a dynamic programming based reduction procedure for the multidimensional 0 1 knapsack problem abstract this paper presents a preprocessing procedure for the 0 1 multidimensional knapsack problem first a non increasing sequence of upper bounds is generated by solving lp relaxations then a non decreasing sequence of lower bounds is built using dynamic programming the comparison of the two sequences allows either to prove that the best feasible solution obtained is optimal or to fix a subset of variables to their optimal values in addition a heuristic solution is obtained computational experiments with a set of large scale instances show the efficiency of our reduction scheme particularly it is shown that our approach allows to reduce the cpu time of a leading commercial software

capacity constraints across nests in assortment optimization under the nested logit model we consider assortment optimization problems when customers choose according to the nested logit model and there is a capacity constraint limiting the total capacity consumption of all products offered in all nests when each product consumes one unit of capacity our capacity constraint limits the cardinality of the offered assortment for the cardinality constrained case we develop an efficient algorithm to compute the optimal assortment when the capacity consumption of each product is arbitrary we give an algorithm to obtain a 4 approximate solution we show that we can compute an upper bound on the optimal expected revenue for an individual problem instance by solving a linear program in our numerical experiments we consider problem instances involving products with arbitrary capacity consumptions comparing the expected revenues from the assortments obtained by our 4 approximation algorithm with the upper bounds on the optimal expected revenues our numerical results indicate that the 4 approxim

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 multiple knapsack problem the multiple knapsack problem mkp is a natural and well known generalization of the single knapsack problem and is defined as follows we are given a set of n items and m bins knapsacks such that each item i has a profit p i and a size s i and each bin j has a capacity c j the goal is to find a subset of items of maximum profit such that they have a feasible packing in the bins mkp is a special case of the generalized assignment problem gap where the profit and the size of an item can vary based on the specific bin that it is assigned to gap is apx hard and a 2 approximation for it is implicit in the work of shmoys and tardos math program a 62 1993 pp 461 474 and thus far this was also the best known approximation for mkp the main result of this paper is a polynomial time approximation scheme ptas for mkp apart from its inherent theoretical interest as a common generalization of the well studied knapsack and bin packing problems it appears to be the strongest special case of gap that is not apx hard we substantiate this by showing that slight generalizations of mkp are apx hard thus our results help demarcate the boundary at which instances of gap become apx hard an interesting aspect of our approach is a ptas preserving reduction from an arbitrary instance of mkp to an instance with o log n distinct sizes and profits

a population based approach for hard global optimization problems based on dissimilarity measures when dealing with extremely hard global optimization problems i e problems with a large number of variables and a huge number of local optima heuristic procedures are the only possible choice in this situation lacking any possibility of guaranteeing global optimality for most problem instances it is quite difficult to establish rules for discriminating among different algorithms we think that in order to judge the quality of new global optimization methods different criteria might be adopted like e g ol gt efficiency measured in terms of the computational effort necessary to obtain the putative global optimum robustness measured in terms of percentage of successes i e of the number of times the algorithm re started with different seeds or starting points is able to end up at the putative global optimum discovery capability measured in terms of the possibility that an algorithm discovers for the first time a putative optimum for a given problem which is better than the best known up to now ol gt of course the third criterion cannot be considered as a compulsory one as it might be the case that for a given problem the best known putative global optimum is indeed the global one so that no algorithm will ever be able to discover a better one in this paper we present a computational framework based on a population based stochastic method in which different candidate solutions for a single problem are maintained in a population which evolves in such a way as to guarantee a sufficient diversity among solutions this diversity enforcement is obtained through the definition of a dissimilarity measure whose definition is dependent on the specific problem class we show in the paper that for some well known and particularly hard test classes the proposed method satisfies the above criteria in that it is both much more efficient and robust when compared with other published approaches moreover for the very hard problem of determining the minimum energy conformation of a cluster of particles which interact through short range morse potential our approach was able to discover four new putative optima

running errands in time approximation algorithms for stochastic orienteering in the stochastic orienteering problem we are given a finite metric space where each node contains a job with some deterministic reward and a random processing time the processing time distributions are known and independent across nodes however the actual processing time of a job is not known until it is completely processed the objective is to compute a nonanticipatory policy to visit nodes and run the corresponding jobs so as to maximize the total expected reward subject to the total distance traveled plus the total processing time being at most a given budget of b this problem combines aspects of the stochastic knapsack problem with uncertain item sizes as well as the deterministic orienteering problem in this paper we consider both nonadaptive and adaptive policies for stochastic orienteering we present a constant factor approximation algorithm for the nonadaptive version and an olog log b approximation algorithm for the adaptive version we extend both these results to directed metrics and a more general sequence orienteering problem finally we address the stochastic orienteering problem when the node rewards are also random and possibly correlated with the processing time and obtain an olog n log b approximation algorithm here n is the number of nodes in the metric all our results for adaptive policies also bound the corresponding adaptivity gaps

local and global lifted cover inequalities for the 0 1 multidimensional knapsack problem abstract the 0 1 multidimensional knapsack problem 0 1 mkp is a well known and strongly np hard combinatorial optimization problem with many applications up to now the majority of upper bounding techniques for the 0 1 mkp have been based on lagrangian or surrogate relaxation we show that good upper bounds can be obtained by a cutting plane method based on lifted cover inequalities lcis as well as using traditional lcis we use some new global lcis which take the whole constraint matrix into account

nonlinear oblique boundary value problems for nonlinear elliptic equations on considere le probleme aux valeurs limites a derivee oblique non lineaire pour des equations uniformement elliptiques quasi lineaires et totalement non lineaires d ordre 2 les operateurs elliptiques satisfont des conditions de structure naturelle on donne des theoremes d existence pour les solutions

the asymptotic value to capacity ratio for the multi class stochastic knapsack problem we consider a stochastic knapsack problem that packs multiple classes of random items the pairs of profit and resource requirement for items of the same class are independent and identically distributed however such pairs for different classes of items are independent but not necessarily identically distributed we investigate convergence and the asymptotic value of the ratio of the knapsack value function to the capacity when the capacity increases we consider two growth models for the multi class stochastic knapsack problem one where the number of available items grows proportionally to the capacity and the other one where the available items are sampled until their total resource requirement does not exceed the capacity by extending the results of meanti et al on a single class of random items we show that for each growth model the ratio converges almost surely to a computable constant for special cases where resource requirements or profit coefficients are deterministic we present simple interpretations on the asymptotic values finally we present an exponential order upperbound on the tail probability of the ratio

two dimensional online bin packing with rotation in two dimensional bin packing problems the input items are rectangles which need to be packed in a non overlapping manner the goal is to assign the items into unit squares using an axis parallel packing most previous work on online packing concentrated on items of fixed orientation which must be assigned such that their bottom side is parallel to the bottom of the bin in this paper we study the case of rotatable items which can be rotated by ninety degrees we give almost tight bounds on the asymptotic competitive ratio of bounded space bin packing of rotatable items and introduce a new unbounded space algorithm this improves the results of fujita and hada

coordination complexity of parallel price directive decomposition the general block angular convex resource sharing problem in k blocks and m nonnegative block separable coupling constraints is considered applications of this model are in combinatorial optimization network flows scheduling communication networks engineering design and finance this paper studies the coordination complexity of approximate price directive decomposition pdd for this problem i e the number of iterations required to solve the problem to a fixed relative accuracy as a function of k and m first a simple pdd method based on the classical logarithmic potential is shown to be optimal up to a logarithmic factor in m in the class of all pdd methods that work with the original unrestricted blocks it is then shown that logarithmic and exponential potentials generate a polylogarithmically optimal algorithm for a wider class of pdd methods which can restrict the blocks by the coupling constraints as an application the fastest currently known deterministic approximation algorithm for minimum cost multicommodity flows is obtained

global solutions of shock reflection by large angle wedges for potential flow when a plane shock hits a wedge head on it experiences a reflectiondiffraction process and then a self similar reflected shock moves outward as the original shock moves forward in time experimental computational and asymptotic analysis has shown that various patterns of shock reflection may occur including regular and mach reflection however most of the fundamental issues for shock reflection have not been understood including the global structure stability and transition of the different patterns of shock reflection therefore it is essential to establish the global existence and structural stability of solutions of shock reflection in order to understand fully the phenomena of shock reflection on the other hand there has been no rigorous mathematical result on the global existence and structural stability of shock reflection including the case of potential flow which is widely used in aerodynamics such problems involve several challenging difficulties in the analysis of nonlinear partial differential equations such as mixed equations of elliptic hyperbolic type free boundary problems and corner singularity where an elliptic degenerate curve meets a free boundary in this paper we develop a rigorous mathematical approach to overcome these difficulties i nvolved and establish a global theory of existence and stability for shock reflection by large angle wedges for potential flow the techniques and ideas developed here will be useful for other nonlinear problems involving similar difficulties

probabilistic properties of the dual structure of the multidimensional knapsack problem and fast statistically efficient algorithms properties of several dual characteristics of the multidimensional knapsack problem such as the probability of existence of optimal and optimal feasible lagrange function generalized saddle points magnitude of relative duality gap etc are investigated for different probabilistic models sufficient conditions of good asymptotic behavior of the dual characteristics are given a fast statistically efficient approximate algorithm with linear running time complexity for problems with random coefficients is presented

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

approximate max min resource sharing for structured concave optimization we present a lagrangian decomposition algorithm which uses logarithmic potential reduction to compute an varepsilon approximate solution of the general max min resource sharing problem with m nonnegative concave constraints on a convex set b we show that this algorithm runs in o m varepsilon 2 rm ln m iterations a data independent bound which is optimal up to polylogarithmic factors for any fixed relative accuracy varepsilon in 0 1 in the block angular case b is the product of k convex sets blocks and each constraint function is block separable for such models an iteration of our method requires a theta varepsilon approximate solution of k independent block maximization problems which can be computed in parallel

approximate minimum cost multicommodity flows in o e 2 knm time we show that an e approximate solution of the cost constrainedk commodity flow problem on ann nodem arc network g can be computed by sequentially solving o k 2 loggk loggm log g 1 gk single commodity minimum cost flow problems on the same network in particular an approximate minimum cost multicommodity flow can be computed in tilde o g 2 gknm running time where the notation o means up to logarithmic factors this result improves the time bound mentioned by grigoriadis and khachiyan 4 by a factor ofm n and that developed more recently by karger and plotkin 8 by a factor of 1 we also provide a simple tilde o nm time algorithm for single commodity budget constrained minimum cost flows which is tilde o 3 times faster than the algorithm developed in the latter paper

a near optimal solution to a two dimensional cutting stock problem we present an asymptotic fully polynomial approximation scheme for strip packing or packing rectangles into a rectangle of fixed width and minimum height a classicalnp hard cutting stock problem the algorithm based on a new linear programming relaxation finds a packing ofn rectangles whose total height is within a factor of 1 e of optimal up to an additive term and has running time polynomial both in n and in 1 e

an asymptotic competitive scheme for online bin packing in the online bin packing problem a list of items with integral sizes between 1 to b arrive one by one each item must be irrevocably assigned into a bin of capacity b upon its arrival without any information on the subsequent items and the goal is to minimize the number of used bins in this paper we deal with online bin packing from a new sight we present an asymptotic competitive scheme i e for any 0 the asymptotic competitive ratio is at most where is the smallest possible asymptotic competitive ratio among all online algorithms apart from the technical results the analysis to bridge the online and the off line approaches might be of particular interests

a cutting plane algorithm for convex programming that uses analytic centers anoracle for a convex sets n accepts as input any pointz in n and ifz s then it returns yes while ifz s then it returns no along with a separating hyperplane we give a new algorithm that finds a feasible point ins in cases where an oracle is available our algorithm uses the analytic center of a polytope as test point and successively modifies the polytope with the separating hyperplanes returned by the oracle the key to establishing convergence is that hyperplanes judged to be unimportant are pruned from the polytope if a ball of radius 2 l is contained ins ands is contained in a cube of side 2l 1 then we can show our algorithm converges after o nl2 iterations and performs a total of o n4l3 tnl2 arithmetic operations wheret is the number of arithmetic operations required for a call to the oracle the bound is independent of the number of hyperplanes generated in the algorithm an important application in which an oracle is available is minimizing a convex function overs

using central prices in the decomposition of linear programs abstract this paper deals with a decomposition technique for linear programs which proposes a new treatment of the master program in the classical dantzig wolfe algorithm this approach exploits a the relationship between the master program and the minimization of a convex piecewise linear function and b a recently proposed cutting plane algorithm for convex nondifferentiable optimization the new algorithm seeks to achieve deep cuts by generating them at or near to the analytic centre of a set of localization containing the solution therefore each iteration entails a sizeable decrease of the set of localization and the overall algorithm shows fast convergence this algorithm has been used to solve reputedly difficult convex nondifferentiable optimization problems its implementation as a decomposition algorithm for large scale structured linear program seems to be a promising alternative to the standard dantzig wolfe approach which suffers from very slow convergence in many practical problems the new approach differs from the dantzig wolfe algorithm through the fact that it does not use prices corresponding to an optimal combination of the past proposals of the subproblem but rather central prices i e analytic centres which balance them all this property seems to explain the good behaviour of the new algorithm

fast approximation schemes for two stage two dimensional bin packing we present an asymptotic fully polynomial time approximation scheme for the two dimensional generalization of bin packing which requires packing or cutting a given set of rectangles from the minimum number of square bins with the further restriction that packing the rectangles in the bins is done in two stages as is frequently the case in real world applications to the best of our knowledge this is the first approximation scheme for a nontrivial two dimensional and real world generalization of a classical one dimensional packing problem in which rectangles have to be packed in finite squares in addition to the approximability result we obtain as a byproduct of our analysis an interesting structural result namely the asymptotic optimality of the lower bound provided by the natural linear programming relaxation of the problem having one variable for each bin pattern that can be solved effectively by column generation techniques besides its practical relevance a main reason to concentrate on the two stage version of the problem is that its study continues to shed light on the general version in particular we strongly conjecture that the approximation guarantee of our algorithm for two dimensional bin packing without the two stage restriction is better than the best known ones a simplification of the method leads to an approximation scheme for the two stage version of two dimensional strip packing

fully dynamic algorithms for bin packing being mostly myopic helps the problem of maintaining an approximate solution for one dimensional bin packing when items may arrive and depart dynamically is studied in accordance with various work on fully dynamic algorithms and in contrast to prior work on bin packing it is assumed that the packing may be arbitrarily rearranged to accommodate arriving and departing items in this context our main result is a fully dynamic approximation algorithm for bin packing mmp that is frac 5 4 competitive and requires theta log n time per operation i e for an insert or a delete of an item this competitive ratio of frac 5 4 is nearly as good as that of the best practical off line algorithms our algorithm utilizes the technique introduced here whereby the packing of an item is done with a total disregard for already packed items of a smaller size this myopic packing of an item may then cause several smaller items to be repacked in a similar fashion with a bit of additional sophistication to avoid certain bad cases the number of items either individual items or bundles of very small items treated as a whole that needs to be repacked is bounded by a constant

coupled and k sided placements generalizing generalized assignment in modern data centers and cloud computing systems jobs often require resources distributed across nodes providing a wide variety of services motivated by this we study the coupled placement problem in which we place jobs into computation and storage nodes with capacity constraints so as to optimize some costs or profits associated with the placement the coupled placement problem is a natural generalization of the widely studied generalized assignment problem which concerns the placement of jobs into single nodes providing one kind of service we also study a further generalization the k sided placement problem in which we place jobs into k tuples of nodes each node in a tuple offering one of k services for both the coupled and k sided placement problems we consider minimization and maximization versions in the minimization versions mincpand min k ksp the goal is to achieve minimum placement cost while incurring a minimum blowup in the capacity of the individual nodes our first main result is an algorithm for min k ksp that achieves optimal cost while increasing capacities by at most a factor of k 1 k 1 also yielding the first constant factor approximation for mincp in the maximization versions maxcp and max k ksp the goal is to maximize the total weight of the jobs that are placed under hard capacity constraints max k ksp can be expressed as a k column sparse integer program and can be approximated to within a factor of o k factor using randomized rounding of a linear program relaxation we consider alternative combinatorial algorithms that are much more efficient in practice our second main result is a local search based combinatorial algorithm that yields a 15 approximation and o k 2 o k2 approximation for maxcp and max k ksp respectively finally we consider an online version of max k ksp and present algorithms that achieve logarithmic competitive ratio under certain necessary technical assumptions

approximation schemes for deal splitting and covering integer programs with multiplicity constraints we consider the problem of splitting an order for r goods r 1 among a set of sellers each having bounded amounts of the goods so as to minimize the total cost of the deal in deal splitting with packages dsp the sellers offer packages containing combinations of the goods in deal splitting with price tables dst the buyer can generate such combinations using price tables our problems which often occur in online reverse auctions generalize covering integer programs with multiplicity constraints cip where we must fill up an r dimensional bin by selecting with a bounded number of repetitions from a set of r dimensional items such that the overall cost is minimized thus both dsp and dst are np hard already for a single good and hard to approximate for arbitrary number of goods in this paper we focus on finding efficient approximations for dsp and dst instances where the number of goods is some fixed constant in particular we develop polynomial time approximation schemes ptas for several subclasses of instances of practical interest our results include a ptas for cip in fixed dimension and a more efficient combinatorial scheme for cip where the multiplicity constraints are omitted our approximation scheme for cip is based on a non trivial application of the fast scheme for the fractional covering problem proposed by fleischer l fleischer a fast approximation scheme for fractional covering problems with variable upper bounds in proc of the 15th acm siam symposium on discrete algorithm 2004 pp 994 1003

approximation algorithms for knapsack problems with cardinality constraints abstract we address a variant of the classical knapsack problem in which an upper bound is imposed on the number of items that can be selected this problem arises in the solution of real life cutting stock problems by column generation and may be used to separate cover inequalities with small support within cutting plane approaches to integer linear programs we focus our attention on approximation algorithms for the problem describing a linear storage polynomial time approximation scheme ptas and a dynamic programming based fully polynomial time approximation scheme fptas the main ideas contained in our ptas are used to derive ptas s for the knapsack problem and its multi dimensional generalization which improve on the previously proposed ptas s we finally illustrate better ptas s and fptas s for the subset sum case of the problem in which profits and weights coincide

local search based heuristics for global optimization atomic clusters and beyond abstract finding good solutions to large scale hard global optimization problems is a demanding task with many relevant applications it is well known that in order to tackle a difficult problem an algorithm has to incorporate all of the available information on the problem domain however as we will show in this paper some general purpose methods and the ideas on which they are built can provide guidance towards the efficient solution of difficult instances most of this paper will be devoted to heuristic techniques applied to the problem of finding a minimum energy configuration of a cluster of atoms in r 3 this is a very relevant problem with a large set of applications which has triggered considerable research efforts in the last decade we will show how some relatively simple ideas can be used to generate fairly efficient methods which have been employed to discover many new cluster structures in this paper we will introduce some of the main algorithmic ideas which have proven to be particularly successful in the field of global optimization applied to atomic cluster conformation problems we will mainly discuss basin hopping methods as well as their population based variant and some specific technique based on direct moves these methods although designed for the specific problem have a much wider applicability in fact one of the aims of this paper is also that of suggesting that similar ideas can be employed in order to design innovative methods even for totally different global optimization problems like e g circle packing and space trajectory planning

two dimensional on line bin packing problem with rotatable items in this paper we consider a two dimensional version of the on line bin packing problem in which each rectangular item that should be packed into unit square bins is rotatable by 90 two on line algorithms for solving the problem are proposed the second algorithm is an extension of the first algorithms and the worst case ratio of the second one is at least 2 25 and at most 2 565

scheduling selfish jobs on multidimensional parallel machines abstract we study the multidimensional vector scheduling problem with selfish jobs both in non cooperative and in cooperative versions in two variants these variants differ in the private costs of the jobs in the first variant the cost is the l norm maximum over components while in the second variant it is the l 1 norm sum of components of the load we show existence of assignments that are nash strong nash weakly and strictly pareto optimal nash equilibria in these settings for the first variant we improve upon the previous bounds on the price of anarchy for the non cooperative case and find tight bounds for every number of machines and dimension for the cooperative case we provide tight bounds on the strong prices of anarchy and stability as well as tight bounds on weakly and strictly pareto optimal prices of anarchy and stability for every number of machines and dimension for the second variant which was not considered before we again consider all the aforementioned measures and find tight bounds each of which being a function of the number of machines and the dimension showing cardinal differences in the behavior of these measures both between the two variants and in comparison to the one dimensional case

fast approximation schemes for convex programs with many blocks and coupling constraints this paper presents block coordinate descent algorithms for the approximate solution of large structured convex programming problems the constraints of such problems consist of k disjoint convex compact sets b k called blocks and m nonnegative valued convex block separable inequalities called coupling or resource constraints the algorithms are based on an exponential potential function reduction technique it is shown that feasibility as well as min mix resource sharing problems for such constraints can be solved to a relative accuracy varepsilon in o k ln m varepsilon 2 ln k iterations each of which solves k block problems to a comparable accuracy either sequentially or in parallel the same bound holds for the expected number of iterations of a randomized variant of the algorithm which uniformly selects a random block to process at each iteration an extension to objective and constraint functions of arbitrary sign is also presented the above results yield fast approximatio

bidimensional packing by bilinear programming we consider geometric problems in which rectangles have to be packed into identical squares these problems turn out to be very hard in practice and ilp formulations in which variables specify the coordinates in the packing perform very poorly while most methods developed so far are based on simple geometric considerations a recent landmark result of fekete and schepers suggests to put these geometric aspects aside and use the most advanced tools for the one dimensional case in this paper we make additional progress in this direction especially on the basic question does a given set of rectangles fit into a square which turns out to be the bottleneck of all the approaches known given a set of rectangles and the associated convex hull of rectangle subsets that fit into a square we derive a wide class of valid inequalities for this convex hull from a complete description of the two knapsack polytopes associated with the widths and the heights of the rectangles respectively in addition we illustrate how to solve the associated separation problem as a bilinear program for which we develop a solution method that turns out to be fast in practice and show that the integer solutions that satisfy all these constraints generally correspond to vertices of the original convex hull for the benchmark instances in the literature the same tools are used to derive lower bounds for the two dimensional bin packing problem corresponding to the determination of an optimal pair of so called dual feasible functions these lower bounds in many cases equal those obtained by the customary set covering formulation for which column generation is very hard but are computable within a time that is smaller by some orders of magnitude this allows us to close a few of the benchmark instances in the literature

tight results for next fit and worst fit with resource augmentation it is well known that the two simple algorithms for the classic bin packing problem nf and wf both have an approximation ratio of 2 however wf seems to be a more reasonable algorithm since it never opens a new bin if an existing bin can still be used using resource augmented analysis where the output of an approximation algorithm which can use bins of size b 1 is compared to an optimal packing into bins of size 1 we give a complete analysis of the asymptotic approximation ratio of wf and of nf and use it to show that wf is strictly better than nf for any 1 2 and for b 1

two dimensional nonlinear boundary value problems for elliptic equations on considere la regularite limite des solutions du probleme aux valeurs limites non lineaire f x u du d 2 u 0 dans g x u du 0 sur pour des domaines a 2 dimensions f est uniformement elliptique et g depend de du on demontre des estimations de continuite pour des derivees premieres et secondes de u sous des hypotheses faibles pour la regularite de f g et

bin packing with general cost structures following the work of anily et al we consider a variant of bin packing called bin packing with general cost structures gcbp and design an asymptotic fully polynomial time approximation scheme afptas for this problem in the classic bin packing problem a set of one dimensional items is to be assigned to subsets of total size at most 1 that is to be packed into unit sized bins however in gcbp the cost of a bin is not 1 as in classic bin packing but it is a non decreasing and concave function of the number of items packed in it where the cost of an empty bin is zero the construction of the afptas requires novel techniques for dealing with small items which are developed in this work in addition we develop a fast approximation algorithm which acts identically for all non decreasing and concave functions and has an asymptotic approximation ratio of 1 5 for all functions simultaneously

parameterized approximation scheme for the multiple knapsack problem the multiple knapsack problem mkp is a well known generalization of the classical knapsack problem we are given a set a of n items and set b of m bins knapsacks such that each item a in a has a size size a and a profit value profit a and each bin b in b has a capacity c b the goal is to find a subset u subset a of maximum total profit such that u can be packed into b without exceeding the capacities the decision version of mkp is strongly np complete since it is a generalization of the classical knapsack and bin packing problem furthermore mkp does not admit a fully time polynomial time approximation scheme fptas even if the number m of bins is two kellerer gave a polynomial time approximation scheme ptas for mkp with identical capacities and chekuri and khanna presented a ptas for mkp with general capacities with running time n o log 1 epsilon epsilon 8 in this paper we propose an efficient ptas eptas with parameterized running time 2 o log 1 epsilon epsilon 5 cdot poly n o m for mkp this also solves an open question by chekuri and khanna

optimal online algorithms for multidimensional packing problems we solve an open problem in the literature by providing an online algorithm for multidimensional bin packing that uses only bounded space to achieve this we introduce a new technique for classifying the items to be packed we show that our algorithm is optimal among bounded space algorithms for any dimension d 1 its asymptotic performance ratio is pi infty d where pi infty approx1 691 is the asymptotic performance ratio of the one dimensional algorithm harm a modified version of this algorithm for the case where all items are hypercubes is also shown to be optimal its asymptotic performance ratio is sublinear in d furthermore we extend the techniques used in these algorithms to give optimal algorithms for online bounded space variable sized packing and resource augmented packing

fast approximation algorithms for fractional packing and covering problems this paper presents fast algorithms that find approximate solutions for a general class of problems which we call fractional packing and covering problems the only previously known algorithms for solving these problems are based on general linear programming techniques the techniques developed in this paper greatly outperform the general methods in many applications and are extensions of a method previously applied to find approximate solutions to multicommodity flow problems our algorithm is a lagrangian relaxation technique an important aspect of our results is that we obtain a theoretical analysis of the running time of a lagrangian relaxation based algorithm we give several applications of our algorithms the new approach yields several orders of magnitude of improvement over the best previously known running times for algorithms for the scheduling of unrelated parallel machines in both the preemptive and the nonpreemptive models for the job shop problem for the held and karp bound for the tr

an approximation algorithm for the general max min resource sharing problem we propose an approximation algorithm for the general max min resource sharing problem with m nonnegative concave constraints on a convex set b the algorithm is based on a lagrangian decomposition method and it uses a c approximation algorithm called approximate block solver for a simpler maximization problem over the convex set b we show that our algorithm achieves within iterations or calls to the approximate block solver a solution for the general max min resource sharing problem with approximation ratio the algorithm is faster and simpler than the previous known approximation algorithms for the problem 12 13

free boundary and mixed boundary value problems for quasilinear elliptic equations tangential oblique derivative and degenerate dirichlet boundary problems abstract we present holder estimates and holder gradient estimates for a class of free boundary problems with tangential oblique derivative boundary conditions provided the oblique vector does not vanish at any point on the boundary we also establish the existence result for a general class of quasilinear degenerate problems of this type including nonlinear wave systems and the unsteady transonic small disturbance equation

packing resizable items with application to video delivery over wireless networks motivated by fundamental optimization problems in video delivery over wireless networks we consider the following problem of packing resizable items pri given is a bin of capacity b 0 and a set i of items each item j i has a size s j 0 a packed item must stay in the bin for a fixed common time interval to accommodate additional items in the bin each item j can be compressed to a given size p j 0 s j for at most a fraction q j 0 1 of the packing interval during one continuous sub interval the goal is to pack in the bin for the given time interval a subset of items of maximum cardinality pri is a generalization of the well known knapsack problem and it is strongly np hard already for highly restricted instances our main result is an approximation algorithm that packs for any instance i of pri at least 2 3 opt i 2 items where opt i is the number of items packed in an optimal solution our algorithm yields a better performance ratio for instances in which the maximum compression time of an item is q max 0 1 2 for subclasses of instances arising in realistic scenarios we give an algorithm that packs at least opt i 2 items finally we show that a non trivial subclass of instances admits an asymptotic fully polynomial time approximation scheme afptas

a new algorithm for minimizing convex functions over convex sets let s subseteq mathbb r n be a convex set for which there is an oracle with the following property given any pointz n the oracle returns a yes ifz s whereas ifz s then the oracle returns a no together with a hyperplane that separatesz froms the feasibility problem is the problem of finding a point ins the convex optimization problem is the problem of minimizing a convex function overs we present a new algorithm for the feasibility problem the notion of a volumetric center of a polytope and a related ellipsoid of maximum volume inscribable in the polytope are central to the algorithm our algorithm has a significantly better global convergence rate and time complexity than the ellipsoid algorithm the algorithm for the feasibility problem easily adapts to the convex optimization problem

a harmonic algorithm for the 3d strip packing problem in the three dimensional 3d strip packing problem we are given a set of 3d rectangular items and a 3d box b the goal is to pack all the items in b such that the height of the packing is minimized we consider the most basic version of the problem where the items must be packed with their edges parallel to the edges of b and cannot be rotated building upon caprara s work for the two dimensional 2d bin packing problem we obtain an algorithm that given any epsilon 0 achieves an approximation of t infty epsilon approx1 69103 epsilon where t infty is the well known number that occurs naturally in the context of bin packing our key idea is to establish a connection between bin packing solutions for an arbitrary instance i and the strip packing solutions for the corresponding instance obtained from i by applying the harmonic transformation to certain dimensions based on this connection we also give a simple alternate proof of the t infty epsilon approximation for 2d

on the limited memory bfgs method for large scale optimization we study the numerical performance of a limited memory quasi newton method for large scale optimization which we call the l bfgs method we compare its performance with that of the method developed by buckley and lenir 1985 which combines cycles of bfgs steps and conjugate direction steps our numerical tests indicate that the l bfgs method is faster than the method of buckley and lenir and is better able to use additional storage to accelerate convergence we show that the l bfgs method can be greatly accelerated by means of a simple scaling we then compare the l bfgs method with the partitioned quasi newton method of griewank and toint 1982a the results show that for some problems the partitioned quasi newton method is clearly superior to the l bfgs method however we find that for other problems the l bfgs method is very competitive due to its low iteration cost we also study the convergence properties of the l bfgs method and prove global convergence on uniformly convex problems

approximation schemes for knapsack problems with shelf divisions given a knapsack of size k non negative values d and and a set s of items each item e s with size se and value ve we define a shelf as a subset of items packed inside a bin with total items size at most two subsequent shelves must be separated by a shelf divisor of size d the size of a shelf is the total size of its items plus the size of the shelf divisor the shelf knapsack problem sk is to find a subset s s partitioned into shelves with total shelves size at most k and maximum value the class constrained shelf knapsack ccsk is a generalization of the problem sk where each item in s has a class and each shelf in the solution must have only items of the same class we present approximation schemes for the sk and the ccsk problems to our knowledge these are the first approximation results where shelves of non null size are used in knapsack problems

a set covering based approach to find the reduct of variable precision rough set abstract attribute reduction is one of the core problems in rough set rs theory in the variable precision rough set vprs model attribute reduction faces two difficulties firstly in the vprs model a reduct anomaly problem may arise and it may cause an inconsistency of positive regions and decision rules after attribute reduction secondly the attribution reduction problem has been proved an np hard problem accordingly we would need to find a tradeoff between calculating the minimal reduct and reducing computing complexity to avoid the combinatorial explosion problem we propose a new approach to calculate the reduct in vprs model this new method focuses on calculating a distribution reduct while avoiding the anomaly problem in the vprs model the basic idea of the proposed approach is to convert the reduct problem into a set covering problem scp according to the positive regions in the vprs model and consequently a set covering heuristic function schf algorithm is applied to calculate the reduct after this conversion this approach keeps the positive regions consistent after the attribute reduction and moreover based on the scp the performance ratio of the proposed method to calculate the minimal reduct ranges between ln u ln ln u o 1 and 1 o 1 ln u with a computational complexity having an upper bound as o mn m n 2 finally we demonstrate the practical application of the vprs model using real case scenario from china s electricity power yield to verify the validity of our proposed approach we then apply statistical evaluation to explain the economic significance of the attributes

faster and simpler approximation algorithms for mixed packing and covering problems we propose an algorithm for approximately solving the mixed packing and covering problem given a convex compact set 0 b r n either compute x b such that f x 1 e a and g x 1 e b or decide that x b f x a g x b 0 here f g b r m are vectors whose components are m non negative convex and concave functions respectively and a b r m are constant positive vectors our algorithm requires an efficient feasibility oracle or block solver which given vectors c d r m and a r computes x b such that c tf x d tg x a or correctly decides that no such x b exists our algorithm which is based on the lagrangian or price directive decomposition method generalizes the result from k jansen approximation algorithm for the mixed fractional packing and covering problem in proceedings of 3rd ifip conference on theoretical computer science ifip tcs 2004 kluwer 2004 pp 223 236 siam journal on optimization 17 2006 331 352 and needs only o m lnm e 2ln e 1 iterations or calls to the feasibility oracle furthermore we show that a more general block solver can be used to obtain a more general approximation within the same runtime bound

decomposition and nondifferentiable optimization with the projective algorithm this paper deals with an application of a variant of karmarkar s projective algorithm for linear programming to the solution of a generic nondifferentiable minimization problem this problem is closely related to the dantzig wolfe decomposition technique used in large scale convex programming the proposed method is based on a column generation technique defining a sequence of primal linear programming maximization problems associated with each problem one defines a weighted potential function which is minimized using a variant of the projective algorithm when a point close to the minimum of the potential function is reached a corresponding point in the dual space is constructed which is close to the analytic center of a polytope containing the solution set of the nondifferentiable optimization problem an admissible cut of the polytope corresponding to a new supporting hyperplane of the epigraph of the function to minimize is then generated at this approximate analytic center in the primal space this new cut translates into a new column for the associated linear programming problem the algorithm has performed well on a set of convex nondifferentiable programming problems

variable sized bin packing tight absolute worst case performance ratios for four approximation algorithms in this paper we consider a one dimensional bin packing problem where the bins do not have identical sizes or a variable sized bin packing problem to minimize the bin consumption i e the total size of the opened bins the identical size problem has been extensively studied in the literature both for static and dynamic settings the worst case or average case performance has been analyzed our problem setting particularly arises in metal cutting industries therefore it presents a great practical relevance four greedy approximation algorithms based on a construction approach called largest object first with least absolute waste lflaw largest object first with least relative waste lflrw least absolute waste law and least relative waste lrw are examined their absolute worst case performances are analyzed the worst case bounds are 2 for lflaw and lflrw 3 for law and 2 ln 2 for lrw we also show that these worst case bounds are tight

approximation algorithms for general packing problems and their application to the multicast congestion problem in this paper we present approximation algorithms based on a lagrangian decomposition via a logarithmic potential reduction to solve a general packing or min max resource sharing problem with m non negative convex constraints on a convex set b we generalize a method by grigoriadis et al to the case with weak approximate block solvers i e with only constant logarithmic or even worse approximation ratios given an accuracy varepsilon in 0 1 we show that our algorithm needs o m ln m varepsilon 2 ln varepsilon 1 calls to the block solver a bound independent of the data and the approximation ratio of the block solver for small approximation ratios the algorithm needs o m ln m varepsilon 2 calls to the block solver as an application we study the problem of minimizing the maximum edge congestion in a multicast communication network interestingly the block problem here is the classical steiner tree problem that can be solved only approximately we show how to use approximation algorithms for the steiner tree problem to solve the multicast congestion problem approximately

new bounds for variable sized online bin packing in the variable sized online bin packing problem one has to assign items to bins one by one the bins are drawn from some fixed set of sizes and the goal is to minimize the sum of the sizes of the bins used we present new algorithms for this problem and show upper bounds for them which improve on the best previous upper bounds we also show the first general lower bounds for this problem the case in which bins of two sizes 1 and alpha in 0 1 are used is studied in detail this investigation leads us to the discovery of several interesting fractal like curves

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

on the online bin packing problem a new framework for analyzing online bin packing algorithms is presented this framework presents a unified way of explaining the performance of algorithms based on the harmonic approach within this framework it is shown that a new algorithm harmonic has asymptotic performance ratio at most 1 58889 it is also shown that the analysis of harmonic 1 presented in richey 1991 is incorrect this is a fundamental logical flaw not an error in calculation or an omitted case the asymptotic performance ratio of harmonic 1 is at least 1 59217 thus harmonic provides the best upper bound for the online bin packing problem to date

bounds for online bin packing with cardinality constraints abstract we study a bin packing problem in which a bin can contain at most k items of total size at most 1 where k 2 is a given parameter items are presented one by one in an online fashion we analyze the best absolute competitive ratio of the problem and prove tight bounds of 2 for any k 4 additionally we present bounds for relatively small values of k with respect to the asymptotic competitive ratio and the absolute competitive ratio in particular we provide tight bounds on the absolute competitive ratio of first fit for k 2 3 4 and improve the known lower bounds on asymptotic competitive ratios for multiple values of k our method for obtaining a lower bound on the asymptotic competitive ratio using a certain type of an input is general and we also use it to obtain an alternative proof of the known lower bound on the asymptotic competitive ratio of standard online bin packing

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

a new approximation method for set covering problems with applications to multidimensional bin packing in this paper we introduce a new general approximation method for set covering problems based on the combination of randomized rounding of the near optimal solution of the linear programming lp relaxation leading to a partial integer solution and the application of a well behaved approximation algorithm to complete this solution if the value of the solution returned by the latter can be bounded in a suitable way as is the case for the most relevant generalizations of bin packing the method leads to improved approximation guarantees along with a proof of tighter integrality gaps for the lp relaxation for d dimensional vector packing we obtain a polynomial time randomized algorithm with asymptotic approximation guarantee arbitrarily close to ln d 1 for d 2 this value is 1 693 dots i e we break the natural 2 barrier for this case moreover for small values of d this is a notable improvement over the previously known o ln d guarantee by chekuri and khanna siam j comput 33 2004 pp 837 851 for two dimensional bin packing with and without rotations we obtain polynomial time randomized algorithms with asymptotic approximation guarantee 1 525 dots improving upon previous algorithms with asymptotic performance guarantees arbitrarily close to 2 by jansen and van stee on strip packing with rotations in proceedings of the 37th annual acm symposium on the theory of computing 2005 pp 755 761 for the problem with rotations and 1 691 ldots by caprara math oper res 33 2008 pp 203 215 for the problem without rotations the previously unknown key property used in our proofs follows from a retrospective analysis of the implications of the landmark bin packing approximation scheme by fernandez de la vega and lueker combinatorica 1 1981 pp 349 355 we prove that their approximation scheme is subset oblivious which leads to numerous applications

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

lower bounds on time accuracy trade offs for the 0 1 knapsack problem the 0 1 knapsack optimization problem is well known to be np hard the best known approximation algorithms for this problem are approximation schemes which exhibit trade offs between the running time of the algorithm and the guaranteed accuracy of the solution this paper establishes lower bounds on the time required to find approximate solutions of guaranteed accuracy for a class of knapsack algorithms which must use oracles to perform feasibility tests and dominance tests

approximation algorithm for the mixed fractional packing and covering problem we propose an approximation algorithm based on the lagrangian or price directive decomposition method to compute an epsilon approximate solution of the mixed fractional packing and covering problem find x in b such that f x le 1 epsilon a g x ge 1 epsilon b where f x g x are vectors with m nonnegative convex and concave functions a and b are m dimensional nonnegative vectors and b is a convex set that can be queried by an optimization oracle we propose an algorithm that needs only o m epsilon 2 ln m epsilon 1 iterations or calls to the oracle the main contribution is that the algorithm solves the general mixed fractional packing and covering problem in contrast to pure fractional packing and covering problems and to the special mixed packing and covering problem with b rz n and runs in time independent of the so called width of the problem

the growth of m constraint random knapsacks the author computes the asymptotic value of a particular m constraint n variable 0 1 random integer programming problem as n increases m remaining fixed this solves a problem of frieze and clarke 1984

online algorithms for 1 space bounded 2 dimensional bin packing and square packing in this paper we study 1 space bounded 2 dimensional bin packing and square packing a sequence of rectangular items square items arrive one by one each item must be packed into a square bin of unit size on its arrival without any information about future items when packing items 90 rotation is allowed 1 space bounded means there is only one active bin if the active bin cannot accommodate the coming item it will be closed and a new bin will be opened the objective is to minimize the total number of bins used for packing all items in the sequence our contributions are as follows for 1 space bounded 2 dimensional bin packing we propose an online packing algorithm with a tight competitive ratio of 5 06 a lower bound of 3 17 on the competitive ratio is proven moreover we study 1 space bounded square packing where each item is a square with side length no more than 1 a 4 3 competitive algorithm is achieved and a lower bound of 2 94 on the competitive ratio is given all these bounds surpass the previously best known results

a robust aptas for the classical bin packing problem bin packing is a well studied problem which has many applications in this paper we design a robust aptas for the problem the robust aptas receives a single input item to be added to the packing at each step it maintains an approximate solution throughout this process by slightly adjusting the solution for each new item at each step the total size of items which may migrate between bins must be bounded by a constant factor times the size of the new item we show that such a property cannot be maintained with respect to optimal solutions

fully dynamic bin packing revisited we consider the fully dynamic bin packing problem where items arrive and depart in an online fashion and repacking of previously packed items is allowed the goal is of course to minimize both the number of bins used as well as the amount of repacking a recently introduced way of measuring the repacking costs at each timestep is the migration factor defined as the total size of repacked items divided by the size of an arriving or departing item concerning the trade off between number of bins and migration factor if we wish to achieve an asymptotic competitive ratio of 1 epsilon for the number of bins a relatively simple argument proves a lower bound of omega 1 epsilon for the migration factor we establish a nearly matching upper bound of o 1 epsilon 4 log 1 epsilon using a new dynamic rounding technique and new ideas to handle small items in a dynamic setting such that no amortization is needed the running time of our algorithm is polynomial in the number of items nand in 1 epsilon the previous best trade off was for an asymptotic competitive ratio of 5 4 for the bins rather than 1 epsilon and needed an amortized number of o log n repackings while in our scheme the number of repackings is independent of n and non amortized

class constrained bin packing revisited we study the following variant of the bin packing problem we are given a set of items where each item has a non negative size and a color we are also given an integer parameter k and the goal is to partition the items into a minimum number of subsets such that for each subset s in the solution the total size of the items in s is at most 1 as in the classical bin packing problem and the total number of colors of the items in s is at most k which distinguishes our problem from the classical version we follow earlier work on this problem and study the problem in both offline and online scenarios

the effects of coefficient correlation structure in two dimensional knapsack problems on solution procedure performance this paper presents the results of an empirical study of the effects of coefficient correlation structure and constraint slackness settings on the performance of solution procedures on synthetic two dimensional knapsack problems 2kp the population correlation structure among 2kp coefficients the level of constraint slackness and the type of correlation product moment or rank are varied in this study representative branch and bound and heuristic solution procedures are used to investigate the influence of these problem parameters on solution procedure performance population correlation structure and in particular the interconstraint component of the correlation structure is found to be a significant factor influencing the performance of both the algorithm and the heuristic in addition the interaction between constraint slackness and population correlation structure is found to influence solution procedure performance

higher regularity for free boundary and tangential derivative problems and global solutions to regular reflections for the unsteady transonic small disturbance utsd equations abstract we establish c 2 estimates for solutions of a class of quasilinear elliptic equations with free boundary and tangential derivative boundary problems using this regularity result we show the existence of global solutions to regular shock reflections for the unsteady transonic small disturbance utsd equation we also present lipschitz estimates near the degenerate dirichlet boundary the sonic boundary for the utsd equation

the growth of multi constraint random knapsacks with various right hand sides of the constraints abstract the asymptotic value of the m constraint n variable 0 1 random knapsack problem with b j n the right hand side of the j th constraint m fixed n is computed previously known results have been generalized

packing d dimensional bins in d stages we consider the d dimensional bin packing problem the most relevant generalization of classical bin packing and show a general result about the asymptotic worst case ratio of a wide class of approximation algorithms that construct solutions in d stages containing many heuristics previously considered in the literature moreover we give the exact value of the asymptotic worst case ratio between the optimal solution and the best solution obtainable by packing the items in d stages showing how to achieve such a ratio efficiently the key property behind our result is the asymptotic optimality of the fractional relaxation of one dimensional bin packing an important by product of the approximation schemes for the problem from the 1980s which to the best of our knowledge is used here for the first time only for the sake of the analysis for the two dimensional case we push the approximablity threshold below 2 after more than 20 years reaching 1 691 but also improve and widely simplify the analysis of the previous best method from the 1980s moreover for d 3 we improve the previous approximability threshold by a factor of 1 691 a notable improvement when d is small

improved approximation algorithms for minimum power covering problems given an undirected graph with edge costs the power of a node is the maximum cost of an edge incident to it and the power of a graph is the sum of the powers of its nodes motivated by applications in wireless networks we consider two network design problems under the power minimization criteria in both problems we are given a graph g v e with edge costs and a set t subseteq v of terminals the goal is to find a minimum power edge subset f subseteq e such that the graph h v f satisfies some prescribed requirements in the min power edge cover problem h should contain an edge incident to every terminal using the iterative randomized rounding irr method we give an algorithm with expected approximation ratio 1 41 the ratio is reduced to 73 60 1 217 when t is an independent set in g in the case of unit costs we also achieve ratio 73 60 and in addition give a simple efficient combinatorial algorithm with ratio 5 4 for all these np hard problems the previous best known ratio was 3 2 in the related min power terminal backup problem h should contain a path from every t in t to some node in t setminus t we obtain ratio 3 2 for this np hard problem improving the trivial ratio of 2

approximate strip packing in this paper we establish an algorithmic framework between bin packing and strip packing with which strip packing can be very well approximated by applying some bin packing algorithms more precisely we obtain the following results 1 any off line bin packing algorithm can be applied to strip packing maintaining almost the same asymptotic worst case ratio 2 a class of harmonic based algorithms for bin packing such as refined harmonic modified harmonic harmonic can be applied to online strip packing in particular we show that online strip packing admits an upper bound of 1 58889 on the asymptotic competitive ratio for any arbitrarily small 0 this significantly improves the previously best bound of 1 6910 and affirmatively answers an open question posed by csirik and woeginger 1997 moreover the time complexity mainly depends on a sorting procedure and the bin packing algorithms employed

the class constrained bin packing problem with applications to video on demand in this paper we present approximation results for the class constrained bin packing problem that has applications to video on demand systems in this problem we are given bins of size b with c compartments and n items of q different classes each item i 1 n with class c i and size s i the problem is to pack the items into bins where each bin contains at most c different classes and has total items size at most b we present several approximation algorithms for offline and online versions of the problem

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

bin packing with controllable item sizes we consider a natural resource allocation problem in which we are given a set of items where each item has a list of pairs associated with it each pair is a configuration of an allowed size for this item together with a nonnegative penalty and an item can be packed using any configuration in its list the goal is to select a configuration for each item so that the number of unit bins needed to pack the sizes plus the sum of penalties is minimized this problem has applications in operating systems bandwidth allocation and discrete time cost tradeoff planning problems for the offline version of the problem we design an augmented asymptotic ptas that is an asymptotic approximation scheme that uses bins of size slightly larger than 1 we further consider the online bounded space variant where only a constant number of bins can be open simultaneously we design a sequence of algorithms where the sequence of their competitive ratios tends to the best possible asymptotic competitive ratio finally we study the bin covering problem in which a bin is covered if the sum of sizes allocated to it is at least 1 in this setting penalties are interpreted as profits and the goal is to maximize the sum of profits plus the number of covered bins we design an algorithm of best possible competitive ratio which is 2 we generalize our results for online algorithms and unit sized bins to the case of variable sized bins where there may be several distinct sizes of bins available for packing or covering and get that the competitive ratios are again the same as for the more standard online problems

a simple on line bin packing algorithm the one dimensional on line bin packing problem is considered a simple o 1 space and o n time algorithm called harmonic m is presented it is shown that this algorithm can achieve a worst case performance ratio of less than 1 692 which is better than that of the o n space and o n log n time algorithm first fit also shown is that 1 691 is a lower bound for all 0 1 space on line bin packing algorithms finally a revised version of harmonic m an o n space and o n time algorithm is presented and is shown to have a worst case performance ratio of less than 1 636 d

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

tightening simple mixed integer sets with guaranteed bounds in this paper we study how to reformulate knapsack sets and simple mixed integer sets in order to obtain provably tight polynomially large formulations

locality preserving allocations problems and coloured bin packing we study the following problem introduced by chung et al in 2006 we are given online or offline a set of coloured items of different sizes and wish to pack them into bins of equal size so that we use few bins in total at most times optimal and that the items of each colour span few bins at most times optimal we call such allocations approximate as usual in bin packing problems we allow additive constants and consider as the asymptotic performance ratios we prove that for e 0 if we desire small no scheme can beat 1 e 1 e approximate allocations and similarly as we desire small no scheme can beat 1 69103 1 e approximate allocations we give offline schemes that come very close to achieving these lower bounds for the online case we prove that no scheme can even achieve o 1 o 1 approximate allocations however a small restriction on item sizes permits a simple online scheme that computes 2 e 1 7 approximate allocations

on lazy bin covering and packing problems in this paper we study two interesting variants of the classical bin packing problem called lazy bin covering lbc and cardinality constrained maximum resource bin packing ccmrbp problems for the offline lbc problem we first prove the approximation ratio of the first fit decreasing and first fit increasing algorithms then present an aptas for the online lbc problem we give a competitive analysis for the algorithms of next fit worst fit first fit and a modified harmonic m algorithm the ccmrbp problem is a generalization of the maximum resource bin packing mrbp problem boyar et al 2006 1 for this problem we prove that its offline version is no harder to approximate than the offline mrbp problem