consequence several proposals for this formalism can be stated and on each one of them the differentiable structures needed for setting the formalism are obtained in different ways in this work we make an accurate study of some of these hamiltonian formalisms showing their equivalence in particular the geometrical structures canonical or not needed for the hamiltonian formalism are introduced and compared and the derivation of hamiltonian field equations from the corresponding variational principle is shown in detail furthermore the hamiltonian formalism of systems described by lagrangians is performed both for the hyper regular and almost regular cases finally the role of connections in the construction of hamiltonian field theories is clarified

pyramidal tours and the traveling salesman problem a traveling salesman problem is studied containing a shortest hamiltonian tour that is as long as a shortest pyramidal tour a tour is pyramidal if it consists of a path from city 1 to n with cities in between visited in ascending order and a path from n to 1 with cities in between visited in descending order if the distance matrix satisfies the so called demidenko conditions in which case it is called a demidenko matrix then there exists an optimal tour that is pyramidal a new proof of this theorem is given for symmetric demidenko matrices the proof cannot be used for the nonsymmetric case if however the demidenko conditions are replaced by somewhat stronger conditions it is possible to give a similar proof for the nonsymmetric case a method to construct demidenko matrices is presented and finally several tsp heuristics are tested on the class of demidenko matrices

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

k cosymplectic manifolds and lagrangian field theories a geometrical description of classical field theories of first order is given the underlying k cosymplectic structure permits to derive the corresponding field equations

genetic local search for multi objective combinatorial optimization abstract the paper presents a new genetic local search gls algorithm for multi objective combinatorial optimization moco the goal of the algorithm is to generate in a short time a set of approximately efficient solutions that will allow the decision maker to choose a good compromise solution in each iteration the algorithm draws at random a utility function and constructs a temporary population composed of a number of best solutions among the prior generated solutions then a pair of solutions selected at random from the temporary population is recombined local search procedure is applied to each offspring results of the presented experiment indicate that the algorithm outperforms other multi objective methods based on gls and a pareto ranking based multi objective genetic algorithm ga on travelling salesperson problem tsp

a method for generating all efficient solutions of 0 1 multi objective linear programming problem in this paper a method is proposed to find all efficient solutions of 0 1 multi objective linear programming molp problem these solutions are specified without generating all feasible solutions corresponding to a feasible solution of 0 1 molp problem a vector is constructed which its components are the values of objective functions the method consists of a one stage algorithm in each iteration of this algorithm for obtaining some efficient solutions of 0 1 molp problem a 0 1 single objective linear programming problem is solved

a probabilistic analysis of the switching algorithm for the euclidean tsp the well known switching algorithm proposed by lin and kernighan for the euclidean travelling salesman problem has proved to be a simple efficient algorithm for medium size problems though it often gets trapped in local optima although its complexity status is still open it has been observed to be polynomially bounded in practice when applied to uniformly distributed points in the unit square in this paper this polynomial behaviour is derived theoretically however we will come up with a bound of o n18 with probability 1 c n whereas in practice the algorithm works slightly better

covariant field theory on frame bundles of fibered manifolds we show that covariant field theory for sections of e m lifts in a natural way to the bundle of vertically adapted linear frames l e our analysis is based on the fact that l e is a principal fiber bundle over the bundle of 1 jets j1 on l e the canonical soldering 1 forms play the role of the contact structure of j1 a lifted lagrangian l l e r is used to construct modified soldering 1 forms which we refer to as the cartan hamilton poincare 1 forms these 1 forms on l e pass to the quotient to define the standard cartan hamilton poincare m form on j1 we derive generalized hamilton jacobi and hamilton equations on l e and show that the hamilton jacobi and canonical equations of caratheodory rund and de donder weyl are obtained as special cases

the graphical relaxation a new framework for the symmetric traveling salesman polytope a present trend in the study of thesymmetric traveling salesman polytope stsp n is to use as a relaxation of the polytope thegraphical relaxation gtsp n rather than the traditionalmonotone relaxation which seems to have attained its limits in this paper we show the very close relationship between stsp n and gtsp n in particular we prove that every non trivial facet of stsp n is the intersection ofn 1 facets of gtsp n n of which are defined by the degree inequalities this fact permits us to define a standard form for the facet defining inequalities for stsp n that we calltight triangular and to devise a proof technique that can be used to show that many known facet defining inequalities for gtsp n define also facets of stsp n in addition we give conditions that permit to obtain facet defining inequalities by composition of facet defining inequalities for stsp n and general lifting theorems to derive facet defining inequalities for stsp n k from inequalities defining facets of stsp n

dynamic programming and the graphical traveling salesman problem in this paper we study cases of polynomial resolution of the traveling salesman problem first we define a generalization of the traveling salesman problem where the cost of an edge depends on the number of times this edge is visited in a tour here an edge may be visited 0 1 or 2 times by this generalization we can find polynomial algorithms to solve the traveling salesman problem on certain classes of graphs that are built from basic graphs by operations called r sum with r fixed the underlying ideas of these algorithms are similar to dynamic programming techniques our classes contain most of the known instances for which the graphical and classical salesman problems are polynomial

pareto memetic algorithm with path relinking for bi objective traveling salesperson problem abstract the paper presents an effective version of the pareto memetic algorithm with path relinking and efficient local search for multiple objective traveling salesperson problem in multiple objective traveling salesperson problem tsp multiple costs are associated with each arc link the multiple costs may for example correspond to the financial cost of travel along a link time of travel or risk in the case of hazardous materials the algorithm searches for new good solutions along paths in the decision space linking two other good solutions selected for recombination instead of a simple local search it uses short runs of tabu search based on the steepest version of the lin kernighan algorithm the efficiency of local search is further improved by the techniques of candidate moves and locked arcs in the final step of the algorithm the neighborhood of each potentially pareto optimal solution is searched for new solutions that could be added to this set the algorithm is compared experimentally to the state of the art algorithms for multiple objective tsp

polynomially solvable cases of the bipartite traveling salesman problem abstract given two sets r and b consisting of n cities each in the bipartite traveling salesman problem one looks for the shortest way of visiting alternately the cities of r and b returning to the city of origin this problem is known to be np hard for arbitrary sets r and b in this paper we provide an o n 6 algorithm to solve the bipartite traveling salesman problem if the quadrangle property holds in particular this algorithm can be applied to solve in o n 6 time the bipartite traveling salesman problem in the following cases s r b is a convex point set in the plane s r b is the set of vertices of a simple polygon and v r b is the set of vertices of a circular graph for this last case we also describe another algorithm which runs in o n 2 time

k symplectic formalism on lie algebroids in this paper we introduce a geometric description of lagrangian and hamiltonian classical field theories on lie algebroids in the framework of k symplectic geometry we discuss the relation between the lagrangian and hamiltonian descriptions through a convenient notion of legendre transformation the theory is a natural generalization of the standard one in addition other interesting examples are studied in particular systems with symmetry and poisson sigma models

reduced classical field theories k cosymplectic formalism on lie algebroids in this paper we introduce a geometric description of lagrangian and hamiltonian classical field theories on lie algebroids in the framework of k cosymplectic geometry we discuss the relation between lagrangian and hamiltonian descriptions through a convenient notion of legendre transformation the theory is a natural generalization of the standard one in addition other interesting examples are studied mainly on reduction of classical field theories

gunther s formalism k symplectic formalism in classical field theory skinner rusk approach and the evolution operator the first aim of this paper is to extend the skinner rusk formalism on classical mechanics for first order field theories the second is to generalize the definition and properties of the evolution k operator on classical mechanics for first order field theories using in both cases gunther s formalism k symplectic formalism

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

covariant hamilton equations for field theory we study the relations between the equations of first order lagrangian field theory on fibre bundles and the covariant hamilton equations on the finite dimensional polysymplectic phase space of covariant hamiltonian field theory if a lagrangian is hyperregular these equations are equivalent a degenerate lagrangian requires a set of associated hamiltonian forms in order to exhaust all solutions of the euler lagrange equations the case of quadratic degenerate lagrangians is studied in detail

on cycle cones and polyhedra given an undirected graph c and a cost associated with each edge the weighted girth problem is to find a simple cycle of g having minimum total cost we consider several variants of the weighted girth problem some of which are np hard and some of which are solvable in polynomial time we also consider the polyhedra associated with each of these problems two of these polyhedra are the cycle cone of 6 which is the cone generated by the incidence vectors of cycles of 6 and the cycle polytope of g which is the convex hull of the incidence vectors of cycles of 6 first we give a short proof of seymour s characterization of the cycle cone of g next we give a polyhedral composition result for the cycle polytope of 6 in particular we prove that if g decomposes via a 3 edge cut into graphs 6 and g say then defining linear systems for the cycle polytopes of g and g can be combined in a certain way to obtain a defining linear system for the cycle polytope of g we also describe a

hamiltonian systems on k cosymplectic manifolds the hamiltonian framework on symplectic and cosymplectic manifolds is extended in order to consider classical field theories to do this the notion of k cosymplectic manifold is introduced and a suitable hamiltonian formalism is developed so that the field equations for scalar and vector hamiltonian functions are derived

the constraint algorithm for time dependent lagrangians the aim of this paper is to develop a constraint algorithm for time dependent lagrangian systems which permit us to solve the motion equations this algorithm extends the gotay and nester algorithm for autonomous lagrangians which is in fact a particular case to do this the almost stable tangent geometry of the evolution space and the notion of cosymplectic structure are used

average value of solutions for the bipartite boolean quadratic programs and rounding algorithms we consider domination analysis of approximation algorithms for the bipartite boolean quadratic programming problem bbqp with m n variables a closed form formula is developed to compute the average objective function value a of all solutions in o m n time however computing the median objective function value of the solutions is shown to be np hard also we show that any solution with objective function value no worse than a dominates at least 2 m n 2 solutions and this bound is the best possible further we show that such a solution can be identified in o m n time and hence the domination ratio of this algorithm is at least 1 4 we then show that for any fixed natural numbers a and b such that a b 1 no polynomial time approximation algorithm exists for bbqp with domination ratio larger than 1 2 1 m n unless p np it is shown that some powerful local search algorithms can get trapped at a local maximum with objective function value less than a one of our approximation algorithms has an interesting rounding property which provides a data dependent lower bound on the optimal objective function value a new integer programming formulation of bbqp is also given and computational results with our rounding algorithms are reported

small and large tsp two polynomially solvable cases of the traveling salesman problem abstract the problem of minimizing and maximizing the objective function of the traveling salesman problem tsp is discussed in particular we consider two special cases of the tsp namely the small tsp and the large tsp for which both a minimum and a maximum of the objective function can be obtained in polynomial time the cost matrix c c ij corresponding to the small large tsp is defined by c ij min a i b j c ij max a i b j for each 1 j 1 n and n dimensional real vector a and ba the main purpose of this paper is to show the relationship between the subtour elimination algorithm of gilmore and gomory for the large tsp and gabovich s algorithm for the small tsp the recognition problem for these two types of matrices is solved by using deineko and filonenko s algorithm for recognizing permuted distribution matrices finally as an application a problem from machine scheduling theory is considered

permutational extreme values of autocorrelation coefficients and a pitman test against serial dependence somewhat more difficult this problem which is shown to reduce to a particular case of the well known travelling salesman problem is explicitly solved here for the autocorrelation coefficient of order one allowing for a simple computation of permutational critical values for pitman tests against serial dependence the case of higher order autocorrelations is however of a different nature and requires another approach

on a kind of noether symmetries and conservation laws in k cosymplectic field theory this paper is devoted to studying symmetries of certain kinds of k cosymplectic hamiltonian systems in first order classical field theories thus we introduce a particular class of symmetries and study the problem of associating conservation laws to them by means of a suitable generalization of noether s theorem

computational geometry an introduction from the reviews this book offers a coherent treatment at the graduate textbook level of the field that has come to be known in the last decade or so as computational geometry the book is well organized and lucidly written a timely contribution by two founders of the field it clearly demonstrates that computational geometry in the plane is now a fairly well understood branch of computer science and mathematics it also points the way to the solution of the more challenging problems in dimensions higher than two

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

hamilton paths in grid graphs a grid graph is a node induced finite subgraph of the infinite grid it is rectangular if its set of nodes is the product of two intervals given a rectangular grid graph and two of its nodes we give necessary and sufficient conditions for the graph to have a hamilton path between these two nodes in contrast the hamilton path and circuit problem for general grid graphs is shown to be np complete this provides a new relatively simple proof of the result that the euclidean traveling salesman problem is np complete

an exact e constraint method for bi objective combinatorial optimization problems application to the traveling salesman problem with profits abstract this paper describes an exact constraint method for bi objective combinatorial optimization problems with integer objective values this method tackles multi objective optimization problems by solving a series of single objective subproblems where all but one objectives are transformed into constraints we show in this paper that the pareto front of bi objective problems can be efficiently generated with the constraint method furthermore we describe heuristics based on information gathered from previous subproblems that significantly speed up the method this approach is used to find the exact pareto front of the traveling salesman problem with profits a variant of the traveling salesman problem in which a profit or prize value is associated with each vertex the goal here is to visit a subset of vertices while addressing two conflicting objectives maximize the collected prize and minimize the travel costs we report the first exact results for this problem on instances derived from classical vehicle routing and traveling salesman problem instances with up to 150 vertices results on approximations of the pareto front obtained from a variant of our exact algorithm are also reported

the polysymplectic hamiltonian formalism in field theory and calculus of variations i the local case on presente un formalisme hamiltonien geometrique invariant pour des problemes variationnels a integrale multiple et des theories de champ le formalisme est base sur la notion de forme polysymplectique qui est une generalisation a valeur vectorielle des formes symplectiques on etudie des equations hamiltoniennes des transformations canoniques des systemes de lagrange des symetries des applications en pulsion de theorie du champ et une classification des systemes de theorie du champ g homogenes sur une generalisation des orbites coadjointes

the bilinear assignment problem complexity and polynomially solvable special cases in this paper we study the bilinear assignment problem bap with size parameters m and n m le n bap is a generalization of the well known quadratic assignment problem and the three dimensional assignment problem and hence np hard we show that bap cannot be approximated within a constant factor unless p np even if the associated quadratic cost matrix q is diagonal further we show that bap remains np hard if m o root r of n for some fixed r but is solvable in polynomial time if m o left frac log n log log n right when the rank of q is fixed bap is observed to admit fptas and when this rank is one it is solvable in polynomial time under some additional restrictions we then provide a necessary and sufficient condition for bap to be equivalent to two linear assignment problems a closed form expression to compute the average of the objective function values of all solutions is presented whereas the median of the solution values cannot be identified in polynomial time unless p np we then provide polynomial time heuristic algorithms that find a solution with objective function value no worse than that of m 1 n 1 solutions however computing a solution whose objective function value is no worse than that of m n lceil frac m beta rceil lceil frac n beta rceil solutions is np hard for any fixed rational number beta 1

the traveling salesman problem with many visits to few cities we study the version of the traveling salesman problem in which a relatively small number of cities say six must be visited a huge number of times e g several hundred times each it costs to go

the x and y axes travelling salesman problem the x and y axes travelling salesman problem forms a special case of the euclidean tsp where all cities are situated on the x axis and on the y axis of an orthogonal coordinate system of the euclidean plane by carefully analyzing the underlying combinatorial and geometric structures we show that this problem can be solved in polynomial time the running time of the resulting algorithm is quadratic in the number of cities

the gunther s formalism in classical field theory momentum map and reduction the polysymplectic formalism in local field theory developed by gunther j diff geom 25 23 1987 is revised a new approach and new results on momentum maps and reduction are given

construction heuristics for the asymmetric tsp abstract non euclidean traveling salesman problem tsp construction heuristics and especially asymmetric tsp construction heuristics have been neglected in the literature by comparison with the extensive efforts devoted to studying euclidean tsp construction heuristics this state of affairs is at odds with the fact that asymmetric models are relevant to a wider range of applications and indeed are uniformly more general that symmetric models moreover common construction approaches for the euclidean tsp have been shown to produce poor quality solutions for non euclidean instances motivation for remedying this gap in the study of construction approaches is increased by the fact that such methods are a great deal faster than other tsp heuristics which can be important for real time problems requiring continuously updated response the purpose of this paper is to describe two new construction heuristics for the asymmetric tsp and a third heuristic based on combining the other two extensive computational experiments are performed for several different families of tsp instances disclosing that our combined heuristic clearly outperforms well known tsp construction methods and proves significantly more robust in obtaining relatively high quality solutions over a wide range of problems

the traveling salesman problem in graphs with 3 edge cutsets this paper analyzes decomposition properties of a graph that when they occur permit a polynomial solution of the traveling salesman problem and a description of the traveling salesman polytope by a system of linear equalities and inequalities the central notion is that of a 3 edge cutset namely a set of 3 edges that when removed disconnects the graph conversely our approach can be used to construct classes of graphs for which there exists a polynomial algorithm for the traveling salesman problem the approach is illustrated on two examples halin graphs and prismatic graphs

a new ilp based refinement heuristic for vehicle routing problems in this paper we address the distance constrained capacitated vehicle routing problem dcvrp where k minimum cost routes through a central depot have to be constructed so as to cover all customers while satisfying for each route both a capacity and a total distance travelled limit

two phase algorithms for the bi objective assignment problem in this paper we present several algorithms for the bi objective assignment problem the algorithms are based on the two phase method which is a general technique to solve multi objective combinatorial optimisation moco problems we give a description of the original two phase method for the bi objective assignment problem including an implementation of the variable fixing strategy of the original method we propose several enhancements for the second phase i e improved upper bounds and a combination of the two phase method with a population based heuristic using path relinking to improve computational performance finally we describe a new technique for the second phase with a ranking approach which outperforms all other tested algorithms all of the algorithms have been tested on instances of varying size and range of objective function coefficients we discuss the results obtained and explain our observations based on the distribution of objective function values

extended neighborhood definition and characterization we consider neighborhood search defined on combinatorial optimization problems suppose that n is a neighborhood for combinatorial optimization problem x we say that n is lo equivalent locally optimal to n if for any instance of x the set of locally optimal solutions with respect to n and n are the same the union of two lo equivalent neighborhoods is itself lo equivalent to the neighborhoods the largest neighborhood that is lo equivalent to n is called the extended neighborhood of n and denoted as n we analyze some basic properties of the extended neighborhood we provide a geometric characterization of the extended neighborhood n when the instances have linear costs defined over a cone for the tsp we consider 2 opt the extended neighborhood for the 2 opt i e 2 exchange neighborhood structure we show that number of neighbors of each tour t in 2 opt is at least n 2 2 we show that finding the best tour in the 2 opt neighborhood is np hard we also show that the extended neighborhood for the graph partition problem is the same as the original neighborhood regardless of the neighborhood defined this result extends to the quadratic assignment problem as well this result on extended neighborhoods relies on a proof that the convex hull of solutions for the graph partition problem has a diameter of 1 that is every two corner points of this polytope are adjacent

sorting jordan sequences in linear time using level linked search trees for a jordan curve c in the plane nowhere tangent to the x axis let x1 x2 xn be the abscissas of the intersection points of c with the x axis listed in the order the points occur on c we call x1 x2 xn a jordan sequence in this paper we describe an o n time algorithm for recognizing and sorting jordan sequences the problem of sorting such sequences arises in computational geometry and computational geography our algorithm is based on a reduction of the recognition and sorting problem to a list splitting problem to solve the list splitting problem we use level linked search trees

an effective implementation of the lin kernighan traveling salesman heuristic abstract this paper describes an implementation of the lin kernighan heuristic one of the most successful methods for generating optimal or near optimal solutions for the symmetric traveling salesman problem tsp computational tests show that the implementation is highly effective it has found optimal solutions for all solved problem instances we have been able to obtain including a 13 509 city problem the largest non trivial problem instance solved to optimality today

an asymmetric analog of van der veen conditions and the traveling salesman problem ii abstract j a a van der veen a new class of pyramidally solvable symmetric traveling salesman problems siam j discrete math 7 1994 585 592 proved that for the traveling salesman problem tsp which satisfies some symmetric conditions called van der veen conditions a shortest pyramidal tour is optimal that is an optimal tour can be computed in polynomial time in this paper we prove that a class satisfying an asymmetric analog of van der veen conditions is polynomially solvable an optimal tour of the instance in this class forms a tour which is an extension of pyramidal ones moreover this class properly includes some known polynomially solvable classes

extended hamiltonian systems in multisymplectic field theories we consider hamiltonian systems in first order multisymplectic field theories we review the properties of hamiltonian systems in the so called restricted multimomentum bundle including the variational principle which leads to the hamiltonian field equations in an analogous way to how these systems are defined in the so called extended symplectic formulation of nonautonomous mechanics we introduce hamiltonian systems in the extended multimomentum bundle the geometric properties of these systems are studied the hamiltonian equations are analyzed using integrable multivector fields the corresponding variational principle is also stated and the relation between the extended and the restricted hamiltonian systems is established all these properties are also adapted to certain kinds of submanifolds of the multimomentum bundles in order to cover the case of almost regular field theories

multiobjective traveling salesperson problem on halin graphs in this paper we study traveling salesperson tsp and bottleneck traveling salesperson btsp problems on special graphs called halin graphs although both problems are np hard on general graphs they are polynomially solvable on halin graphs we address the multiobjective versions of these problems we show computational complexities of finding a single nondominated point as well as finding all nondominated points for different objective function combinations we develop algorithms for the polynomially solvable combinations

traveling salesman based curve reconstruction in polynomial time an instance of the curve reconstruction problem is a finite sample set v of an unknown collection of curves gamma the task is to connect the points in v in the order in which they lie on gamma giesen proceedings of the 15th annual acm symposium on computational geometry scg 99 1999 pp 207 216 showed recently that the traveling salesman tour of v solves the reconstruction problem for single closed curves under otherwise weak assumptions on gamma and v gamma must be a single closed curve we extend his result along several directions we weaken the assumptions on the sample we show that traveling salesman based reconstruction also works for single open curves with and without specified endpoints and for collections of closed curves we give alternative proofs and we show that in the context of curve reconstruction the traveling salesman tour can be constructed in polynomial time

order picking in a rectangular warehouse a solvable case of the traveling salesman problem this paper addresses the problem of order picking in a rectangular warehouse that contains crossovers only at the ends of aisles an algorithm is presented for picking an order in minimum time the computational effort required is linear in the number of aisles the procedure has been implemented on a microcomputer a 50 aisle problem requires only about 1 minute to solve

multi criteria branch and bound a vector maximization algorithm for mixed 0 1 multiple objective linear programming the paper describes the multi criteria branch and bound mcbb algorithm a vector maximization algorithm capable of deriving all efficient extreme points for small and medium sized mixed 0 1 multiple objective linear programming mixed 0 1 molp particular emphasis is given to computational aspects aiming principally at accelerating the solution procedure for facilitating the decision maker s search toward the most preferred efficient solution the notion of efficient combinations of the binary variables is further exploited it is also shown that the mcbb algorithm can be used in single objective problems mixed integer lp problems in order to determine all alternative optima as well as in mixed integer molp problems and pure 0 1 molp problems that frequently arise in practice a computational experiment is included in the paper in order to illustrate the performance of the algorithm

sequencing jobs that require common resources on a single machine a solvable case of the tsp in this paper a one machine scheduling model is analyzed wheren different jobs are classified intok groups depending on which additional resource they require the change over time from one job to another consists of the removal time or of the set up time of the two jobs it is sequence dependent in the sense that the change over time is determined by whether or not the two jobs belong to the same group the objective is to minimize the makespan this problem can be modeled as an asymmetric traveling salesman problem tsp with a specially structured distance matrix for this problem we give a polynomial time solution algorithm that runs in o n logn time 1998 the mathematical programming society inc published by elsevier science b v

optimal wire ordering and spacing in low power semiconductor design a key issue for high integration circuit design in the semiconductor industry are power constraints that stem from the need for heat removal and reliability or battery lifetime limitations as the power consumption depends heavily on the capacitances between adjacent wires determining the optimal ordering and spacing of parallel wires is an important issue in the design of low power chips as it turns out optimal wire spacing is a convex optimization problem whereas the optimal wire ordering is combinatorial in nature containing a special class of the minimum hamilton path problem while the latter is mathcal np hard in general the present paper provides an mathcal o n log n algorithm that solves the coupled ordering and spacing problem for n parallel wires to optimality