time domain reciprocity theorems for acoustic wave fields in fluids with relaxation time domain reciprocity theorems of the time convolution and the time correlation type for acoustic wave fields in linear time invariant and locally reacting fluids are discussed inhomogeneity inertial anisotropy and arbitrary relaxation effects in inertia and compressibility properties both of the active and the passive type are included the theorems also apply to the equivalent fluid model of a solid in which only compressional waves are considered and shear is neglected the analysis is entirely carried out in space time without intermediate recourse to the frequency or the wave vector domains the application to inverse source and inverse constituency or inverse profiling scattering or imaging problems is briefly indicated

tutorial on seismic interferometry part 2 underlying theory and new advances in the 1990s the method of time reversed acoustics was developed this method exploits the fact that the acoustic wave equation for a lossless medium is invariant for time reversal when ultrasonic responses recorded by piezoelectric transducers are reversed in time and fed simultaneously as source signals to the transducers they focus at the position of the original source even when the medium is very complex in seismic interferometry the time reversed responses are not physically sent into the earth but they are convolved with other measured responses the effect is essentially the same the time reversed signals focus and create a virtual source which radiates waves into the medium that are subsequently recorded by receivers a mathematical derivation based on reciprocity theory formalizes this principle the crosscorrelation of responses at two receivers integrated over different sources gives the green s function emitted by a virtual source at the position of one of the receivers and observed by the other receiver this green s function representation for seismic interferometry is based on the assumption that the medium is lossless and nonmoving recent developments circumventing these assumptions include interferometric representations for attenuating and or moving media as well as unified representations for waves and diffusion phenomena bending waves quantum mechanical scattering potential fields elastodynamic electromagnetic poroelastic and electroseismic waves significant improvements in the quality of the retrieved green s functions have been obtained with interferometry by deconvolution a trace by trace deconvolution process compensates for complex source functions and the attenuation of the medium interferometry by multidimensional deconvolution also compensates for the effects of one sided and or irregular illumination

reciprocity theorems for two way and one way wave vectors a comparison for acoustic applications in which there is a preferred direction of propagation the axial direction it is useful to arrange the two way and one way wave equations into the same matrix vector formalism in this formalism axial variations of the wave vector are expressed in terms of lateral variations of the same wave vector the two way wave vector contains the field quantities pressure and velocity axial component only whereas the one way wave vector contains waves propagating in the positive and negative axial direction by exploiting the equivalent form of the two way and one way matrix vector equations it appears to be possible to derive two way and one way reciprocity theorems that have an equivalent form but a different interpretation the main differences appear in the boundary integrals for unbounded media in the contrast terms and for the correlation type theorems in the handling of evanescent waves

commitment under uncertainty two stage stochastic matching problems we define and study two versions of the bipartite matching problem in the framework of two stage stochastic optimization with recourse in one version the uncertainty is in the second stage costs of the edges and in the other version the uncertainty is in the set of vertices that needs to be matched we prove lower bounds and analyze efficient strategies for both cases these problems model real life stochastic integral planning problems such as commodity trading reservation systems and scheduling under uncertainty

on seismic interferometry the generalized optical theorem and the scattering matrix of a point scatterer we have analyzed the far field approximation of the green s function representation for seismic interferometry by writing each of the green s functions involved in the correlation process as a superposition of a direct wave and a scattered wave the green s function representation is rewritten as a superposition of four terms when the scattered waves are modeled with the born approximation it appears that a three term approximation of the green s function representation omitting the term containing the crosscorrelation of the scattered waves yields a nearly exact retrieval whereas the full four term expression leads to a significant nonphysical event this is because the born approximation does not conserve energy and therefore is an insufficient model to explain all aspects of seismic interferometry we use the full four term expression of the green s function representation to derive the generalized optical theorem unlike other recent derivations which use stationary phase analysis our derivation uses reciprocity theory from the generalized optical theorem we derive the nonlinear scattering matrix of a point scatterer this nonlinear model accounts for primary and multiple scattering at the point scatterer and conforms with well established scattering theory of classical waves the model is essential to explain fully the results of seismic interferometry even when it is applied to the response of a single point scatterer the nonlinear scattering matrix also has implications for modeling inversion and migration

some unconditionally stable time stepping methods for the 3d maxwell s equations almost all the difficulties that arise in the numerical solution of maxwell s equations are due to material interfaces in case that their geometrical features are much smaller than a typical wavelength one would like to use small space steps with large time steps the first time stepping method which combines a very low cost per time step with unconditional stability was the adi fdtd method introduced in 1999 the present discussion starts with this method and with an even more recent crank nicolson based split step method with similar properties we then explore how these methods can be made even more efficient by combining them with techniques that increase their temporal accuracies

the two dimensional cutting stock problem revisited in the strip packing problem a standard version of the two dimensional cutting stock problem the goal is to pack a given set of rectangles into a vertical strip of unit width so as to minimize the total height of the strip needed the k stage guillotine packings form a particularly simple and attractive family of feasible solutions for strip packing we present a complete analysis of the quality of k stage guillotine strip packings versus globally optimal packings k 2 stages cannot guarantee any bounded asymptotic performance ratio k 3 stages lead to asymptotic performance ratios arbitrarily close to 1 69103 this bound is tight finally k 4 stages yield asymptotic performance ratios arbitrarily close to 1

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

green s function representations for seismic interferometry the term seismic interferometry refers to the principle of generating new seismic responses by crosscorrelating seismic observations at different receiver locations the first version of this principle was derived by claerbout 1968 who showed that the reflection response of a horizontally layered medium can be synthesized from the autocorrelation of its transmission response for an arbitrary 3d inhomogeneous lossless medium it follows from rayleigh s reciprocity theorem and the principle of time reversal invariance that the acoustic green s function between any two points in the medium can be represented by an integral of crosscorrelations of wavefield observations at those two points the integral is along sources on an arbitrarily shaped surface enclosing these points no assumptions are made with respect to the diffusivity of the wavefield the rayleigh betti reciprocity theorem leads to a similar representation of the elastodynamic green s function when a part of the enclosing surface is the earth s free surface the integral needs only to be evaluated over the remaining part of the closed surface in practice not all sources are equally important the main contributions to the reconstructed green s function come from sources at stationary points when the sources emit transient signals a shaping filter can be applied to correct for the differences in source wavelets when the sources are uncorrelated noise sources the representation simplifies to a direct crosscorrelation of wavefield observations at two points similar as in methods that retrieve green s functions from diffuse wavefields in disordered media or in finite media with an irregular bounding surface

retrieving the green s function in an open system by cross correlation a comparison of approaches l we compare two approaches for deriving the fact that the green s function in an arbitrary inhomogeneous open system can be obtained by cross correlating recordings of the wave field at two positions one approach is based on physical arguments exploiting the principle of time reversal invariance of the acoustic wave equation the other approach is based on rayleigh s reciprocity theorem using a unified notation we show that the result of the time reversal approach can be obtained as an approximation of the result of the reciprocity approach

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

stability and dispersion analysis for higher order 3 d adi fdtd method in this paper the stability and dispersion analysis for higher order three dimensional 3 d alternating direction implicit adi finite difference time domain fdtd methods is presented starting with the stability and dispersion analysis of the fourth order 3 d adi fdtd method which adopts the fourth order cell centered finite difference scheme for the spatial differential operator we generalize the analysis to other higher order methods based on the sixth and tenth order cell centered finite difference schemes to the best of our knowledge this is the first time that a comprehensive study of the stability and dispersion characteristics for one series of higher order 3 d adi fdtd methods is presented our analysis results show that all the higher order adi fdtd methods that are based on cell centered finite difference schemes are unconditionally stable the generalized form of the dispersion relations for these unconditionally stable adi fdtd methods is also presented using the relations attained the effects of the order of schemes mesh size and time step on the dispersion are illustrated through numerical results this study will be useful for the selection and evaluation of various higher order adi methods

on unified numerical algorithm for 3 d scattering from dielectric and pec random rough surfaces we present a unified fast scattering algorithm for dielectric random rough surfaces that asymptotically reduces to the perfect electric conductor pec case when the loss tangent grows extremely large the coifman wavelets are employed to implement galerkin s procedure in the method of moments mom the coiflets based surface integral equations ies consist of both the tangential and normal components of electromagnetic fields as unknowns the inherited mathematical superiority e g local multiresolution analysis and high regularity with holder index 1 449 in smoothness allows efficiently implementing both electric field ie and magnetic field ie due to the high precision one point quadrature the coiflets yield fast evaluations of the most off diagonal entries reducing the matrix fill effort from o n2 to o n the orthogonality and riesz basis of the coiflets generate well conditioned impedance matrix with rapid convergence for the conjugate gradient solver in addition a semianalytical expression of the tapered wave carried power is derived which speeds up computations of the normalization factor of scattering coefficients numerical results demonstrate that the reduced pec model does not suffer from ill posed problems namely matrix condition numbers are kept small and solutions are stable under extremely large loss tangent where normal components of h field and tangential e field have vanished compared with the previous publications and laboratory measurements good agreement is observed

two dimensional packing problems a survey we consider problems requiring to allocate a set of rectangular items to larger rectangular standardized units by minimizing the waste in two dimensional bin packing problems these units are finite rectangles and the objective is to pack all the items into the minimum number of units while in two dimensional strip packing problems there is a single standardized unit of given width and the objective is to pack all the items within the minimum height we discuss mathematical models and survey lower bounds classical approximation algorithms recent heuristic and metaheuristic methods and exact enumerative approaches the relevant special cases where the items have to be packed into rows forming levels are also discussed in detail

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

a hierarchical generalization of the acoustic reciprocity theorem involving higher order derivatives and interaction quantities an acoustic reciprocity theorem is generalized for a smoothly varying perturbed medium to a hierarchy of reciprocity theorems including higher order derivatives of acoustic fields the standard reciprocity theorem is the first member of the hierarchy it is shown that the conservation of higher order interaction quantities is related closely to higher order derivative distributions of perturbed media then integral reciprocity theorems are obtained by applying gauss s divergence theorem which give explicit integral representations connecting higher order interactions and higher order derivative distributions of perturbed media some possible applications to an inverse problem are also discussed

an updated review of general dispersion relation for conditionally and unconditionally stable fdtd algorithms we present a general numerical dispersion equation which is applicable to all known conditionally and unconditionally stable finite difference time domain fdtd algorithms on staggered rectangular grid including yee s fdtd wavelet based fdtd extended curl fdtd alternating direction implicit adi fdtd crank nicolson cn fdtd crank nicolson split step cnss fdtd and their modifications of higher order spatial stencils the real part of the complex eigenvalue of the total amplification matrix defines and distinguishes the dispersion relation for each individual scheme easy to check conditions are provided under which the numerical dispersion of a particular time domain scheme is governed by the proposed dispersion equation these conditions are on the amplification matrix eigenvalues the proposed dispersion equation includes each considered dispersion relation as a special case and presents itself a general governing equation to estimate 3 d numerical dispersion of the aforementioned schemes in the frame of plane waves

migration of primaries and multiples using an imaging condition for amplitude normalized separated wavefields abstractmigration of primary and multiple reflections leads to enhanced subsurface illumination and to increased image resolution a joint migration approach using the complete wavefield requires properly imaged primaries and multiples of all orders in recent works primaries and multiples have therefore been imaged separately using the upgoing and downgoing pressure wavefields obtained by decomposing dual sensor streamer data the matches between the corresponding depth images are still not found to be sufficiently accurate so new and more appropriate imaging conditions need to be found we reviewed the classical imaging approach used in one way wave equation migration with the aim of extending it for simultaneous migration of primaries and all orders of multiples based on rayleigh s reciprocity theorem and well known theoretical developments we derived a new imaging condition described in terms of the upgoing pressure wavefield and a filtered version of the downgoing vertical velocity wavefield t

nonreciprocal green s function retrieval by cross correlation the cross correlation of two recordings of a diffuse acoustic wave field at different receivers yields the green s function between these receivers in nearly all cases considered so far the wave equation obeys time reversal invariance and the green s function obeys source receiver reciprocity here the theory is extended for nonreciprocal green s function retrieval in a moving medium it appears that the cross correlation result is asymmetric in time the causal part represents the green s function from one receiver to the other whereas the acausal part represents the time reversed version of the green s function along the reverse path

microseismic event localization by acoustic time reversal extrapolation traditional ray based methods for microseismic event localization require picking of p and s wave first arrivals which is often time consuming polarization analysis for each event is often also needed to determine its absolute location location methods based on reverse time extrapolation avoid the need for first arrival time picking traditional reverse time extrapolation only incorporates particle velocity or displacement wavefields this is an incomplete approximation of the acoustic representation theorem which leads to artifacts in the back propagation process for instance if the incomplete approximation is used for microseismic event locations using three component 3c borehole recordings it produces a ghost event on the opposite side of the well which leads to ambiguous interpretations we have developed representation theorem based reverse time extrapolation for microseismic event localization combining the 3c particle velocities displacements and the pressure wavefield the unwanted ghost location is removed by explicitly incorporating a wavefield and its spatial derivative moreover polarization analysis is not needed because wavefields will focus at its absolute location during back propagation determination of microseismic event locations using wavefield extrapolation also necessitates a robust focusing criterion the hough transform allows for accurate determination of source timing and location by summing wavefront energy in the time space domain synthetic examples demonstrated the good performance of the wavefield extrapolation scheme and focusing criterion in complex velocity fields for borehole acquisition geometries

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

wavelet based deconvolution algorithm for time domain near field isar imaging based on correlation receiver in communication theory the measured backscattering waveform of a calibration conducting sphere is adopted as the system impulse response this antenna target system response is then approximated by the derivatives of a gaussian function dgf as well as the coiflets to restore the impulse response of the target and to reconstruct the isar images both the dgf and coiflets are applied to the deconvolution and their results are compared the gaussian and coiflet function parameters are derived by a best fit to the measured time domain system response employing the parametric fits a lowpass filter has been automatically embedded which performs regularization of the conjugate gradient cg optimization and speeds up the cg convergence numerical results show that the image quality from the coiflet system is similar to or better than from the dgf system owing to the continuity smoothness orthogonality and compact support of the coiflets the size of the system matrix has been reduced from 1024 times 1024 for dgf to 261 times 261 and the deconvolution process is accelerated more than 22 fold without sacrificing image characters

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

representation theorems and green s function retrieval for scattering in acoustic media reciprocity theorems for perturbed acoustic media are provided in the form of convolution and correlation type theorems these reciprocity relations are particularly useful in the general treatment of both forward and inverse scattering problems using green s functions to describe perturbed and unperturbed waves in two distinct wave states representation theorems for scattered waves are derived from the reciprocity relations while the convolution type theorems can be manipulated to obtain scattering integrals that are analogous to the lippmann schwinger equation the correlation type theorems can be used to retrieve the scattering response of the medium by cross correlations unlike previous formulations of green s function retrieval the extraction of scattered wave responses by cross correlations does not require energy equipartitioning allowing for uneven energy radiation brings experimental advantages to the retrieval of fields scattered by remote lossless and or attenuative scatterers these concepts are illustrated with a number of examples including analytic solutions to a one dimensional scattering problem and a numerical example in the context of seismic waves recorded on the ocean bottom

a new lower bound for online strip packing in this paper we consider the online strip packing problem in which a list of online rectangles has to be packed without overlap or rotation into a strip of width 1 and infinite length so as to minimize the required height of the packing we derive a new improved lower bound of 3 5 2 2 618 for the competitive ratio for this problem this result improves the best known lower bound of 2 589

performance bounds for orthogonal oriented two dimensional packing algorithms two orthogonal oriented two dimensional packing algorithms are presented and their behavior is analyzed under several assumptions for one of the algorithms we show that the ratio of the height used by the algorithm to the optimal height is asymptotically bounded by frac 4 3 this bound is an improvement over similar bounds for previously proposed algorithms

examination clarification and simplification of stability and dispersion analysis for adi fdtd and cnss fdtd schemes we describe a rigorous analysis of unconditional stability for the alternating direction implicit finite difference time domain adi fdtd and crank nicholson split step cnss fdtd schemes avoiding use of the von neumann spectral criterion the proof is performed in the spectral domain and uses skew hermitivity of matrix terms in the adi and cnss total amplification matrices a bound for the total adi amplification matrix is provided while the cn and cnss fdtd amplification matrices are unitary the bound on the adi fdtd depends on the courant number importantly we have found that the adi fdtd amplification matrix is not normal i e the unit spectral radius alone cannot be used to prove the adi fdtd unconditional stability the paper also shows that the adi fdtd and cnss fdtd schemes share the same dispersion equation by a similarity of their total amplification matrices and the two schemes have identical numerical dispersion in the frame of plane waves

parameterized complexity of strip packing and minimum volume packing abstract we study the parameterized complexity of minimum volume packing and strip packing in the two dimensional version the input consists of a set of rectangles s with integer side lengths in the minimum volume packing problem given a set of rectangles s and a number k the goal is to decide if the rectangles can be packed in a bounding box of volume at most k in the strip packing problem we are given a set of rectangles s numbers w and k the objective is to find if all the rectangles can be packed in a box of dimensions w k we prove that the 2 dimensional volume packing is in fpt by giving an algorithm that runs in 2 2 k k o 1 time we also show that strip packing is w 1 hard even in two dimensions and give an fpt algorithm for a special case of strip packing some of our results hold for the problems defined in higher dimensions as well

a synchronized multigrid time domain method via huygens subgridding and implicit algorithms based upon huygens subgridding hsg technique and unconditionally stable time domain algorithms we present a synchronized multigrid time domain method for 2 and 3 d scattering problems in this method the implicit algorithms are employed in the fine grid region while the yee fdtd is marching on the coarse mesh as a result the electromagnetic fields in both coarse and fine grid regions are updated synchronously the synchronization has greatly simplified the hsg implementation and reduced updating substeps numerical examples demonstrate extremely low spurious reflection at the interface robust stability for arbitrarily large spatial ratios and suppression of late time instability

a strip packing algorithm with absolute performance bound 2 this paper proposes a new approximation algorithm m for packing rectangles into a strip with unit width and unbounded height so as to minimize the total height of the packing it is shown that for any list l of rectangles m l leq 2 cdot mbox opt l where m l is the strip height actually used by the algorithm m when applied to l and opt l is the minimum possible height within which the rectangles in l can be packed

wave field reciprocity and optimization in remote sensing a unified approach to local optimization techniques and wave field reciprocity as applied to constructing solutions to remote sensing imaging and inversion problems in acoustic elastic and electromagnetic wave theory is presented the starting point is a system of linear first order partial differential equations in space time of the class to which the indicated wave phenomena gives rise for this system three types of remote sensing problems the inverse source the inverse scattering and the inverse transducted wave field problems are formulated and the construction of their solutions via local optimization techniques is discussed emphasis is placed on iterative algorithms that are based on a guaranteed decrease in the mismatch between modelled and observed data at each update of the medium subsequently the wave field reciprocity theorems of the time convolution and the time correlation types are derived and their occurrence in the optimization procedures is discussed also attention is paid to approximate methods in particular to the rayleigh gans born approximation approximations of this sort provide the means to invoke the method of preconditioning in the process of inverting the operator equations exotic media for example chiral media in electromagnetics are included in the analysis

analysis of the divergence properties for the three dimensional leapfrog adi fdtd method analysis of the divergence properties for the three dimensional 3 d leapfrog alternating direction implicit finite difference time domain adi fdtd method is presented in this paper the analysis of divergence properties is based on the fourier domain plane wave method for dispersion analysis but extended with additional procedure to determine the homogeneous solutions the singular value decomposition svd is used to obtain the solutions of the homogeneous system in a reliable manner there are two linearly independent null modes in the homogeneous system corresponding to the repeated zero singular values to provide more physical insights proper linear combinations of the two null modes are determined as te and tm modes with respect to the direction of discrete numerical space wavevector the divergence of e fields is analyzed by examining the tm mode while the divergence of h fields is analyzed by examining the te mode numerical results for the divergence properties of the leapfrog adi fdtd method are presented and compared with the conventional adi fdtd method along with numerical experiments to illustrate the existence of divergent waves the leapfrog adi fdtd method is not divergence free in the source free region as is the case for the conventional adi fdtd method furthermore both methods have different divergence properties in general

strip packing with precedence constraints and strip packing with release times the strip packing problem seeks to tightly pack a set of n rectangles into a strip of fixed width and arbitrary height the rectangles model tasks and the height models time this paper examines two variants of strip packing when the rectangles to be placed have precedence constraints and when the rectangles have release times strip packing is used to model scheduling problems in which tasks require a contiguous subset of identical resources that are arranged in a linear topology the variants studied here are motivated by scheduling tasks for dynamically reconfigurable field programmable gate arrays fpgas comprised of a linear arrangement of k homogeneous computing resources where k is a fixed positive integer and each task occupies a contiguous subset of these resources for the case in which tasks have precedence constraints we give an o logn approximation algorithm we then consider the special case in which all the rectangles have uniform height and reduce it to the resource constrained scheduling studied by garey graham johnson and yao thereby extending their asymptotic results to our special case problem we also give an absolute 3 approximation for this special case problem for strip packing with release times we provide an asymptotic polynomial time approximation scheme we make the standard assumption that the rectangles have height at most 1 in addition we also require widths to be in 1k 1 for the fpga application this would imply that the rectangles are at least as wide as a column our running time is polynomial in n and 1 e but exponential in k

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

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

combinatorial benders cuts for the strip packing problem we study the strip packing problem in which a set of two dimensional rectangular items has to be packed in a rectangular strip of fixed width and infinite height with the aim of minimizing the height used the problem is important because it models a large number of real world applications including cutting operations where stocks of materials such as paper or wood come in large rolls and have to be cut with minimum waste scheduling problems in which tasks require a contiguous subset of identical resources and container loading problems arising in the transportation of items that cannot be stacked one over the other the strip packing problem has been attacked in the literature with several heuristic and exact algorithms nevertheless benchmark instances of small size remain unsolved to proven optimality in this paper we propose a new exact method that solves a large number of the open benchmark instances within a limited computational effort our method is based on a benders decomposition in whi