a branch and bound algorithm for symmetric 2 peripatetic salesman problems abstract the symmetric 2 peripatetic salesman problem 2 psp is an extension of the well known symmetric traveling salesman problem tsp in the sense that 2 edge disjoint hamiltonian cycles of minimum total length are required in a graph g with n vertices lower bound solutions are provided by 2 minimum length edge disjoint 1 trees which can be constructed in at most o n 2 log n operations if g is complete generating upper bound solutions and executing edge elimination tests takes o n 2 time a branch and bound algorithm has been developed based on these procedures to solve symmetric problems computational results are available for euclidean problems up to 60 vertices and problems with randomly generated distance matrices up to 130 vertices

the efficiency of ballstep subgradient level methods for convex optimization we study subgradient methods for convex optimization that use projections onto successive approximations of level sets of the objective corresponding to estimates of the optimal value we establish convergence and efficiency estimates for simple ballstep level controls without requiring that the feasible set be compact our framework may handle accelerations based on cheap projections surrogate constraints and conjugate subgradient techniques

tree based picture generation the concept of tree based picture generation is introduced it is shown that there are equivalent tree based definitions of four picture generating devices known from the literature namely collage

tree based generation of languages of fractals the notion of p interpreted top down tree generators is introduced combining the nondeterministic nature of grammars as known from formal language theory with the infinite refinement of pictures studied in fractal geometry

on dual solutions of the linear assignment problem abstract for the linear assignment problem we describe how to obtain different dual solutions it turns out that a shortest path algorithm can be used to compute such solutions with several interesting properties that enable to do better post optimality analysis two examples illustrate how different dual solutions can be used in the context of the traveling salesman problem

cancellation in context free languages enrichment by reduction the following problem is shown to be decidable given a context free grammar g and a string w x does there exist a string u l g such that w is obtained from u by deleting all substrings ui that are elements of the symmetric dyck set d 1 the intersection of any two context free languages can be obtained from only one context free language by cancellation either with the smaller semi dyck set d 1 d 1 or with d 1 itself also the following is shown here for the first time if the set eq xnx n n n d 1 is used for this cancellation then each recursively enumerable set can be obtained from linear context free languages previous work has shown that cancellation of substrings from the semi dyck language d 1 or from any of the former languages d 1 or eq allows one to obtain the following from the context free languages any terminal petri net language 14 16 the intersections of any context free with a terminal petri net language and the nested iterated substitution thereof 15 we generalize this characterization by showing that linear context free languages suffice for generating terminal petri net languages in proving this we obtain a new closure property of the family of petri net languages which is not shared by the context free sets

linearized oscillation theory for a nonlinear nonautonomous delay differential equation oscillation properties of two following equations are compared a scalar nonlinear delay differential equation y t k 1m rk t f k y hk t 0 with rk t o hk t t and a linear delay differential equation x t k 1m rk t x hk t 0 coefficients rk t and delays are not assumed to be continuous as an application explicit oscillation and nonoscillation conditions are established for nonlinear equations arising in population dynamics

existence of global solutions to nonlinear mixed type functional differential equations abstract the paper considers systems of nonlinear mixed type functional differential equations it aims to find conditions for the existence of global solutions with graphs staying in a prescribed domain a relevant result is proved by schauder tychonoff fixed point technique with illustrative examples shown a linear variant of the derived result is formulated comparisons with known results are discussed and some open problems are formulated

a branch and bound algorithm for the symmetric traveling salesman problem based on the 1 tree relaxation abstract in 1970 held and karp introduced the lagrangean approach to the symmetric traveling salesman problem we use this 1 tree relaxation in a new branch and bound algorithm it differs from other algorithms not only in the branching scheme but also in the ascent method to calculate the 1 tree bounds urthermore we determine heuristic solutions throughout the computations to provide upperbounds we present computational results for both a depth first and a breadth first version of our algorithm on the average our results on a number of euclidean problems from the literature are obtained in about 60 less 1 trees than the best known algorithm based on the 1 tree relaxation for random table problems up to 100 cities the average results are also satisfactory

a lagrangean approach to the degree constrained minimum spanning tree problem abstract a known branch and bound algorithm for the degree constrained minimum spanning tree problem is adapted for the use of lagrangean multipliers they improve the bounds from which the edge elimination analysis in the algorithm benefits in particular computational results are given for problems up to 150 nodes both random drawn coordinate problems as well drawn table problems these results are much better than other results reported in literature and they show the lagrangean approach to be more effective when the problems are more difficult to solve

generating self affine fractals by collage grammars abstract self affinity and self similarity are fundamental concepts in fractal geometry in this paper they are related to collage grammars syntactic devices based on hyperedge replacement that generate sets of collages essentially a collage is a picture consisting of geometric parts like line segments circles polygons polyhedra etc the overlay of all collages in a collage language yields a fractal pattern we show that collage grammars of a special type so called increasing generalized sierpinski grammars yield self affine fractals if one replaces the overlay by an intersection of all generated collages the same result holds for decreasing generalized sierpinski grammars here the converse also holds every self affine fractal can be generated by a decreasing generalized sierpinski grammar which provides a characterization of this class of fractals

two undecidability results for chain code picture languages it is undecidable whether or not two 1 retreat bounded regular languages describe exactly the same set of pictures or they describe a picture in common

embedding learning capability in lagrangean relaxation an application to the travelling salesman problem abstract this paper presents an effective procedure that finds lower bounds for the travelling salesman problem based on the 1 tree using a learning based lagrangian relaxation technique the procedure can dynamically alter its step size depending upon its previous iterations along with having the capability of expansion contraction the procedure performs a learning process in which lagrange multipliers are influenced by a weighted cost function of their neighbouring nodes in analogy with simulated annealing paradigm here a learning process is equivalent to escaping local optimality via exploiting the structure of the problem computational results conducted on euclidean benchmarks from the tsplib library show that the procedure is very effective

leftmove bound picture languages let fu d r lg be the chain code picture alphabet such that u d r l denotes the graphics command to move the drawing pen up down right left in the 2d cartesian plane it is known that the picture membership problem can be solved in polynomial time for each context free language over fu d rg and is np complete for a so called retreat bounded regular or reversalbounded linear language over imposing both retreat and reversal bounds on languages over results in the leftmove bounded languages whose words describe pictures by making no more than a bounded number of left moves the picture membership problem can be solved in polynomial time for each leftmove bounded context free language over and is np complete for a leftmoveunbounded but retreat bounded linear language over fu d lrg there exists a context sensitive language over fu d rg or fu d lrg for which the picture membership problem is undecidable c 2000 elsevier science b v all rights reserved

the symmetric clustered traveling salesman problem abstract the clustered traveling salesman problem is the well known standard problem with additional constraints the cities are divided into groups and each group of cities cluster must be visited contiguously we describe an extensive adaptation of an existing traveling salesman algorithm based on the 1 tree relaxation in the lagrangean approach a new multiplier is introduced to improve the lower bounds furthermore we describe a heuristic to find satisfactory upper bounds and discuss the elimination of nonoptimal edges the computational results are satisfying and include random euclidean problems up to 150 cities

existence of global solutions of delayed differential equations via retract approach abstract the purpose of this contribution is to give sufficient conditions for the existence of global solutions or left semi global solutions for some classes of delayed functional differential equations the topological approach known as the topological retract principle is used inequalities for coordinates of global solutions are derived as a consequence of used method examples illustrate the results

a useful transform of standard input data for a classical np complete problem abstract the archetypal symmetric travelling salesman problem can be seen in a new and interesting way by using first a standard preparatory phase of input data and then by applying a transform from the set d of distances among cities and the set b of loss of optimality the specific form of d b transform is introduced and discussed in order to show in realistic terms the interest of the approach proposed a class of diffusive heuristic procedures operating from b is defined an example of solution by an algorithm included in this class is completely worked out an outline of computational tests done on the same algorithm is also given

using logical surrogate information in lagrangean relaxation an application to symmetric traveling salesman problems abstract the traveling salesman problem tsp is a classical combinatorial optimization problem which has been intensively studied the lagrangean relaxation was first applied to the tsp in 1970 the lagrangean relaxation limit approximates what is known today as held and karp hk bound a very good bound less than 1 from optimal for a large class of symmetric instances it became a reference bound for new heuristics mainly for the very large scale instances where the use of exact methods is prohibitive a known problem for the lagrangean relaxation application is the definition of a convenient step size control in subgradient like methods even preserving theoretical convergence properties a wrong defined control can affect the performance and increase computational times we show in this work how to accelerate a classical subgradient method while conserving good approximations to the hk bounds the surrogate and lagrangean relaxations are combined using the local information of the relaxed constraints it results in a one dimensional search that corrects the possibly wrong step size and is independent of the used step size control comparing with the ordinary subgradient method and beginning with the same initial multiplier the computational times are almost twice as fast for medium instances and greatly improved for some large scale tsplib instances

bounds for maximin latin hypercube designs latin hypercube designs lhds play an important role when approximating computer simulation models to obtain good space filling properties the maximin criterion is frequently used unfortunately constructing maximin lhds can be quite time consuming when the number of dimensions and design points increase in these cases we can use heuristical maximin lhds in this paper we construct bounds for the separation distance of certain classes of maximin lhds these bounds are useful for assessing the quality of heuristical maximin lhds until now only upper bounds are known for the separation distance of certain classes of unrestricted maximin designs i e for maximin designs without a latin hypercube structure the separation distance of maximin lhds also satisfies these unrestricted bounds by using some of the special properties of lhds we are able to find new and tighter bounds for maximin lhds within the different methods used to determine the upper bounds a variety of combinatorial optimization techniques are employed mixed integer programming the traveling salesman problem and the graph covering problem are among the formulations used to obtain the bounds besides these bounds also a construction method is described for generating lhds that meet baer s bound for the l distance measure for certain values of n

valuations of languages with applications to fractal geometry abstract valuations morphisms from to 0 1 are a simple generalization of bernoulli morphisms distributions measures as introduced by berstel and perrin 1988 berstel and reutenauer 1988 blanchard and hansel 1984 eilenberg 1974 and hansel and perrin 1983 1989 this paper shows that valuations are useful not only within the theory of codes but also when dealing with ambiguity especially in context free grammars or for defining outer measures on the space of words which are of some importance to the theory of fractals these connections yield new formulae to determine the hausdorff dimension of fractal sets especially in euclidean spaces defined via formal languages the class of fractals describable with context free languages strictly includes that of mrfs fractals introduced by culik and dube 1990 and mauldin and williams 1988 some of the results of this paper also appear as part of the author s ph d thesis and other works fernau 1992a b 1993

complexity and decidability for restricted classes of picture languages abstract the paper investigates complexity and decidability questions for two restricted classes of picture languages called three way picture languages and stripe picture languages it is shown that 1 the picture membership problem can be solved in deterministic polynomial time for three way context free picture languages 2 the equivalence and containment problems for picture languages are undecidable for a regular language l1 and a linear language l2 where both l1 and l2 describe three way stripe picture languages and 3 the intersection emptiness problem for picture languages is undecidable for two three way stripe linear picture languages and is also undecidable for a regular language l1 and a linear language l2 where l1 describes a three way stripe picture language and l2 describes a stripe picture language

nonoptimal edges for the symmetric traveling salesman problem for the symmetric traveling salesman problem we identify a set of undirected edges that can be eliminated while still retaining at least one optimal solution of the problem the simple identification rules are based on the fact that a solution can be optimal only if it is 2 optimal the rules are less stringent than those formulated for nonoptimal arcs directed applied to the symmetric case so in general they identify more nonoptimal edges application of the theory in a 1 tree based traveling salesman algorithm roughly halves the average computation time for euclidean problems we indicate to what extent the theorems presented here can be adapted for variants of the traveling salesman problem

l2 perturbation of a linear delay differential equation abstract in this paper we present an asymptotic formula for the solutions of the nonautonomous delay differential equation t a 0 a t x t b 0 b t x t assuming that it is an l 2 perturbation of the autonomous equation t a 0 x t b 0 x t our theorem is an essential generalization of the corresponding result in the classical book by r bellman and k l cooke the proof is based on a new result on the asymptotic constancy of the solutions of a linear delay equation

some inverse optimization problems under the hamming distance given a feasible solution to a particular combinatorial optimization problem defined on a graph and a cost vector defined on the arcs of the graph the corresponding inverse problem is to disturb the cost vector such that the feasible solution becomes optimal the aim is to optimize the difference between the initial cost vector and the disturbed one this difference can be measured in several ways we consider the hamming distance measuring in how many components two vectors are different where weights are associated to the components general algorithms for the bottleneck or minimax criterion are described and after modification applied to the inverse minimum spanning tree problem the inverse shortest path tree problem and the linear assignment problem

perfectly quilted rectangular snake tilings we introduce a particular form of snake tilings to define picture languages and relate the obtained family to the recognizable picture languages as defined by wang tiles the correspondence for substitution tilings is even closer and hence is applicable to the hilbert curve

a transformation system for generating description languages of chain code pictures abstract a picture is a set of unit lines from the cartesian plane considered as a square grid to each connected picture a description is given by a word over the alphabet r l u d where r means go one unit line right from the current point and draw this unit line and l u d are interpreted analogously with left up and down respectively for a picture p it was shown in 9 that the description language i e all the words which describe the picture p is a regular one to construct such a regular language one needs geometric information about the picture in this paper we present a system of transformations which map one word over the alphabet r l u d to another word which describes the same picture it is also shown that exactly all the picture descriptions can be generated started by some word

undecidability of existential properties in picture languages abstract we are interested in the description of a set of pictures by string languages by using several semantics segments 12 segments with blank moves 8 and pixels 10 we give a method to code the post correspondence problem in rational picture languages in order to show the undecidability of existence of words satisfying a given property in a rational language in particular we study the properties like is a self avoiding word is a minimal picture word or describes a connected picture

a branch and bound algorithm for the singly constrained assignment problem the singly constrained assignment problem scap is a linear assignment problem lap with one extra side constraint e g due to a time restriction the scap is in contrast to the lap difficult to solve a branch and bound algorithm is presented to solve the scap to optimality lower bounds are obtained by lagrangean relaxation computational results show that the algorithm is able to solve different types of scap instances up to size n 1000 within short running times on a standard personal computer

complexity and decidability for chain code picture languages abstract a picture description is a word over the alphabet u d r l where u means go one unit line up from the current point and d r and l are interpreted analogously with down right and left instead of up by this a picture description describes a walk in the plane its trace is the picture it describes a set of picture descriptions describes a chain code picture language this paper investigates complexity and decidability questions for these picture languages thus it is shown that the membership problem is np complete for regular picture languages i e picture languages described by regular languages of picture descriptions and that it is undecidable whether two regular picture description languages describe a picture in common after this we investigate so called stripe picture languages all pictures are within a stripe defined by two parallel lines providing better complexity and decidability results membership is decidable in linear time for regular stripe picture languages emptiness of intersection and equivalence is decidable for regular stripe picture languages

the asymptotic bounds of solutions of linear delay systems abstract our aim is to obtain conditions under which every solutionyof the vector functional differential equation formula can be estimated in terms of a solution of a scalar functional equation formula wherea x is a suitable real positive function andbis a suitable real negative constant moreover ifb x is diagonal matrix we derive more precise estimates ofyin terms of a solution of a vector functional equation formula where the elements ofa x are formed by the absolute values of the corresponding elements ofa x these asymptotic results enable us to obtain sharp estimates of solutions of certain linear delay systems

an empirical study of a new metaheuristic for the traveling salesman problem abstract we present a deceptively simple yet empirically powerful metaheuristic called jump search for solving traveling salesman problems that has been found to be more effective than tabu search on both random and benchmark test problems from the literature while the underlying philosophy of jump search applying local search from different starting points has been discussed in the literature previously using random starting points the use of construction based heuristic solutions has heretofore not been considered extensive empirical analysis shows the method to be surprisingly effective vis a vis a randomized strategy and in comparison with tabu search the approach is quite robust and suggests that a systematic search among neighborhoods of good not random solutions provides distinct advantages this suggests that further research be focused on better construction heuristics and hybrid schemes that incorporate jump search in for example tabu search

an improved transformation of the symmetric multiple traveling salesman problem we improve the standard transformation of the symmetric single depot multiple traveling salesman problem mtsp to one on a sparser edge configuration the improvement tends to suppress to a large extent the degeneracy of the resulting symmetric traveling salesman problem tsp as a result a 1 tree based tsp algorithm solves large mtsps of the type we consider as easily as it solves standard tsps we demonstrate the improvement by presenting computational results that are currently the best available

the symmetric traveling salesman problem and edge exchanges in minimal 1 trees abstract in this article the effect of exchanging edges inside a minimal 1 tree with edges outside is analysed in combination with an upper bound this analysis enables the elimination of variables in the symmetric traveling salesman problem after discussion on a number of improvements for this analysis the implementation is described in a traveling salesman algorithm based on the 1 tree relaxation computational results show the advantages of the edges exchanges for euclidean problems up to 120 cities as well as for random table problems up to 200 cities

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

the black and white traveling salesman problem the black and white traveling salesman problem bwtsp is defined on a graph g whose vertex set is partitioned into black and white vertices the aim is to design a shortest hamiltonian tour on g subject to cardinality and length constraints both the number of white vertices as well as the length of the tour between two consecutive black vertices are bounded above the bwtsp has applications in airline scheduling and in telecommunications this paper proposes an integer linear formulation for the undirected bwtsp as well as several classes of valid inequalities an exact branch and cut algorithm is then developed extensive tests show that it can solve exactly instances involving up to 100 vertices the algorithm can also be applied directly to solve unit demand vehicle routing problems of similar sizes

a note on finding a shortest complete cycle in an undirected graph abstract how to transform a shortest complete cycle problem into a traveling salesman problem is well known although these problems appear ill conditioned a 1 tree relaxation based traveling salesman algorithm solves them rather easily up to 150 nodes provided non optimal edges are eliminated the computational results are considerably better than known results for a linear programming approach

the membership and equivalence problems for picture languages abstract the paper describes two principal results 1 there is a nondeterministic polynomial time algorithm to determine whether an arbitrary picture is a member of a context free picture language and 2 the equivalence problem for regular picture languages is not decidable not even partially decidable this solves open questions described in earlier publications furthermore these results are best possible that is the membership problem for context free picture languages was earlier shown to be np hard and the complement of the equivalence problem for regular picture languages is easily seen to be partially decidable

solving some lexicographic multi objective combinatorial problems abstract the lexicographic multi objective optimization problem r lmop with r sum or bottleneck objectives optimizes a first objective then as far as a choice remains a second one then a third one and so on a general solution scheme from literature is based on scaling for the case that only sum criteria are involved the complexity is o rlogn t n m with t n m the complexity of optimization on combinatorial sum problem instances with m arcs and n nodes for cases with at least one bottleneck criterion the complexity is at least o rnlog2n t n m we describe a solution scheme that is suited to solve the shortest path r lmop the minimum spanning tree r lmop and the linear assignment r lmop the complexity is in all cases o r t n m

new lower bounds for the symmetric travelling salesman problem in this paper new lower bounds for the symmetric travelling salesman problem are proposed and combined in additive bounding procedures efficient implementations of the algorithms are given in particular fast procedures for computing the linear programming reduced costs of the shortest spanning tree sst problem and for finding all ther sst of a given graph are presented computational results on randomly generated test problems are reported

retreat bounded picture languages a well known string encoding of 2d pictures is to use the picture alphabet u d r l where u d r and l means draw a unit line by moving the pen up down right and left from the current point a word over is k retreat bounded k 0 if it describes a picture in such a manner that the maximum distance of left moves ignoring up and down moves from a rightmost point of any partially drawn picture is bounded by k a set of such words forms a k retreat bounded language a k retreat bounded picture language is a set of pictures described by a k retreat bounded language there is a 1 retreat bounded regular picture language 1 retreat bounded vertical stripe linear picture language for which the membership problem is np complete the membership problem can be solved in o n4 time for each k retreat bounded nonvertical stripe context free picture language the inclusion and intersection emptiness problems are undecidable for a 1 retreat bounded regular picture language and a 2 retreat bounded regular picture language the equivalence and ambiguity problems are undecidable for 2 retreat bounded regular picture languages the picture recognition algorithm presented is the first polynomial time algorithm in the literature for a reasonably large subclass of context free picture languages and the np completeness and undecidability results improve the previously known such results for regular picture languages with no structural restriction imposed

positive solutions of the equation ẋ t c t x t τ in the critical case abstract the equation t c t x t is considered in the critical case for it the asymptotic behavior of dominant and subdominant solutions is studied a generalization is made and connections with known results are discussed

using string languages to describe picture languages a picture is a set of unit lines from the cartesian plane considered as a square grid a word over the alphabet l r u d is a picture description in the sense that it represents a traversal of a picture where the interpretation of the symbols l r u d is l go one unit line to the left of the current point r go one unit line to the right of the current point u go one unit line up from the current point and d go one unit line down from the current point a set of picture descriptions forms a picture description language this paper investigates the basic properties of pictures and picture description languages from the formal language theory point of view

on reversal bounded picture languages abstract for an integer k 0 a k reversal bounded picture language is a chain code picture language which is described by a language l over the alphabet u d r l such that for every word x in l the number of alternating occurrences of r s and l s in x is bounded by k it is shown that the membership problem can be solved in o n4k 4 time for k reversal bounded regular picture languages for every k 1 and is np complete for 1 reversal bounded stripe linear picture languages the membership problem is known to be np complete for regular and context free picture languages without restriction on the number of reversals and solvable in o n time o n12 time for 0 reversal bounded regular context free picture languages whether the membership problem for stripe context free picture languages could be solved in polynomial time has been an open problem other basic properties of reversal bounded picture languages are also presented