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

a polynomial time approximation scheme for the two stage multiprocessor flow shop problem in this paper we investigate the two stage multiprocessor flow shop scheduling problem f2 p cmax where the numbers m1 and m2 of machines available in the two stages are part of the input we demonstrate the existence of a polynomial time approximation scheme for this problem this result solves the simplest case of an open problem that has been posed by leslie hall in a recent paper hall 1995 an extension of our algorithm yields an approximation scheme for the closely related two stage multiprocessor job shop problem

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

extension of the ablowitz ladik method to the derivative nonlinear schro dinger equation abstract the derivative nonlinear schrodinger equation is solved by application of the ablowitz ladik scheme to an equivalent equation the variations of the results due to modifications in the spatial grid size and time step are analyzed the scheme maintains the main properties of the original equation and allows the use of rather large time steps

a numerical energy conserving method for the dnls equation an implicit numerical energy conserving method is developed for the derivative nonlinear schrodinger dnls equation for periodic boundary conditions we find no numerical high frequency modulational instabilities in addition to the modulational instability from a linear analysis around a nonlinear state for the dnls equation if the modulation is small and k0 a22 2 t k0 is the wavenumber and a the amplitude the numerical scheme is used to follow the nonlinear behavior of the dnls modulational instability the numerical code is also tested by the evolution for one soliton initial data these tests show that if the modulation is not small compared to the background wave amplitude new nonlinear numerical instabilities are introduced

on surfaces of class vii 0 with numerically anticanonical divisor we consider minimal compact complex surfaces s with betti numbers b 1 and n bi 0 a theorem of donaldson gives n exceptional line bundles we prove that if in a deformation these line bundles have sections s is a degeneration of blown up hopf surfaces besides if there exists an integer m 1 and a flat line bundle f such that h s mk 0 these surfaces admit no nonconstant mero morphic functions the major problem in classification of non kahlerian surfaces is to achieve the classification of surfaces s of class viij all known surfaces of this class contain global spherical shells gss i e admit a biholomorphic map ip u v from a neighbourhood u c c2 0 of the sphere 53 db2 onto an open set v such that i f s3 does not disconnect s are there other surfaces in first section we investigate the general situation a theorem of donaldson 13 gives a z base of 2 s z such that b of s these line bundles form families we propose the following conjecture which can be easily checked for surfaces with gss

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

approximation schemes for job shop scheduling problems with controllable processing times in this paper we study the job shop scheduling problem under the assumption that the jobs have controllable processing times the fact that the jobs have controllable processing times means that it is possible to reduce the processing time of the jobs by paying a certain cost we consider two models of controllable processing times continuous and discrete for both models we present polynomial time approximation schemes when the number of machines and the number of operations per job are fixed

an algorithm for solving the job shop problem in this paper we propose a branch and bound method for solving the job shop problem it is based on one machine scheduling problems and is made more efficient by several propositions which limit the search tree by using immediate selections it solved for the first time the famous 10 10 job shop problem proposed by muth and thompson in 1963

analytical and numerical aspects of certain nonlinear evolution equations ii numerical nonlinear schrodinger equation abstract various numerical methods are employed in order to approximate the nonlinear schrodinger equation namely i the classical explicit method ii hopscotch method iii implicit explicit method iv crank nicolson implicit scheme v the ablowitz ladik scheme vi the split step fourier method f tappert and vii pseudospectral fourier method fornberg and whitham comparisons between the ablowitz ladik scheme which was developed using notions of the inverse scattering transform and the other utilized schemes are obtained

une construction elementaire des surfaces d inoue hirzebruch 0 introduction 663 1 surfaces d inoue hirzebruch et th6or6me principal 664 2 application aux champs de vecteurs globaux et fibr6 canonique 668 3 rev ements et automorphismes des surfaces de kato 673 4 d monstration du th or me principal 681 r6f6rences 682

analytical and numerical aspects of certain nonlinear evolution equations i analytical abstract nonlinear partial difference equations are obtained which have as limiting forms the nonlinear schrodinger korteweg devries and modified korteweg devries equations these difference equations have a number of special properties they are constructed by methods related to the inverse scattering transform they can be used as a basis for numerical schemes to the associated nonlinear evolution equations experiments have shown that they compare very favorably with other known numerical methods papers ii iii in paper ii the ablowitz ladik scheme for the nonlinear schrodinger equation is compared to other known numerical schemes and generally proved to be faster than all utilized finite difference schemes but somewhat slower than the finite fourier pseudospectral methods in paper iii a proposed scheme for the korteweg de vries equation proved to be faster than both the finite difference and finite fourier methods already considered

an fptas for the parallel two stage flowshop problem we consider the np hard m parallel two stage flowshop problem abbreviated as the m 2 pfs problem where we need to schedule n jobs to m parallel identical two stage flowshops in order to minimize the makespan i e the maximum completion time of all the jobs on the m flowshops the m 2 pfs problem can be decomposed into two subproblems to assign the n jobs to the m parallel flowshops and for each flowshop to schedule the jobs assigned to the flowshop we first present a pseudo polynomial time dynamic programming algorithm to solve the m 2 pfs problem optimally for any fixed m based on an earlier idea for solving the 2 2 pfs problem using the dynamic programming algorithm as a subroutine we design a fully polynomial time approximation scheme fptas for the m 2 pfs problem

an approximation scheme for minimizing the makespan of the parallel identical multi stage flow shops in the parallel k stage flow shops problem we are given m identical k stage flow shops and a set of jobs each job can be processed by any one of the flow shops but switching between flow shops is not allowed the objective is to minimize the makespan which is the finishing time of the last job this problem generalizes the classical parallel identical machine scheduling where k 1 k 1 and the classical flow shop scheduling where m 1 m 1 problems and thus it is np hard we present a polynomial time approximation scheme ptas for the problem when m and k are fixed constants the key technique is to partition the jobs into big jobs and small jobs enumerate over all feasible schedules for the big jobs and handle the small jobs by solving a linear program and employing a sliding method such a technique has been used in the design of ptas for several flow shop scheduling variants our main contributions are the non trivial application of this technique and a valid answer to the open question in the literature

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

donaldson theory on non kahlerian surfaces and class vii surfaces with b2 1 we prove that any class vii surface with b2 1 has curves this implies the global spherical shell conjecture in the case b2 1

job shop scheduling by simulated annealing we describe an approximation algorithm for the problem of finding the minimum makespan in a job shop the algorithm is based on simulated annealing a generalization of the well known iterative improvement approach to combinatorial optimization problems the generalization involves the acceptance of cost increasing transitions with a nonzero probability to avoid getting stuck in local minima we prove that our algorithm asymptotically converges in probability to a globally minimal solution despite the fact that the markov chains generated by the algorithm are generally not irreducible computational experiments show that our algorithm can find shorter makespans than two recent approximation approaches that are more tailored to the job shop scheduling problem this is however at the cost of large running times

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

instantons and curves on class vii surfaces we develop a general strategy based on gauge theoretical methods to prove existence of curves on class vii surfaces we prove that for b 2 2 every minimal class vii surface has a cycle of rational curves hence by a result of nakamura is a global deformation of a one parameter family of blown up primary hopf surfaces the case b 2 1 was solved in a previous article the fundamental object intervening in our strategy is the moduli space m pst 0 k of polystable bundles with c 2 0 det k for large b 2 the geometry of this moduli space becomes very complicated the case b 2 2 treated here in detail requires new ideas and difficult techniques of both complex geometric and gauge theoretical nature we explain the substantial obstacles which must be overcome in order to extend our methods to the case b 2 3

smooth deformations of singular contractions of class vii surfaces we consider normal compact surfaces y obtained from a minimal class vii surface x by contraction of a cycle c of r rational curves with c 2 0 our main result states that if the obtained cusp is smoothable then y is globally smoothable the proof is based on a vanishing theorem for h 2 theta y if r b 2 x any smooth small deformation of y is rational and if r b 2 x i e when x is a half inoue surface any smooth small deformation of y is an enriques surface the condition the cusp is smoothable in our main theorem can be checked in terms of the intersection numbers of the cycle using the looijenga conjecture which has recently become a theorem therefore this is a decidable condition we prove that this condition is always satisfied if r b 2 x leqslant 11 therefore the singular surface y obtained by contracting a cycle c of r rational curves in a minimal class vii surface x with r b 2 x leqslant 11 is always smoothable by rational surfaces the statement holds even for unknown class vii surfaces