structure preserving approximation of dissipative evolution problems we present a framework for the systematic numerical approximation of nonlinear evolution problems with dissipation the approach is based on rewriting the problem in a canonical form that complies with the underlying energy dissipation structure we show that the corresponding weak formulation then allows for a dissipation preserving approximation by galerkin methods in space and discontinuous galerkin methods in time the proposed methodology is rather general and can be applied to a wide range of applications this is demonstrated by discussion of some typical examples ranging from diffusive partial differential equations to dissipative hamiltonian systems

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

approximation of pareto optima in multiple objective shortest path problems we study methods for approximating the set of pareto optimal paths in multiple objective shortest path problems known generalizations of standard shortest path methods will compute this set but can suffer from rapidly increasing computational and storage demands as problem size increases in an effort to avoid such difficulties we develop approximation methods that can estimate the pareto optima to any required degree of accuracy the approximation methods are fully polynomial that is they operate in time and space bounded by a polynomial in problem size and accuracy of approximation the greater the accuracy the more time required to reach a solution we show how approximation methods may be applied to yield fully polynomial approximation schemes for a variety of np complete single objective problems

min max min robust combinatorial optimization the idea of k adaptability in two stage robust optimization is to calculate a fixed number k of second stage policies here and now after the actual scenario is revealed the best of these policies is selected this idea leads to a min max min problem in this paper we consider the case where no first stage variables exist and propose to use this approach to solve combinatorial optimization problems with uncertainty in the objective function we investigate the complexity of this special case for convex uncertainty sets we first show that the min max min problem is as easy as the underlying certain problem if k is greater than the number of variables and if we can optimize a linear function over the uncertainty set in polynomial time we also provide an exact and practical oracle based algorithm to solve the latter problem for any underlying combinatorial problem on the other hand we prove that the min max min problem is np hard for every fixed number k even when the uncertainty set is a polyhedron given by an inner description for the case that k is smaller or equal to the number of variables we finally propose a fast heuristic algorithm and evaluate its performance

regularity of the moments of the solution of a transport equation let u u x v satisfy the transport equation u v xu f x rn v rnwhere f belongs to some space of type lp dx d v where is a positive bounded measure on rn we study the resulting regularity of the moment u x v d v in terms of fractional sobolev spaces for example counter examples are given in order to test the optimality of our results

bounded query classes polynomial time machines having restricted access to an np oracle are investigated restricted access means that the number of queries to the oracle is restricted or the way in which the queries are made is restricted e g queries made during truth table reductions very different kinds of such restrictions result in the same or comparable complexity classes in particular the class p text np o log n can be characterized in very different ways furthermore the boolean hierarchy is generalized in such a way that it is possible to characterize p text np and p text np o log n in terms of the generalization

solving multiobjective multiconstraint knapsack problems using mathematical programming and evolutionary algorithms abstract in this paper we solve instances of the multiobjective multiconstraint or multidimensional knapsack problem momckp from the literature with three objective functions and three constraints we use exact as well as approximate algorithms the exact algorithm is a properly modified version of the multicriteria branch and bound mcbb algorithm which is further customized by suitable heuristics three branching heuristics and a more general purpose composite branching and construction heuristic are devised comparison is made to the published results from another exact algorithm the adaptive e constraint method laumanns m thiele l zitzler e 2006 an efficient adaptive parameter variation scheme for metaheuristics based on the epsilon constraint method european journal of operational research 169 932 942 using the same data sets furthermore the same problems are solved using standard multiobjective evolutionary algorithms moea namely the spea2 and the nsgaii the results from the exact case show that the branching heuristics greatly improve the performance of the mcbb algorithm which becomes faster than the adaptive e constraint regarding the performance of the moea algorithms in the specific problems spea2 outperforms nsgaii in the degree of approximation of the pareto front as measured by the coverage metric especially for the largest instance

reference points and approximation algorithms in multicriteria discrete optimization abstract mathematical research on multicriteria optimization problems predominantly revolves around the set of pareto optimal solutions in practice on the other hand methods that output a single solution are more widespread reference point methods are a successful example of this approach and are widely used in real world multicriteria optimization a reference point solution is the solution closest to a given reference point in the objective space we study the connection between reference point methods and approximation algorithms for multicriteria optimization problems over discrete sets in particular we establish that in terms of computational complexity computing approximate reference point solutions is polynomially equivalent to approximating the pareto set complementing these results we show for a number of general algorithmic techniques in single criteria optimization how they can be lifted to reference point optimization in particular we lift the link between dynamic programming and fptas as well as certain lp rounding techniques the latter applies e g to set cover and several machine scheduling problems

an improved version of the augmented e constraint method augmecon2 for finding the exact pareto set in multi objective integer programming problems generation or a posteriori methods in multi objective mathematical programming momp is the most computationally demanding category among the momp approaches due to the dramatic increase in computational speed and the improvement of mathematical programming algorithms the generation methods become all the more attractive among today s decision makers in the current paper we present the generation method augmecon2 which is an improvement of our development augmecon although augmecon2 is a general purpose method we will demonstrate that augmecon2 is especially suitable for multi objective integer programming moip problems specifically augmecon2 is capable of producing the exact pareto set in moip problems by appropriately tuning its running parameters in this context we compare the previous and the new version in a series of new and old benchmarks found in the literature we also compare augmecon2 s performance in the generation of the exact pareto sets with established methods and algorithms based on specific moip problems knapsack set packing and on published results except from other mathematical programming methods augmecon2 is found to be competitive also with multi objective meta heuristics momh in producing adequate approximations of the pareto set in multi objective combinatorial optimization moco problems

weak entropy boundary conditions for isentropic gas dynamics via kinetic relaxation abstract we consider a kinetic bgk model relaxing to isentropic gas dynamics previously introduced by the authors but with dirichlet boundary condition on the incoming velocities we pass to the limit as the relaxation parameter tends to zero by compensated compactness inside the domain and obtain that the limit satisfies entropy inequalities on the boundary involving weak traces of entropy fluxes our method is very general and could be applied to any entropy satisfying bgk model as soon as we have strong compactness of the macroscopic variables inside the domain

min max and min max regret versions of combinatorial optimization problems a survey abstract min max and min max regret criteria are commonly used to define robust solutions after motivating the use of these criteria we present general results then we survey complexity results for the min max and min max regret versions of some combinatorial optimization problems shortest path spanning tree assignment min cut min s t cut knapsack since most of these problems are np hard we also investigate the approximability of these problems furthermore we present algorithms to solve these problems to optimality

analysis of a multiobjective evolutionary algorithm on the 0 1 knapsack problem multiobjective evolutionary algorithms moeas are increasingly being used for effectively solving many real world problems and many empirical results are available however theoretical analysis is limited to a few simple toy functions in this work we select the well known knapsack problem for the analysis the multiobjective knapsack problem in its general form is np complete moreover the size of the set of pareto optimal solutions can grow exponentially with the number of items in the knapsack thus we formalize a 1 e approximate set of the knapsack problem and attempt to present a rigorous running time analysis of a moea to obtain the formalized set the algorithm used in the paper is based on a restricted mating pool with a separate archive to store the remaining population we call the algorithm a restricted evolutionary multiobjective optimizer remo we also analyze the running time of remo on a special bi objective linear function known as lotz leading ones trailing zeros whose pareto set is shown to be a subset of the knapsack an extension of the analysis to the simple evolutionary multiobjective optimizer semo is also presented a strategy based on partitioning of the decision space into fitness layers is used for the analysis

divergence measure fields and hyperbolic conservation laws we analyze a class of l infty vector fields called divergence measure fields we establish the gauss green formula the normal traces over subsets of lipschitz boundaries and the product rule for this class of l infty fields then we apply this theory to analyze l infty entropy solutions of initial boundary value problems for hyperbolic conservation laws and to study the ways in which the solutions assume their initial and boundary data the examples of conservation laws include multidimensional scalar equations the system of nonlinear elasticity and a class of m x m systems with affine characteristic hypersurfaces the analysis in l infty also extends to l p

approximation algorithms for scheduling unrelated parallel machines we consider the following scheduling problem there arem parallel machines andn independent jobs each job is to be assigned to one of the machines the processing of jobj on machinei requires timep ij the objective is to find a schedule that minimizes the makespan our main result is a polynomial algorithm which constructs a schedule that is guaranteed to be no longer than twice the optimum we also present a polynomial approximation scheme for the case that the number of machines is fixed both approximation results are corollaries of a theorem about the relationship of a class of integer programming problems and their linear programming relaxations in particular we give a polynomial method to round the fractional extreme points of the linear program to integral points that nearly satisfy the constraints in contrast to our main result we prove that no polynomial algorithm can achieve a worst case ratio less than 3 2 unlessp np we finally obtain a complexity classification for all special cases with a fixed number of processing times

complexity restricted advice functions the authors consider uniform subclasses of the nonuniform complexity classes defined by karp and lipton l enseign math 28 1982 via the notion of advice functions these subclasses are obtained by restricting the complexity of computing correct advice also the effect of allowing advice functions of limited complexity to depend on the input rather than on the input s length is investigated among other results using the notions described above new characterizations of a np np cap sparse b np with a restricted access to an np oracle and c the odd levels of the boolean hierarchy are given as a consequence it is shown that every set that is nondeterministically truth table reducible to sat in the sense of rich j comput system sci 38 1989 pp 511 523 is already deterministically truth table reducible to sat furthermore it turns out that the np reduction classes of bounded versions of this reducibility coincide with the odd levels of the boolean hierarchy

approximately dominating representatives we propose and investigate from the algorithmic standpoint a novel form of fuzzy query called approximately dominating representatives or adrs the adrs of a multidimensional point set consist of a few points guaranteed to contain an approximate optimum of any monotone lipschitz continuous combining function of the dimensions adrs can be computed by appropriately post processing pareto or skyline queries kian lee tan pin kwang eng beng chin ooi efficient progressive skyline computation in vldb 2001 pp 301 310 wolf tilo balke ulrich guntzer jason xin zheng efficient distributed skylining for web information systems in edbt 2004 14 we show that the problem of minimizing the number of points returned for a user specified desired approximation can be solved in polynomial time in two dimensions for three and more it is np hard but has a polynomial time logarithmic approximation finally we present a polynomial time constant factor approximation algorithm for three dimensions

renormalized solutions to the vlasov equation with coefficients of bounded variation we prove that weak bounded solutions to the vlasov equation with bv coefficients have the renormalization property and we show that when the renormalization property holds for a general transport equation it also holds for only lipschitz nonlinearities

a convergent finite difference scheme for the navier stokes equations of one dimensional nonisentropic compressible flow convergence is proved and error bounds are derived for a finite difference approximation to discontinuous solutions of the navier stokes equations for nonisentropic compressible flow in one space dimension the scheme is fully implicit and can be implemented under reasonable mesh conditions it is shown that the approximations converge at the rate o delta x 1 2 when the initial data is in h 1 and o delta x a a frac 1 12 when the initial velocity and energy are in l 2 and the initial density is piecewise h 1 the errors are measured in a norm that dominates the sup norm of the error in the density which in general is discontinuous this choice is dictated by the known continuous dependence theory and accounts for the low rate of convergence in the second case

a new bound for the midpoint solution in minmax regret optimization with an application to the robust shortest path problem abstract minmax regret optimization aims at finding robust solutions that perform best in the worst case compared to the respective optimum objective value in each scenario even for simple uncertainty sets like boxes most polynomially solvable optimization problems have strongly np complete minmax regret counterparts thus heuristics with performance guarantees can potentially be of great value but only few such guarantees exist a popular heuristic for combinatorial optimization problems is to compute the midpoint solution of the original problem it is a well known result that the regret of the midpoint solution is at most 2 times the optimal regret besides some academic instances showing that this bound is tight most instances reveal a way better approximation ratio we introduce a new lower bound for the optimal value of the minmax regret problem using this lower bound we state an algorithm that gives an instance dependent performance guarantee for the midpoint solution that is at most 2 the computational complexity of the algorithm depends on the minmax regret problem under consideration we show that our sharpened guarantee can be computed in strongly polynomial time for several classes of combinatorial optimization problems to illustrate the quality of the proposed bound we use it within a branch and bound framework for the robust shortest path problem in an experimental study comparing this approach with a bound from the literature we find a considerable improvement in computation times

on the approximability of robust spanning tree problems in this paper the minimum spanning tree problem with uncertain edge costs is discussed in order to model the uncertainty a discrete scenario set is specified and a robust framework is adopted to choose a solution the min max min max regret and 2 stage min max versions of the problem are discussed the complexity and approximability of all these problems are explored it is proved that the min max and min max regret versions with nonnegative edge costs are hard to approximate within o log 1 en for any e 0 unless the problems in np have quasi polynomial time algorithms similarly the 2 stage min max problem cannot be approximated within o logn unless the problems in np have quasi polynomial time algorithms in this paper randomized lp based approximation algorithms with performance bound of o log 2n for min max and 2 stage min max problems are also proposed

minmax regret combinatorial optimization problems with ellipsoidal uncertainty sets we consider robust counterparts of uncertain combinatorial optimization problems where the difference to the best possible solution over all scenarios is to be minimized such minmax regret problems are typically harder to solve than their nominal non robust counterparts while current literature almost exclusively focuses on simple uncertainty sets that are either finite or hyperboxes we consider problems with more flexible and realistic ellipsoidal uncertainty sets we present complexity results for the unconstrained combinatorial optimization problem the shortest path problem and the minimum spanning tree problem to solve such problems two types of cuts are introduced and compared in a computational experiment

convergence of the viscosity method for isentropic gas dynamics a convergence theorem for the method of artificial viscosity applied to the isentropic equations of gas dynamics is established convergence of a subsequence in the strong topology is proved without uniform estimates on the derivatives using the theory of compensated compactness and an analysis of progressing entropy waves

the boolean hierarchy i structural properties in this paper we study the complexity of sets formed by boolean operations union intersection and complement on np sets these are the sets accepted by trees of hardware with np predicates as leaves and together these form the boolean hierarchy we present many results about the structure of the boolean hierarchy separation and immunity results natural complete languages and structural asymmetries between complementary classes we show that in some relativized worlds the boolean hierarchy is infinite and that for every k there is a relativized world in which the boolean hierarchy extends exactly k levels we prove natural languages variations of vertex cover complete for the various levels of the boolean hierarchy we show the following structural asymmetry though no set in the boolean hierarchy is text d text p immune there is a relativized world in which the boolean hierarchy contains text cod text p immune sets thus this paper explores the structural properties of the

single approximation for the biobjective max tsp we mainly study the max tsp with two objective functions we propose an algorithm which returns a single hamiltonian cycle with performance guarantee on both objectives the algorithm is analyzed in three cases when both respectively at least one objective function s fulfill s the triangle inequality the approximation ratio is 512 e 0 41 respectively 38 e when the triangle inequality is not assumed on any objective function the algorithm is 1 2214 e 0 27 approximate

min max regret version of a scheduling problem with outsourcing decisions under processing time uncertainty abstract we consider the min max regret version of a single machine scheduling problem to determine which jobs are processed by outsourcing under processing time uncertainty the performance measure is expressed as the total cost for processing some jobs in house and outsourcing the rest processing time uncertainty is described through two types of scenarios either an interval scenario or a discrete scenario the objective is to minimize the maximum deviation from optimality over all scenarios we show that when the cost for in house jobs is expressed as the makespan the problem with an interval scenario is polynomially solvable while the one with a discrete scenario is np hard thus for the discrete scenario case we develop a 2 approximation algorithm and investigate when the problem is polynomially solvable since the problem minimizing the total completion time as a performance measure for in house jobs is known to be np hard for both scenarios we consider the problem with a special structure for the processing time uncertainty and develop a polynomial time algorithm for both scenarios

it cemop an iterative co evolutionary algorithm for multiobjective optimization problem with nonlinear constraints over the past few years researchers have developed a number of multiobjective evolutionary algorithms moeas although most studies concentrate on solving unconstrained optimization problems there exit a few studies where moeas have been extended to solve constrained optimization problems most of them were based on penalty functions for handling nonlinear constraints by genetic algorithms however the performance of these methods is highly problem dependent many methods require additional tuning of several parameters in this paper we present a new optimization algorithm which is based on concept of co evolution and repair algorithm for handling nonlinear constraints the algorithm maintains a finite sized archive of nondominated solutions which gets iteratively updated in the presence of new solutions based on the concept of e dominance the use of e dominance also makes the algorithms practical by allowing a decision maker to control the resolution of the pareto set approximation by choosing an appropriate e value which guarantees convergence and diversity the results provided by the proposed algorithm for six benchmark problems are promising when compared with exiting well known algorithms also our results suggest that our algorithm is better applicable for solving real world application problems

approximation of min max and min max regret versions of some combinatorial optimization problems this paper investigates for the first time in the literature the approximation of min max regret versions of classical problems like shortest path minimum spanning tree and knapsack for a constant number of scenarios we establish fully polynomial time approximation schemes for the min max versions of these problems using relationships between multi objective and min max optimization using dynamic programming and classical trimming techniques we construct a fully polynomial time approximation scheme for min max regret shortest path we also establish a fully polynomial time approximation scheme for min max regret spanning tree and prove that min max regret knapsack is not at all approximable for a non constant number of scenarios in which case min max and min max regret versions of polynomial time solvable problems usually become strongly np hard non approximability results are provided for min max regret versions of shortest path and spanning tree

approximation algorithms for combinatorial fractional programming problems we are concerned with a combinatorial optimization problem which has the ratio of two linear functions as the objective function this type of problems can be solved by an algorithm that uses an auxiliary problem with a parametrized linear objective function because of its combinatorial nature however it is often difficult to solve the auxiliary problem exactly in this paper we propose an algorithm which assumes that the auxiliary problems are solved only approximately and prove that it gives an approximate solution to the original problem of which the accuracy is at least as good as that of approximate solutions to the auxiliary problems it is also shown that the time complexity is bounded by the square of the computation time of the approximate algorithm for the auxiliary problem as an example of the proposed algorithm we present a fully polynomial time approximation scheme for the fractional 0 1 knapsack problem

coupling conditions for networked systems of euler equations we are interested in the dynamics of the euler equations on networks we study the situation of a single vertex where the dynamics on the connected arcs are given by the polytropic euler equations such problems arise for example in the context of gas transport in pipe networks for the euler equations in one spatial dimension coupling conditions for the vertex have been proposed recently we present further conditions and introduce a two dimensional representation of the vertex we conduct numerical simulations and compare the obtained results with the theoretical predictions

existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in eulerian and lagrangian coordinates we prove the existence and compactness stability of entropy solutions for the hyperbolic systems of conservation laws corresponding to the isentropic gas dynamics where the pressure and density are related by a law for any 1 our results considerably extend and simplify the program initiated by diperna and provide a complete existence proof our methods are based on the compensated compactness and the kinetic formulation of systems of conservation laws 1996 john wiley sons inc

kinetic relaxation to entropy based coupling conditions for isentropic flow on networks abstract we consider networks for isentropic gas and prove existence of weak solutions for a large class of coupling conditions first we construct approximate solutions by a vector valued bgk model with a kinetic coupling function introducing so called kinetic invariant domains and using the method of compensated compactness justifies the relaxation towards the isentropic gas equations we will prove that certain entropy flux inequalities for the kinetic coupling function remain true for the traces of the macroscopic solution these inequalities define the macroscopic coupling condition our techniques are also applicable to networks with arbitrary many junctions which may possibly contain circles we give several examples for coupling functions and prove corresponding entropy flux inequalities we prove also new existence results for solid wall boundary conditions and pipelines with discontinuous cross sectional area

existence and uniqueness of solutions to the generalized riemann problem for isentropic flow we consider network models for isentropic flow and evaluate the application of three different momentum related coupling constants pressure momentum flux and the bernoulli invariant subsonic solutions to generalized riemann problems with subsonic initial conditions are proved to be unique and the region where such initial conditions yield subsonic solutions is identified the proof is restricted to monotone coupling constants we also prove that the three momentum related coupling constants fulfill the monotonicity constraint further we analyze the existence of entropic solutions using a commonly applied entropy condition the condition states that nonphysical solutions are characterized by production of mechanical energy at a junction both pressure and momentum flux are seen to yield nonphysical solutions in a constructed test case consisting of three pipe sections of equal cross sectional area that are connected at a junction the bernoulli invariant is proved to yield entropic solutions for all

construction of bgk models with a family of kinetic entropies for a given system of conservation laws we introduce a general framework for kinetic bgk models we assume to be given a system of hyperbolic conservation laws with a family of lax entropies and we characterize the bgk models that lead to this system in the hydrodynamic limit and that are compatible with the whole family of entropies this is obtained by a new characterization of maxwellians as entropy minimizers that can take into account the simultaneous minimization problems corresponding to the family of entropies we deduce a general procedure to construct such bgk models and we show how classical examples enter the framework we apply our theory to isentropic gas dynamics and full gas dynamics and in both cases we obtain new bgk models satisfying all entropy inequalities

initial boundary value problem for conservation laws this paper concerns the initial boundary value problems for some systems of quasilinear hyperbolic conservation laws in the space of bounded measurable functions the main assumption is that the system under study admits a convex entropy extension it is proved that then any twicely differentiable entropy fluxes have traces on the boundary if the bounded solutions are generated by either godunov schemes or by suitable viscous approximations furthermore in the case that the weak interior solutions are generated by godunov schemes any lipschitz continuous entropy fluxes corresponding to convex entropies have traces on the boundary and the traces are bounded above by computable numerical boundary values this in particular gives a trace formula for the flux functions in terms of the numerical boundary data we also investigate the formulation of boundary conditions for systems of hyperbolic conservation laws it is shown that the set of expected boundary values derived from the viscous approximation contains the one derived in terms of the boundary riemann problems and the converse is not true in general the general theory is then applied to some specific examples first several new facts are obtained for convex scalar conservation laws for example we give example which show that godunov schemes produce numerical boundary layers it is shown that any continuous functions of density have traces on the boundary instead of only entropy fluxes we also obtain interior and boundary regularity of the weak solutions for bounded measurable initial and boundary data a generalized oleinik entropy condition is also obtained next we prove the existence of a weak solution to the initial boundary value problem for a family of quadratic system with a uniformly characteristic boundary condition

stochastic convergence of random search methods to fixed size pareto front approximations in this paper we investigate to what extent random search methods equipped with an archive of bounded size to store a limited amount of solutions and other data are able to obtain good pareto front approximations we propose and analyze two archiving schemes that allow for maintaining a sequence of solution sets of given cardinality that converge with probability one to an pareto set of a certain quality under very mild assumptions on the process used to sample new solutions the first algorithm uses a hierarchical grid to define a family of approximate dominance relations to compare solutions and solution sets acceptance of a new solution is based on a potential function that counts the number of occupied boxes on various levels and thus maintains a strictly monotonous progress to a limit set that covers the pareto front with non overlapping boxes at finest resolution possible the second algorithm uses an adaptation scheme to modify the current value of based on the information gathered during the run this way it will be possible to achieve convergence to the best smallest value and to a corresponding solution set of k solutions that dominate all other solutions which is probably the best possible result regarding the limit behavior of random search methods or metaheuristics for obtaining pareto front approximations

approximating a class of combinatorial problems with rational objective function in the late seventies megiddo proposed a way to use an algorithm for the problem of minimizing a linear function a 0 a 1 x 1 a n x n subject to certain constraints to solve the problem of minimizing a rational function of the form a 0 a 1 x 1 a n x n b 0 b 1 x 1 b n x n subject to the same set of constraints assuming that the denominator is always positive using a rather strong assumption hashizume et al extended megiddo s result to include approximation algorithms their assumption essentially asks for the existence of good approximation algorithms for optimization problems with possibly negative coefficients in the linear objective function which is rather unusual for most combinatorial problems in this paper we present an alternative extension of megiddo s result for approximations that avoids this issue and applies to a large class of optimization problems specifically we show that if there is an approximation for the problem of minimizing a nonnegative linear function subject to constraints satisfying a certain increasing property then there is an approximation 1 approximation for the problem of minimizing maximizing a nonnegative rational function subject to the same constraints our framework applies to covering problems and network design problems among others

fuzzy approach to multilevel knapsack problems this study proposes a fuzzy approach for solving the multiobjective and multilevel knapsack problems kps the problem was first formulated as a multilevel programming problem with multiple decision makers dms then the degree of satisfaction of each dm was established and represented by their individual membership functions the recursive formulation of dynamic programming was used to solve the decisions of the interrelated stages the overall satisfaction of the decision was obtained through this stage wise operation on the hierarchical structure capacity allocation was developed and a step by step solution procedure was illustrated a detailed comparison between multiobjective and multilevel kps was also carried out finally the possible use of turnpike theorem in kps was scrutinized in the fuzzy domain

on the robust knapsack problem we consider an uncertain variant of the knapsack problem that arises when the exact weight of each item is not exactly known in advance but belongs to a given interval and the number of items whose weight differs from the nominal value is bounded by a constant we analyze the worsening of the optimal solution value with respect to the classical problem and exactly determine its worst case performance depending on uncertainty for all parameter configurations we perform the same analysis for the fractional version of the problem in which one is allowed to pack any fraction of the items in addition we derive the worst case performance ratio with respect to the optimal solution value for both the fractional problem and for a variant of the well known greedy algorithm finally we consider a relevant special case and provide a combinatorial algorithm for solving the fractional problem in an efficient way

quadratic programming and combinatorial minimum weight product problems we present a fully polynomial time approximation scheme fptas for minimizing an objective a t x b t x under linear constraints a x d examples of such problems are combinatorial minimum weight product problems such as the following given a graph g v e and two edge weights a b e rightarrow mathbb r find an s t path p that minimizes a p b p the product of its edge weights relative to a and b

a general framework for designing approximation schemes for combinatorial optimization problems with many objectives combined into one in this paper we present a general framework for designing approximation schemes for combinatorial optimization problems in which the objective function is a combination of more than one function examples of such problems include those in which the objective function is a product or ratio of two linear functions parallel machine scheduling problems with the makespan objective robust versions of weighted multiobjective optimization problems and assortment optimization problems with logit choice models the main idea behind our approximation schemes is the construction of an approximate pareto optimal frontier of the functions that constitute the given objective using this idea we give the first fully polynomial time approximation schemes for the max min resource allocation problem with a fixed number of agents combinatorial optimization problems in which the objective function is the sum of a fixed number of ratios of linear functions or the product of a fixed number of linear functions and assor

approximation algorithms for the m dimensional 0 1 knapsack problem worst case and probabilistic analyses abstract we describe a polynomial approximation scheme for an m constraint 0 1 integer programming problem m fixed based on the use of the dual simplex algorithm for linear programming we also analyse the asymptotic properties of a particular random model

a fully polynomial time approximation scheme for single item stochastic inventory control with discrete demand the single item stochastic inventory control problem is to find an inventory replenishment policy in the presence of independent discrete stochastic demands under periodic review and finite time horizon in this paper we prove that this problem is intractable and design for it a fully polynomial time approximation scheme

a unified approach to uncertain optimization abstract in this paper we consider uncertain scalar optimization problems with infinite scenario sets we apply methods from vector optimization in general spaces set valued optimization and scalarization techniques to develop a unified characterization of different concepts of robust optimization and stochastic programming these methods provide new insights on the interrelation between different concepts for handling uncertainties and naturally lead to new concepts of robustness

a fully polynomial approximation algorithm for the 0 1 knapsack problem abstract a fully polynomial approximation algorithm is developed for the 0 1 knapsack problem the algorithm uses results of lawler and ibarra and kim a pseudo polynomial dynamic programming algorithm is first suggested which solves the problem in o nb log n time and o b space

erratum the polynomial time hierarchy collapses if the boolean hierarchy collapses it is shown that if the boolean hierarchy bh collapses then there exists a sparse set s such that text co np subseteq text np s and therefore the polynomial time hierarchy ph col

a finite difference scheme for the navier stokes equations of one dimensional isentropic compressible flow convergence is proved and error bounds are derived for a finite difference approximation to discontinuous solutions of the navier stokes equations for one dimensional isentropic compressible flow this scheme can be implemented under mesh conditions essentially equivalent to the usual cfl condition it is shown that for initial data which is by and piecewise smooth the error is bounded by o delta x 1 6 in a norm which dominates the sup norm of the density and is bounded by o delta x 1 2 when the initial data is hl

priority based e dominance dominance is a fundamental concept in multiobjective optimization moo where a solution is said to dominate the other if it is better in at least one objective and same as or better in the other objectives given a pair of solutions if neither dominates the other then the solutions are said to be non dominated in real spaces a large number of non dominated solutions can exist in principle making convergence of moo techniques rather slow relaxed forms of dominance have been proposed in the literature for faster convergence one such form is the dominance which in principle partitions the space into sized grids the value of is often user defined and the performance of dominance based multiobjective evolutionary algorithms moeas critically depends on the value of in this article we propose a novel way of determining the value of which is different for each objective function based on the correlation between them we call this approach priority based pbe as the method ranks the objectives according to their priority before calculating the values pbe is incorporated in an archived simulated annealing based moo technique called amosa amosa has been earlier shown to outperform several well known moo techniques especially for many objective optimization pbe based amosa referred to as pbe amosa is found to comprehensively outperform amosa moea d de the conventional based version amosa and moea both in case of benchmark test problems and 0 1 knapsack problem

a robust conservative mixed finite element method for isentropic compressible flow on pipe networks we consider the numerical approximation of compressible flow in a pipe network appropriate coupling conditions are formulated that allow us to derive a variational characterization of solutions an

approximating multiobjective knapsack problems for multiobjective optimization problems it is meaningful to compute a set of solutions covering all possible trade offs between the different objectives the multiobjective knapsack problem is a generalization of the classical knapsack problem in which each item has several profit values for this problem efficient algorithms for computing a provably good approximation to the set of all nondominated feasible solutions the pareto frontier are studied for the multiobjective one dimensional knapsack problem a practical fully polynomial time approximation scheme fptas is derived it is based on a new approach to the single objective knapsack problem using a partition of the profit space into intervals of exponentially increasing length for the multiobjectivem dimensional knapsack problem the first known polynomial time approximation scheme ptas based on linear programming is presented

choosing robust solutions in discrete optimization problems with fuzzy costs in this paper a wide class of discrete optimization problems which can be formulated as a 0 1 linear programming problem is discussed it is assumed that the objective function costs are not precisely known this uncertainty is modeled by specifying a finite set of fuzzy scenarios under every fuzzy scenario the costs are given as fuzzy intervals possibility theory is then applied to chose a solution in such a problem and mixed integer linear programming models are proposed to compute this solution

on multidimensional packing problems we study the approximability of multidimensional generalizations of three classical packing problems multiprocessor scheduling bin packing and the knapsack problem specifically we study the vector scheduling problem its dual problem namely the vector bin packing problem and a class of packing integer programs the vector scheduling problem is to schedule n d dimensional tasks on m machines such that the maximum load over all dimensions and all machines is minimized the vector bin packing problem on the other hand seeks to minimize the number of bins needed to schedule all n tasks such that the maximum load on any dimension across all bins is bounded by a fixed quantity say 1 such problems naturally arise when scheduling tasks that have multiple resource requirements finally packing integer programs capture a core problem that directly relates to both vector scheduling and vector bin packing namely the problem of packing a maximum number of vectors in a single bin of unit height we obtain a variety of new algorithmic as well as inapproximability results for these three problems

small approximate pareto sets for biobjective shortest paths and other problems we investigate the problem of computing a minimum set of solutions that approximates within a specified accuracy epsilon the pareto curve of a multiobjective optimization problem we show that for a broad class of biobjective problems containing many important widely studied problems such as shortest paths spanning tree matching and many others we can compute in polynomial time an epsilon pareto set that contains at most twice as many solutions as the minimum set furthermore we show that the factor of 2 is tight for these problems i e it is np hard to do better we present upper and lower bounds for three or more objectives as well as for the dual problem of computing a specified number k of solutions which provide a good approximation to the pareto curve

boundary conditions for nonlinear hyperbolic systems of conservation laws we propose two formulations of the boundary conditions for nonlinear hyperbolic systems of conservation laws a first approach is based on the vanishing viscosity method and a second one is related to the riemann problem the equivalence between these two conditions is studied the latter formulation is extended to treat numerically physically relevant boundary conditions monodimensional experiments are presented

bounded queries to sat and the boolean hierarchy we study the complexity of decision problems that can be solved by a polynomial time turing machine that makes a bounded number of queries to an np oracle depending on whether we allow some queries to depend on the results of other queries we obtain two probably different hierarchies we present several results relating the bounded np query hierarchies to each other and to the boolean hierarchy we also consider the similarly defined hierarchies of functions that can be computed by a polynomial time turing machine that makes a bounded number of queries to an np oracle we present relations among these two hierarchies and the boolean hierarchy in particular we show for all k that there are functions computable with 2k parallel queries to an np set that are not computable in polynomial time with k serial queries to any oracle unless p np as a corollary k 1 parallel queries to an np set allow us to compute more functions than are computable with only k parallel queries to an np set unless p np the same is true of serial queries similar results hold for all tt self reducible sets using a mind change technique we show that 2k 1 parallel queries to an np set allow us to accept in polynomial time exactly the same sets as can be accepted in polynomial time with k serial queries to an np set in fact the same is true for any class in place of np that is closed under polynomial time positive bounded truth table reductions this contrasts with the expected result for function computations with an np oracle beigel 1988 in addition we show that the boolean hierarchy and the bounded query hierarchies of languages either stand or collapse together finally we show that if the boolean hierarchy collapses to any level but the zeroth deterministic polynomial time then for all k there are functions computable in polynomial time with k parallel queries to an np set that are not computable in polynomial time with k 1 serial queries to any set np complete sets are p superterse

implementing an efficient fptas for the 0 1 multi objective knapsack problem abstract in the present work we are interested in the practical behavior of a new fully polynomial time approximation schemes fptas to solve the approximation version of the 0 1 multi objective knapsack problem the proposed methodology makes use of very general techniques such as dominance relations in dynamic programming and thus may be applicable in the implementation of fptas for other problems as well extensive numerical experiments on various types of instances in the bi and tri objective cases establish that our method performs very well both in terms of cpu time and size of solved instances we point out some reasons for the good practical performance of our algorithm a comparison with an exact method and the fptas proposed in erlebach t kellerer h pferschy u 2002 approximating multiobjective knapsack problems management science 48 12 1603 1612 is also performed

restricted information from nonadaptive queries to np we investigate classes of sets that can be decided by bounded truth table reductions to an np set in which evaluators donothave full access to the answers to the queries but get only restricted information such as the number of queries that are in the oracle set or even just this number modulom for somem 2 we also investigate the case in which evaluators are nondeterministic we show that when we vary the information that the evaluators get this can change the resulting power of the evaluators we locate all these classes within levels of the boolean hierarchy which allows us to compare the complexity of such classes