simultaneous minimization of total completion time and total deviation of job completion times abstract this paper addresses a single machine scheduling problem with the objective of minimizing a linear combination of total job completion times and total deviation of job completion times from a common due date the due date is assumed to be non restrictive i e sufficiently large to have no impact on the optimal sequence when the weights are job independent the problem is shown to have a polynomial time solution and the optimal schedule is fully characterized as a function of the different parameters when job dependent weights are assumed the problem is known to be np hard we introduce a pseudo polynomial dynamic programming algorithm indicating that this case is np hard in the ordinary sense the algorithm is shown experimentally to perform extremely well when tested on high multiplicity instances with up to 1000 jobs

a split step numerical method for the time dependent dirac equation in 3 d axisymmetric geometry abstract a numerical method is developed to solve the time dependent dirac equation in cylindrical coordinates for 3 d axisymmetric systems the time evolution is treated by a splitting scheme in coordinate space using alternate direction iteration while the wave function is discretized spatially on a uniform grid the longitudinal coordinate evolution is performed exactly by the method of characteristics while the radial coordinates evolution uses poisson s integral solution which allows to implement the radial symmetry of the wave function the latter is evaluated on a time staggered mesh by using hermite polynomial interpolation and by performing the integration analytically the cylindrical coordinate singularity problem at r 0 is circumvented by this method as the integral is well defined at the origin the resulting scheme is reminiscent of non standard finite differences in the last step of the splitting the remaining equation has a solution in terms of a time ordered exponential which is approximated to a higher order than the time evolution scheme we study the time evolution of gaussian wave packets and we evaluate the eigenstates of hydrogen like systems by using a spectral method we compare the numerical results to analytical solutions to validate the method in addition we present three dimensional simulations of relativistic laser matter interactions using the dirac equation

linear programming in linear time when the dimension is fixed it is demonstrated that the linear programming problem in d variables and n constraints can be solved in o n time when d is fixed this bound follows from a multidimensional search technique which is applicable for quadratic programming as well there is also developed an algorithm that is polynomial in both n and d provided d is bounded by a certain slowly growing function of n

due window assignment problems with unit time jobs we study a class of due window assignment problems the objective is to find the job sequence and the window starting time and size such that the total cost of earliness tardiness and due window is minimized the study assumes unit time jobs and considers settings of a single machine and of parallel identical machines both the due window starting time and size are decision variables for the single machine setting we study a complete set of problems consisting of all possible combinations of the decision variables and four cost factors earliness tardiness due window size and due window starting time for parallel identical machines we study a complete set of problems consisting of all possible combinations of the decision variables and three cost factors earliness tardiness and due window size all the problems are shown to be solvable in no more than o n 3 time where n is the number of jobs

efficient time integration of the maxwell klein gordon equation in the non relativistic limit regime the maxwell klein gordon equation describes the interaction of a charged particle with an electromagnetic field solving this equation in the non relativistic limit regime i e the speed of light c formally tending to infinity is numerically very delicate as the solution becomes highly oscillatory in time in order to resolve the oscillations standard numerical time integration schemes require severe time step restrictions depending on the large parameter c 2 the idea to overcome this numerical challenge is to filter out the high frequencies explicitly by asymptotically expanding the exact solution with respect to the small parameter c 2 this allows us to reduce the highly oscillatory problem to its corresponding non oscillatory schrodinger poisson limit system on the basis of this expansion we are then able to construct efficient numerical time integration schemes which do not suffer from any c dependent time step restriction

efficient and accurate numerical methods for the klein gordon schrodinger equations in this paper we present efficient unconditionally stable and accurate numerical methods for approximations of the klein gordon schrodinger kgs equations with without damping terms the key features of our methods are based on i the application of a time splitting spectral discretization for a schrodinger type equation in kgs ii the utilization of fourier pseudospectral discretization for spatial derivatives in the klein gordon equation in kgs iii the adoption of solving the ordinary differential equations odes in phase space analytically under appropriate chosen transmission conditions between different time intervals or applying crank nicolson leap frog for linear nonlinear terms for time derivatives the numerical methods are either explicit or implicit but can be solved explicitly unconditionally stable and of spectral accuracy in space and second order accuracy in time moreover they are time reversible and time transverse invariant when there is no damping terms in kgs conserve or keep the same decay rate of the wave energy as that in kgs without or with a linear damping term keep the same dynamics of the mean value of the meson field and give exact results for the plane wave solution extensive numerical tests are presented to confirm the above properties of our numerical methods for kgs finally the methods are applied to study solitary wave collisions in one dimension 1d as well as dynamics of a 2d problem in kgs

a recovering beam search algorithm for the single machine just in time scheduling problem abstract we consider the just in time scheduling problem where the just in time notion is captured by means of multiple conflicting criteria the calculation of any non dominated solution for these criteria is achieved by solving an extension of the single machine problem of minimising the mean weighted deviation from distinct due dates in the extended problem each job to schedule is also constrained by a release date and a deadline this problem is np hard in the strong sense and we propose heuristic algorithms to solve it computational experiments show that among those algorithms the most effective heuristic in terms of quality is a recovering beam search algorithm

algebraic solution for the vector potential in the dirac equation the dirac equation for an electron in an external electromagnetic field can be regarded as a singular set of linear equations for the vector potential radford s method of algebraically solving for the vector potential is reviewed with attention to the additional constraints arising from non maximality of the rank the extension of the method to general spacetimes is illustrated by examples in diverse dimensions with both c and a number wavefunctions pacs numbers 0350d 0210 0220 0365p 1130p

general solutions of maxwell dirac equations in 1 1 dimensional space time and a spatially confined solution the most general solution of the system of massless maxwell dirac equations in 1 1 dimensional space time or classical schwinger theory is obtained in terms of four arbitrary functions and one arbitrary constant a particular example is furnished for which the dirac wave function vanishes completely outside a finite spatial range maxwell dirac equations for a nonzero mass parameter are reduced to a single real fourth order nonlinear partial differential equation a particular class of solutions for this complicated equation is provided

earliness tardiness scheduling problems ii derivation of completion times about a restrictive common due date a companion paper part i considers the problem of minimizing the weighted earliness and tardiness of jobs scheduled on a single machine around a common due date d which is unrestrictively late this paper part ii considers the problem of minimizing the unweighted earliness and tardiness of jobs allowing the possibility that d is early enough to constrain the scheduling decision we describe several optimality conditions the recognition version of the problem is shown to be np complete in the ordinary sense confirming a well known conjecture moreover this complexity definition is precise since we describe a dynamic programming algorithm which runs in pseudopolynomial time this algorithm is also extremely efficient computationally providing an improvement over earlier procedures of almost two orders of magnitude in the size of instance that can be solved finally we describe a special case of the problem which is polynomially solvable

analysis and comparison of numerical methods for the klein gordon equation in the nonrelativistic limit regime we analyze rigourously error estimates and compare numerically temporal spatial resolution of various numerical methods for solving the klein gordon kg equation in the nonrelativistic limit regime involving a small parameter 0 varepsilon ll 1 which is inversely proportional to the speed of light in this regime the solution is highly oscillating in time i e there are propagating waves with wavelength of o varepsilon 2 and o 1 in time and space respectively we begin with four frequently used finite difference time domain fdtd methods and obtain their rigorous error estimates in the nonrelativistic limit regime by paying particularly attention to how error bounds depend explicitly on mesh size h and time step as well as the small parameter varepsilon based on the error bounds in order to compute correct solutions when 0 varepsilon ll1 the four fdtd methods share the same varepsilon scalability tau o varepsilon 3 then we propose new numerical methods by using either fourier pseudospectral or finite difference approximation for spatial derivatives combined with the gautschi type exponential integrator for temporal derivatives to discretize the kg equation the new methods are unconditionally stable and their varepsilon scalability is improved to o 1 and tau o varepsilon 2 for linear and nonlinear kg equations respectively when 0 varepsilon ll1 numerical results are reported to support our error estimates

baire class 1 selectors for upper semicontinuous set valued maps let t be a metric space and x a banach space let f t x be a set valued map assuming arbitrary values and satisfying the upper semicontinuity condition t t f t c o is closed for each weakly closed set c in x then there is a sequence of norm continuous functions converging pointwise in the norm to a selection for f we prove a statement of similar precision and generality when x is a metric space

numerical study of time splitting spectral discretizations of nonlinear schrodinger equations in the semiclassical regimes in this paper we study the performance of time splitting spectral approximations for general nonlinear schrodinger equations nls in the semiclassical regimes where the planck constant varepsilon is small the time splitting spectral approximation under study is explicit unconditionally stable and conserves the position density in l 1 moreover it is time transverse invariant and time reversible when the corresponding nls is extensive numerical tests are presented for weak strong focusing defocusing nonlinearities for the gross pitaevskii equation and for current relaxed quantum hydrodynamics the tests are geared towards the understanding of admissible meshing strategies for obtaining correct physical observables in the semiclassical regimes furthermore comparisons between the solutions of the nls and its hydrodynamic semiclassical limit are presented

scheduling about a common due date with kob dependent asymmetric earliness and tardiness penalties abstract this paper is concerned with the problem of scheduling n jobs with a common due date on a single machine so as to minimize the total cost arising from earliness and tardiness a general model is examined in which earliness penalty and tardiness penalty are respectively arbitrary non decreasing functions moreover the model includes two important features that commonly appear in practical problems namely 1 earliness and tardiness are penalized with different weights which are job dependent and 2 the earliness or tardiness penalty consists of two parts one is a variable cost dependent on the length of earliness or tardiness while the other is a fixed cost incurred when a job is early or tardy this model provides a general and flexible performance measure for earliness tardiness scheduling which has not been addressed before we establish a number of results on the characterizations of optimal and sub optimal solutions and propose two algorithms based on these results the first algorithm can find under an agreeable weight condition an optimum in time o n2 pn and the second algorithm can generate a sub optimum in time o npn where pn is the sum of the processing times further we derive an upper bound on the relative error of the sub optimal solution and show that under certain conditions the error tends to zero as n increases computational results are also reported to demonstrate the effectiveness of the algorithms proposed

two machine flow shop scheduling with two criteria maximum earliness and makespan abstract in this study we address the problem of minimizing makespan and maximum earliness simultaneously in a two machine flow shop environment we develop a branch and bound procedure that generates all efficient solutions with respect to two criteria we propose several lower and upper bounding schemes to enhance the efficiency of the algorithm we also propose a heuristic procedure that generates approximate efficient solutions our computational results reveal that the branch and bound procedure is capable of solving problems with up to 25 jobs and the heuristic procedure produces approximate efficient solutions that are very close to exact efficient solutions in very small computation times

earliness tardiness scheduling problems i weighted deviation of completion times about a common due date this paper and its companion part ii concern the scheduling of jobs with cost penalties for both early and late completion in part i we consider the problem of minimizing the weighted sum of earliness and tardiness of jobs scheduled on a single processor around a common due date d we assume that d is not early enough to constrain the scheduling decision the weight of a job does not depend on whether the job is early or late but weights may vary between jobs we prove that the recognition version of this problem is np complete in the ordinary sense we describe optimality conditions and present a computationally efficient dynamic programming algorithm when the weights are bounded by a polynomial function of the number of jobs a fully polynomial approximation scheme is given we also describe four special cases for which the problem is polynomially solvable part ii provides similar results for the unweighted version of this problem where d is arbitrary

a uniformly accurate ua multiscale time integrator pseudospectral method for the dirac equation in the nonrelativistic limit regime we propose and rigourously analyze a multiscale time integrator fourier pseudospectral mti fp method for the dirac equation with a dimensionless parameter varepsilon in 0 1 which is inversely proportional to the speed of light in the nonrelativistic limit regime i e 0 varepsilon ll 1 the solution exhibits highly oscillatory propagating waves with wavelength o varepsilon 2 and o 1 in time and space respectively due to the rapid temporal oscillation it is quite challenging in designing and analyzing numerical methods with uniform error bounds in varepsilon in 0 1 we present the mti fp method based on properly adopting a multiscale decomposition of the solution of the dirac equation and applying the exponential wave integrator with appropriate numerical quadratures by a careful study of the error propagation and using the energy method we establish two independent error estimates via two different mathematical approaches as h m 0 frac tau 2 varepsilon 2 and h m 0 tau 2 varepsilon 2 where h is the mesh size tau is the time step and m 0 depends on the regularity of the solution these two error bounds immediately imply that the mti fp method converges uniformly and optimally in space with exponential convergence rate if the solution is smooth and uniformly in time with linear convergence rate at o tau for all varepsilon in 0 1 and optimally with quadratic convergence rate at o tau 2 in the regimes when either varepsilon o 1 or 0 varepsilon lesssim tau numerical results are reported to demonstrate that our error estimates are optimal and sharp finally the mti fp method is applied to study numerically the convergence rates of the solution of the dirac equation to those of its limiting models when varepsilon to0

single machine scheduling of unit time jobs with earliness and tardiness penalties the problem of determining a schedule of jobs with unit time lengths on a single machine that minimizes the total weighted earliness and tardiness penalties with respect to arbitrary rational due dates is formulated as an integer programming problem we show that if the penalties meet a certain criterion called the dominance condition then there exists an extremal optimal solution to the lp relaxation that is integral leading to a polynomial time solution procedure the general weighted symmetric penalty structure is one cost structure that satisfies the dominance condition we point out other commonly found penalty structures that also fall into this category

minimizing mean squared deviation of completion times about a common due date this paper addresses a nonpreemptive single machine scheduling problem where all jobs have a common due date and have zero ready time the scheduling objective is to minimize mean squared deviation msd of job completion times about the due date this nonregular measure of performance is appropriate when earliness and tardiness are both penalized and when large deviations of completion time from the due date are undesirable a special case of the msd problem referred to as the unconstrained msd problem is shown to be equivalent to the completion time variance problem ctv studied by merten and muller merten a g m e muller 1972 variance minimization in single machine sequencing problems management sci 18september 518 528 and schrage schrage l 1975 minimizing the time in system variance for a finite jobset management sci 21may 540 543 strong results for this latter problem are combined with several new propositions to develop a reasonably efficient procedure for solving the unconstrained msd problem this enables us to improve the existing procedures for the ctv problem we also propose a branching procedure for the constrained msd problem and present computational results

a numerical method with particle conservation for the maxwell dirac system a numerical method is presented for solving the maxwell dirac systems the maxwell equations with particle and current densities as the source terms are discretized explicitly to guarantee the particle conservation the dirac equations coupled electromagnetic potentials are discretized by the time splitting method and implicit finite difference these numerical schemes are conservative in particle density and have second order accuracy in time and space one dimensional numerical results are given to validate the accuracy and the conservation and three dimensional examples are presented to describe dynamical behaviors of the maxwell dirac system with several external potentials

on time splitting spectral approximations for the schrodinger equation in the semiclassical regime in this paper we study time splitting spectral approximations for the linear schrodinger equation in the semiclassical regime where the planck constant e is small in this regime the equation propagates oscillations with a wavelength of o e and finite difference approximations require the spatial mesh size h o e and the time step k o e in order to obtain physically correct observables much sharper mesh size constraints are necessary for a uniform l2 approximation of the wave function the spectral time splitting approximation under study will be proved to be unconditionally stable time reversible and gauge invariant it conserves the position density and gives uniform l2 approximation of the wave function for k o e and h o e extensive numerical examples in both one and two space dimensions and analytical considerations based on the wigner transform even show that weaker constraints e g k independent of e and h o e are admissible for obtaining correct observables finally we address the application to nonlinear schrodinger equations and conduct some numerical experiments to predict the corresponding admissible meshing strategies

one processor scheduling with symmetric earliness and tardiness penalties we consider one processor scheduling problems having the following form tasks t1 t2 tn are given with each ti having a specified length li and a preferred starting time ai or equivalently a preferred completion time bi the tasks are to be scheduled nonpreemptively i e a task cannot be split on a single processor to begin as close to their preferred starting times as possible we examine two different cost measures for such schedules the sum of the absolute discrepancies from the preferred starting times and the maximum such discrepancy for the first of these we show that the problem of finding minimum cost schedules is np complete however we give an efficient algorithm that finds minimum cost schedules whenever the tasks either all have the same length or are required to be executed in a given fixed sequence for the second cost measure we give an efficient algorithm that finds minimum cost schedules in general with no constraints on the ordering or lengths of the tasks

due dates assignment and jit scheduling with equal size jobs abstract this paper deals with due date assignment and just in time scheduling for single machine and parallel machine problems with equal size jobs where the objective is to minimize the total weighted earliness tardiness and due date cost these two problems but with a common due date to be calculated were shown to be polynomially solvable in o n 4 time we first show that this complexity can be reduced to o n 3 by modeling the single machine scheduling problem as an assignment problem without necessary due date enumeration we next prove that the general case with identical parallel machines and a given set of assignable due dates where the cardinality of this set is bounded by a constant number is still polynomially solvable

efficient and stable numerical methods for the generalized and vector zakharov system we present efficient and stable numerical methods for approximations of the generalized zakharov system gzs and vector zakharov system for multicomponent plasma vzsm with without a linear damping term the key points in the methods are based on i a time splitting discretization of a schrodinger type equation in gzs or vzsm ii discretizing a nonlinear wave type equation by a pseudospectral method for spatial derivatives and iii solving the ordinary differential equations odes in phase space analytically under appropriate chosen transmission conditions between different time intervals or applying crank nicolson leap frog for linear nonlinear terms for time derivatives the methods are explicit unconditionally stable and of spectral order accuracy in space and second order accuracy in time moreover they are time reversible and time transverse invariant when there is no damping term in gzs or vzsm conserve or keep the same decay rate of the wave energy as that in gzs or vzsm without a or with a linear damping term and give exact results for the plane wave solution extensive numerical tests are presented for plane waves and solitary wave collisions in one dimensional gzs and we also give the dynamics of three dimensional vzsm to demonstrate our new efficient and accurate numerical methods furthermore the methods are applied to study the convergence and quadratic convergence rates of vzsm to gzs and of gzs to the nonlinear schrodinger nls equation in the subsonic limit regime 0 varepsilon ll1 where the parameter varepsilon is inversely proportional to the acoustic speed our tests also suggest that the following meshing strategy or varepsilon resolution is admissible in this regime spatial mesh size h o varepsilon and time step k o varepsilon

an efficient and stable numerical method for the maxwell dirac system in this paper we present an explicit unconditionally stable and accurate numerical method for the maxwell dirac system md and use it to study dynamics of md as preparatory steps we take the three dimensional 3d maxwell dirac system scale it to obtain a two parameter model and review plane wave solution of free md then we present a time splitting spectral method tssp for md the key point in the numerical method is based on a time splitting discretization of the dirac system and to discretize nonlinear wave type equations by pseudospectral method for spatial derivatives and then solving the ordinary differential equations odes in phase space analytically under appropriate chosen transmission conditions between different time intervals the method is explicit unconditionally stable time reversible time transverse invariant and of spectral order accuracy in space and second order accuracy in time moreover it conserves the particle density exactly in discretized level and gives exact results for plane wave solution of free md extensive numerical tests are presented to confirm the above properties of the numerical method furthermore the tests also suggest the following meshing strategy or e resolution is admissible in the nonrelativistic limit regime 0 e 1 spatial mesh size h o e and time step t o e2 where the parameter e is inversely proportional to the speed of light

minimizing weighted mean absolute deviation of job completion times from their weighted mean we address a single machine scheduling problem where the objective is to minimize the weighted mean absolute deviation of job completion times from their weighted mean this problem and its precursors aim to achieve the maximum admissible level of service equity it has been shown earlier that the unweighted version of this problem is np hard in the ordinary sense for that version a pseudo polynomial time dynamic program and a 2 approximate algorithm are available however not much except for an important solution property exists for the weighted version in this paper we establish the relationship between the optimal solution to the weighted problem and a related one in which the deviations are measured from the weighted median rather than the mean of the job completion times this generalizes the 2 approximation result mentioned above we proceed to give a pseudo polynomial time dynamic program establishing the ordinary np hardness of the problem in general we then present a fully polynomial time approximation scheme as well finally we report the findings from a limited computational study on the heuristic solution of the general problem our results specialize easily to the unweighted case they also lead to an approximation of the set of schedules that are efficient with respect to both the weighted mean absolute deviation and the weighted mean completion time

a time splitting spectral scheme for the maxwell dirac system we present a time splitting spectral scheme for the maxwell dirac system and similar time splitting methods for the corresponding asymptotic problems in the semi classical and the non relativistic regimes the scheme for the maxwell dirac system conserves the lorentz gauge condition is unconditionally stable and highly efficient as our numerical examples show in particular we focus in our examples on the creation of positronic modes in the semi classical regime and on the electron positron interaction in the non relativistic regime furthermore in the non relativistic regime our numerical method exhibits uniform convergence in the small parameter d which is the ratio of the characteristic speed and the speed of light

minimizing weighted earliness tardiness and due date cost with unit processing time jobs abstract the classical single machine scheduling and due date assignment problem of panwalker et al panwalker s s smith m l seidmann a 1982 common due date assignment to minimize total penalty for the one machine scheduling problem operations research 30 2 1982 391 399 is the following all n jobs share a common due date which is to be determined jobs completed prior to or after the due date are penalized according to a cost function which is linear and job independent the objective is to minimize the total earliness tardiness and due date cost we study a generalized version of this problem in which i the earliness and tardiness costs are allowed to be job dependent and asymmetric and ii jobs are processed on parallel identical machines we focus on the case of unit processing time jobs the problem is shown to be solved in polynomial o n4 time then we study the special case with no due date cost a classical problem known in the literature as twet we introduce an o n3 solution for this case finally we study the minmax version of the problem i e the objective is to minimize the largest cost incurred by any of the jobs which is shown to be solved in polynomial time as well

galerkin method for unsplit 3 d dirac equation using atomically kinetically balanced b spline basis a galerkin method is developed to solve the time dependent dirac equation in prolate spheroidal coordinates for an electron molecular two center system the initial state is evaluated from a variational principle using a kinetic atomic balanced basis which allows for an efficient and accurate determination of the dirac spectrum and eigenfunctions b spline basis functions are used to obtain high accuracy this numerical method is used to compute the energy spectrum of the two center problem and then the evolution of eigenstate wavefunctions in an external electromagnetic field

on structure of upper semicontinuity abstract the refinement of a choquet theorem on strong upper semi continuity and its relation to the vainstein lemma are dealt with here relevance of subcontinuity is discussed consequently an improvement of a characterization theorem of dolecki and rolewicz is achieved

first class selectors for upper semi continuous multifunctions by a multtfunction x y we mean a function defined on the space x and whose values are subsets of the space y a point x0 in x is a point of upper semi continuity for provided x0 belongs to the interior of x q x c g whenever g is an open set in y containing x0 q is said to be upper semi continuous if it is upper semi continuous at ach point of x in this case x g x c g is open in x whenever g is open in y and x x nf is closed in x w h enever f is closed in y interchanging the roles of open and closed sets in this definition we get the concept of a lower semi continuous multifunction a selector for is a single valued function cp x y with q x e cd x for each x in x in this paper we are primarily concerned with the problem of determin ing the best possible selector cp for an upper semi continuous multifunction defined on a metric space x and taking only non empty values in a second metric space y very elementary examples show that such multi functions will in general fail to have continuous selectors e g take 4p x 0 foro x o 1 and x 1 for x 1 hence an optimal selector in our case would be one that is the pointwise limit of a sequence of continuous functions or more generally a first class bore1 function i e cp g is an 9 set in x for each open set g c y here we will prove

linear time algorithms for two and three variable linear programs o n time algorithms for linear programming problems with two or three variables and n constraints are described the approach uses convexity dominance of linear functions and linear time median finding algorithms the algorithms improve the previously known best bounds of o n log n time for both of these problems

a new approach to the maximum flow problem all previously known efficient maximum flow algorithms work by finding augmenting paths either one path at a time as in the original ford and fulkerson algorithm or all shortest length augmenting paths at once using the layered network approach of dinic an alternative method based on the preflow concept of karzanov is introduced a preflow is like a flow except that the total amount flowing into a vertex is allowed to exceed the total amount flowing out the method maintains a preflow in the original network and pushes local flow excess toward the sink along what are estimated to be shortest paths the algorithm and its analysis are simple and intuitive yet the algorithm runs as fast as any other known method on dense graphs achieving an o n 3 time bound on an n vertex graph by incorporating the dynamic tree data structure of sleator and tarjan we obtain a version of the algorithm running in o nm log n 2 m time on an n vertex m edge graph this is as fast as any known method for any graph density and faster on graphs of moderate density the algorithm also admits efficient distributed and parallel implementations a parallel implementation running in o n 2 log n time using n processors and o m space is obtained this time bound matches that of the shiloach vishkin algorithm which also uses n processors but requires o n 2 space

minimization of agreeably weighted variance in single machine systems abstract this paper considers the variance minimization problem with job dependent weights we show that an optimal job sequence must be v shaped in terms of weighted processing time when the problem is agreeably weighted in the sense that p i p j implies w i w j where p i and w i are the processing time and the weight of job i respectively an o nwp algorithm is proposed to find an optimal solution where n is the number of jobs w is the sum of weights and p is the sum of processing times furthermore an o np algorithm is derived to obtain a sub optimal solution the relative error of is guaranteed less than any pre specified amount and tends towards zero if w w min grows at a rate slower than n 3 as n increases where w min is the minimum weight particularly we show that in the special case where all weights are equal the o np algorithm can find a solution with relative error bounded above by 3 n 1 n 2 when n 2 numerical results are reported which indicate that the o np algorithm generated an optimal solution for almost every problem tested finally two fully polynomial approximation schemes are presented for problems in which the weights are bounded above by a polynomial in n the first one solves problems with p max 1 2 p while the second one solves problems with p max 1 2 p where p max is the maximum processing time

numerical methods for the generalized zakharov system we present two numerical methods for the approximation of the generalized zakharov system zs the first one is the time splitting spectral tssp method which is explicit time reversible and time transverse in variant if the generalized zs is keeps the same decay rate of the wave energy as that in the generalized zs gives exact results for the plane wave solution and is of spectral order accuracy in space and second order accuracy in time the second one is to use a local spectral method the discrete singular convolution dsc for spatial derivatives and the fourth order runge kutta rk4 for time integration which is of high the same as spectral order accuracy in space and can be applied to deal with general boundary conditions in order to test accuracy and stability we compare these two methods with other existing methods fourier pseudospectral method fps and wavelet galerkin method wg for spatial derivatives combining with the rk4 for time integration as well as the standard finite difference method fd for solving the zs with a solitary wave solution furthermore extensive numerical tests are presented for plane waves solitary wave collisions in 1d as well as a 2d problem of the generalized zs numerical results show that tssp and dsc are spectralorder accuracy in space and much more accurate than fd and for stability tssp requires k o h dsc rk4 requires k o h2 for fixed acoustic speed where k is the time step and h is the spatial mesh size