using dual approximation algorithms for scheduling problems theoretical and practical results the problem of scheduling a set of n jobs on m identical machines so as to minimize the makespan time is perhaps the most well studied problem in the theory of approximation algorithms for np hard optimization problems in this paper the strongest possible type of result for this problem a polynomial approximation scheme is presented more precisely for each e an algorithm that runs in time o n e 1 e 2 and has relative error at most e is given in addition more practical algorithms for e 1 5 2 k and e 1 6 2 k which have running times o n k log n and o n km 4 log n are presented the techniques of analysis used in proving these results are extremely simple especially in comparison with the baroque weighting techniques used previously the scheme is based on a new approach to constructing approximation algorithms which is called dual approximation algorithms where the aim is to find superoptimal but infeasible solutions and the performance is measured by the degree of infeasibility allowed this notion should find wide applicability in its own right and should be considered for any optimization problem where traditional approximation algorithms have been particularly elusive

oscillatory integrals with quadratic phase function on a real abstract wiener space abstract stochastic oscillatory integrals with quadratic phase function are studied in the case where amplitude functions are multiple wiener integrals we show that a principle of stationary phase holds when kernels of multiple wiener integrals are of trace class on the way we establish a new criterion for a wiener functional to be analytic

bin packing in multiple dimensions inapproximability results and approximation schemes we study the following packing problem given a collection of d dimensional rectangles of specified sizes pack them into the minimum number of unit cubes we show that unlike the one dimensional case the two dimensional packing problem cannot have an asymptotic polynomial time approximation scheme aptas unless pnp on the positive side we give an aptas for the special case of packing d dimensional cubes into the minimum number of unit cubes second we give a polynomial time algorithm for packing arbitrary two dimensional rectangles into at most opt square bins with sides of length 1 where opt denotes the minimum number of unit bins required to pack these rectangles interestingly this result has no additive constant term i e is not an asymptotic result as a corollary we obtain the first approximation scheme for the problem of placing a collection of rectangles in a minimum area encasing rectangle

analysis of wiener functionals malliavin calculus and its applications to heat kernels on donne un expose facilement accessible du calcul de malliavin du calcul differentiel de dimension infinie sur l espace de wiener

short shop schedules we consider the open shop job shop and flow shop scheduling problems with integral processing times we give polynomial time algorithms to determine if an instance has a schedule of length at most 3 and show that deciding if there is a schedule of length at most 4 is x1d4a9 x1d4ab complete the latter result implies that unless x1d4ab x1d4a9 x1d4ab there does not exist a polynomial time approximation algorithm for any of these problems that constructs a schedule with length guaranteed to be strictly less than 5 4 times the optimal length this work constitutes the first nontrivial theoretical evidence that shop scheduling problems are hard to solve even approximately

the weak solution of a functional differential equation in a general banach space abstract the existence of a unique weak lipschitz continuous solution of the abstract functional differential equation u t a t u t g t u t t 0 t fde u t t t r 0 is obtained by a fixed point argument here x is a general banach space a t d a t x x is m accretive t 0 t and g is lipschitz continuous in both variables stability properties of this weak solution are given and a first order hyperbolic pde to which the theory can be applied is considered

ranges of perturbed maximal monotone and m accretive operators in banach spaces a more comprehensive and unified theory is developed for the solvability of the inclusions s c r a b ints c r a b where a x d d a 2y b x d d b y and s c x here x is a real banach space and y x or y x mainly a is either maximal monotone or maccretive and b is either pseudo monotone or compact cases are also considered where a has compact resolvents and b is continuous and bounded these results are then used to obtain more concrete sets in the ranges of sums of such operators a and b various results of browder calvert and gupta gupta gupta and hess and kartsatos are improved and or extended the methods involve the application of a basic result of browder concerning pseudo monotone perturbations of maximal monotone operators and the leray schauder degree theory

on the approximate k controllability of nonlinear systems in banach spaces abstract let x be a real banach space control problems of the type x a t x b t x u t 0 t x 0 0 are considered where for every t 0 t a t x d a x and b 0 t x 2 x are given operators the concept of approximate k controllability or controllability with preassigned responses is introduced for systems of the type it is shown that there exist lipschitz continuous approximating control functions u t t 0 t for a variety of response types evans responses and kato responses are considered for fully nonlinear problems as well as mild ones for semilinear problems it is also shown that the function r which determines the proximity of the response x t with x t a t x t b t x t u t t 0 t x 0 0 to the preassigned response f t x f r is of the type c where c is a positive constant independent of a more natural approximate k controllability concept approximate ek controllability is also introduced and a result is given about it using leray schauder theory an application is given in the field of partial differential equations

improved approximation algorithms for shop scheduling problems in the job shop scheduling problem there are m machines and n jobs a job consists of a sequence of operations each of which must be processed on a specified machine and the aim is to complete all jobs as quickly as possible this problem is strongly cal np hard even for very restrictive special cases the authors give the first randomized and deterministic polynomial time algorithms that yield polylogarithmic approximations to the optimal length schedule these algorithms also extend to the more general case where a job is given not by a linear ordering of the machines on which it must be processed but by an arbitrary partial order comparable bounds can also be obtained when there are m types of machines a specified number of machines of each type and each operation must be processed on one of the machines of a specified type as well as for the problem of scheduling unrelated parallel machines subject to chain precedence constraints

ultimate boundedness and periodicity for some partial functional differential equations with infinite delay abstract in this work we study the existence of periodic solutions for some partial functional differential equation with infinite delay we assume that the linear part is not necessarily densely defined and satisfies the known hille yosida condition firstly we give some estimates of the solutions secondly we prove that the poincare map is condensing which allows us to prove the existence of periodic solutions when the solutions are ultimately bounded

necessary and sufficient conditions for the regularity and stability for some partial functional differential equations with infinite delay abstract in this work we give necessary and sufficient conditions for the regularity and stability of solutions for some partial functional differential equations with infinite delay we establish also a new characterization of the infinitesimal generator of the solution semigroup

some results for impulsive fractional differential inclusions with infinite delay and sectorial operators in banach spaces this paper is concerned with impulsive fractional differential inclusions with sectorial operators nonlocal conditions and infinite delay in banach spaces using tools involving the measure of noncompactness and multi valued fixed point theory we derive some existence results of pc mild solutions when the multivalued map f is convex as well as nonconvex furthermore the compactness of the set of solutions and continuous dependence results are also obtained as an application an impulsive neutral partial functional differential equation is also investigated at last

makespan minimization in open shops a polynomial time approximation scheme in this paper we demonstrate the existence of a polynomial time approximation scheme for makespan minimization in the open shop scheduling problem with an arbitrary fixed numberm of machines for the variant of the problem where the number of machines is part of the input it is known that the existence of an approximation scheme would implyp np hence our result draws a precise separating line between approximable cases i e withm fixed and non approximable cases i e withm part of the input of this shop problem 1998 the mathematical programming society inc published by elsevier science b v

levy s stochastic area and the principle of stationary phase abstract let w 0 h 0 0 be the 2 dimensional classical pinned wiener space over 0 1 a localization phenomenon will be shown for stochastic oscillatory integrals with levy s stochastic area as phase function

the cauchy problem for nonlinear abstract functional differential equations with infinite delay abstract in this paper we use existence families and admissible phase spaces to study the cauchy problem for a class of nonlinear abstract functional differential equations with infinite delay in a unified way our main results concern the existence and uniqueness of mild solutions strong solutions and classical solutions

on wiener functionals of order 2 associated with soliton solutions of the kdv equation abstract the eigenvalues and eigenvectors of the hilbert schmidt operators corresponding to the wiener functionals of order 2 which give a rise of soliton solutions of the kdv equation are determined two explicit expressions of the stochastic oscillatory integral with such wiener functional as phase function are given one is of infinite product type and the other is of levy s formula type as an application the asymptotic behavior of the stochastic oscillatory integral will be discussed

existence and continuous dependence results for nonlinear differential inclusions with infinite delay this paper is concerned with nonlinear functional differential inclusions with infinite delay in banach spaces using tools involving the measure of noncompactness and multi valued fixed point theory existence and continuous dependence results are obtained for integral solutions without the assumption of compactness on the associated nonlinear semigroup

approximating a two machine flow shop scheduling under discrete scenario uncertainty abstract this paper deals with the two machine permutation flow shop problem with uncertain data whose deterministic counterpart is known to be polynomially solvable in this paper it is assumed that job processing times are uncertain and they are specified as a discrete scenario set for this uncertainty representation the min max and min max regret criteria are adopted the min max regret version of the problem is known to be weakly np hard even for two processing time scenarios in this paper it is shown that the min max and min max regret versions of the problem are strongly np hard even for two scenarios furthermore the min max version admits a polynomial time approximation scheme if the number of scenarios is constant and it is approximable with performance ratio of 2 and not 4 3 approximable for any 0 unless p np if the number of scenarios is a part of the input on the other hand the min max regret version is not at all approximable even for two scenarios

existence and continuous dependence results for semilinear functional differential inclusions with infinite delay we consider local and global existence results for dierential inclusions with infinite delay in a banach space of the form y 0 t 2 ay t f t yt t ae yae 2 b and continuous dependence of the solutions set on the initial functions

the method of lines and the approximation of zeros of m accretive operators in general banach spaces abstract it is shown that a zero of an m accretive operator t d t x 2 x in a general banach space x can be approximated via methods of lines for associated evolution equations results of browder for single valued locally defined continuous accretive operators t in spaces x with uniformly convex duals or uniformly continuous accretive operators t in general banach spaces x are extended to the present case results of reich for m accretive operators in reflexive banach spaces are also extended to the present setting unlike browder s results our estimates do not use the modulus of continuity of the operators t or the duality mapping of x kobayashi type estimates are used for the bounds involving the normed differences between the endpoints of the methods of lines and a zero of t

approximability of flow shop scheduling shop scheduling problems are notorious for their intractability both in theory and practice in this paper we construct a polynomial approximation scheme for the flow shop scheduling problem with an arbitrary fixed number of machines for the three common shop models open flow and job this result is the only known approximation scheme since none of the three models can be approximated arbitrarily closely in the general case unless p np the result demonstrates the approximability gap between the models in which the number of machines is fixed and those in which it is part of the input of the instance the result can be extended to flow shops with job release dates and delivery times and to flow shops with a fixed number of stages where the number of machines at any stage is fixed 1998 the mathematical programming society inc published by elsevier science b v

a study on several combination problems of classic shop scheduling and shortest path in this paper we study several combinatorial optimization problems which combine the classic open shop or job shop scheduling problem and the shortest path problem our goal is to select a subset of jobs that constitutes a feasible solution of the shortest path problem and then execute the selected jobs on the shop machines to minimize the makespan i e the last completion time of all the jobs we prove that these problems are np hard even if there are two machines if the number of machines is an input we show that it is unlikely to find approximation algorithms with performance ratios better than 2 unless p np we present an intuitive approximation algorithm when the number of machines is an input and an improved approximation algorithm when the number of machines is fixed in addition we propose a polynomial time approximation scheme for the open shop case when the number of machines is fixed

existence results for nonlinear functional evolution equations with delay conditions abstract let x be a real banach space a t d a t x 2 x be an m accretive operator g 0 t l p r 0 x x x be a mapping l t l p r t x x be a mapping u t r 0 x satisfy u t s u t s for every s r 0 and 0 l p r 0 x for 1 p with 0 0 d where d d a t independent of t the local existence of integral solutions of nonlinear functional evolution equation with delay condition d u t d t a t u t g t u t l t u 0 t t u 0 0 t r t 0 is established in the case when the evolution operator u t s generated by a t is equicontinuous

minimizing makespan in no wait job shops in this paper we study polynomial time approximation schemes ptases for the no wait job shop scheduling problem with the makespan objective function it is known that the problem is maxsnp hard in the case when each job is allowed to have three operations or more we show that if each job has at most two operations the problem admits a ptas if the number of machines is a constant i e not part of the input if the number of machines is not a constant we show that the problem is hard to approximate within a factor better than 5 4

open block scheduling in optical communication networks in this paper the process of data transmission in star coupled optical communication networks is modelled as a shop type scheduling problem where channels wavelengths are treated as machines we formulate an open block problem with the minimum makespan objective o b cmax in which a relation of a new type between the operations of each job is introduced any two operations of a job have identical processing times and may be processed either simultaneously in a common block or alternatively at disjoint time intervals we show that the problem is polynomially solvable for 4 machines np hard for 6 machines and strongly np hard for a variable number of machines for the case of a variable number of machines we present a polynomial time 2 approximation algorithm and prove that there is no polynomial time approximation algorithm with 11 10 unless p np for the case when the number of machines is fixed we show that the problem admits a linear time ptas in addition we show that the two machine problem with release dates is np hard in the strong sense

a study of delayed cooperation diffusion system with dirichlet boundary conditions abstract in this paper we study the existence and uniqueness of strong solution of a delayed cooperation diffusion system with dirichlet boundary conditions using the method of semidiscretization in time

solvability of the cauchy problem for infinite delay equations abstract in this paper we study the solvability of the cauchy problem for functional differential equations with infinite delay in a banach space x which are more general than those in many previous publications we undertake our study under suitable hypotheses based on noncompactness measures and kamke type functions and in a general setting of admissible phase space p including the space lp 0 x therefore our results extend and unify many existing results for finite and infinite delay equations as a sample of possible applications we analyze the solvability of the cauchy problem for a schrodinger type partial functional integro differential equation in l1 r

the three machine proportionate open shop and mixed shop minimum makespan problems abstract we first consider the ordinary np hard three machine proportionate open shop minimum makespan o 3 prpt c max problem and show that it is solvable in o n log n time when certain conditions on the total machine load are met when these conditions are not met we derive an approximate solution by implementing the optimal solution for the o 3 prpt c max problem when the two longest jobs are of equal length in that case both absolute and ratio worst case bounds are derived we also consider the more general mixed shop problem m 3 prpt c max in which a given job subset must be processed as in a flow shop while the remaining jobs can be processed as in the o 3 prpt c max problem we show that our open shop solution techniques can be implemented to derive exact and approximate solutions for the m 3 prpt c max problem finally we discuss the applicability of our open shop results to the proportionate open shop problem with unequal machine speeds

on the existence of solutions for functional evolution equations soit x un espace de banach de norme soit pc l espace des fonctions continues par morceaux r 0 x pour un r fixe 0 pour pc on pose pc sup r 0 on considere l equation d evolution fonctionnelle non lineaire abstraite g t u t u t a t u t u 0 t 0 t

rothe s method for a telegraph equation with integral conditions abstract we study the existence and uniqueness of the solution for a telegraph equation with integral condition we apply the rothe time discretization method then we prove its convergence

analytic functions cauchy formula and stationary phase on a real abstract wiener space abstract a new complexification of a real abstract wiener space will be introduced and some analogs of the algebra of analytic functions on finite dimensional euclidean space will be considered analytic functions on the original space their holomorphic prolongation to the complexified space and holomorphic functions and a cauchy formula on the complexified space the cauchy formula is a key tool to study probabilistic quantities via deformation of the contour integration namely it will be applied to establish i an explicit representation of stochastic oscillatory integrals with quadratic phase function and ii a stationary phase estimation of the integrals further the later estimation is applicable to study gevrey type smoothness of density functions an integration by parts formula on a totally real submanifold in the complexified space is also studied

a method of lines for a nonlinear abstract functional evolution equation let x be a real banach space with x uniformly convex a method of lines is introduced and developed for the abstract functional problem lipschitzian like function in its two variables further conditions are given for the convergence of the method to a strong solution of e recent results for perturbed abstract ordinary equations are substantially improved the method applies also to large classes of functional parabolic problems as well as problems of integral perturbations the method is straightforward because it avoids the introduction of the operators a t and the corresponding use of nonlinear evolution operator theory 1 introduction preliminaries let x be a real banach space with norm 11 let