lower bound for the online bin packing problem with restricted repacking in 1996 ivkovic and lloyd a fundamental restriction on fully dynamic maintenance of bin packing inform process lett 59 1996 pp 229 232 gave the lower bound frac 4 3 on the asymptotic worst case ratio for so called fully dynamic bin packing algorithms where the number of repackable items in each step is restricted by a constant in this paper we improve this result to about 1 3871 we present our proof for a semionline case of the classical bin packing but it works for fully dynamic bin packing as well we prove the lower bound by analyzing and solving a specific optimization problem the bound can be expressed exactly using the lambert w function

ends of maps ii versions of the finiteness obstruction and simple homotopy theory within e overx are developed this provides a setting for obstructions to the map analogs of the end ands cobordism theorems for manifolds these are applied to study equivariant mapping cylinder neighborhoods in topological group actions triangulations of locally triangulable spaces and block bundle structures on approximate fibrations

rigidity of coxeter groups let w be a coxeter group acting properly discontinuously and cocompactly on manifolds n and m 9m 0 such that the fixed point sets of finite subgroups are contractible let f n n mx dk m xsk1 be a w homotopy equivalence which restricts to a w homeomorphism on the boundary under an assumption on the three dimensional fixed point sets we show that then f is w homotopic to a w homeomorphism

an effective dynamic programming algorithm for the minimum cost maximal knapsack packing problem abstract given a set of items with profits and weights and a knapsack capacity we study the problem of finding a maximal knapsack packing that minimizes the profit of the selected items we propose an effective dynamic programming dp algorithm which has a pseudo polynomial time complexity we demonstrate the equivalence between this problem and the problem of finding a minimal knapsack cover that maximizes the profit of the selected items in an extensive computational study on a large and diverse set of benchmark instances we demonstrate that the new dp algorithm outperforms a state of the art commercial mixed integer programming mip solver applied to the two best performing mip models from the literature

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

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

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

nonlinear similarity begins in dimension six the real representations pi and p2 of a finite group g are topologically similar written pi t p2 if there is a homeomorphism h v p1 v pa where v pi denotes the vector space of the representation pi such that h p1 g v p2 g h v for v e v p1 and g e g i e the representation spaces are equivariantly homeomorphic de rham dr conjectured that topological similarity implies the linear equivalence of the two representations the first two authors showed that de rham s conjecture is true for representations of dimension less than six cs6 but false for representations of dimension greater than or equal to nine cs2 here we give examples of nonlinear similarity i e topological similarity between linearly inequivalent representations in dimension six which is therefore the minimal dimension in which nonlinear similarity may occur since linear equivalence is detected by cyclic subgroups a minimal counterexample to de rham s conjecture will occur for g cyclic moreover by hp and mr nonlinear similarity may only occur for groups whose order is divisible by four thus we shall restrict attention to g z4q the cyclic group of order 4q we shall construct six dimensional nonlinear similarities of z4q for every q greater than two moreover we shall show in a later paper cssww2 joint with weinberger that the examples here represent all six dimensional nonlinear similarities of cyclic groups actually we shall give the complete topological classification of representations of the form p 6 c where p is free i e z4q acts freely away from the origin in the representation space of p and 6 and e are the nontrivial and trivial one dimensional representations respectively representations of the form p 6 were studied in cs2 suppose that pi 6 c 1t p2 6 e with p1 and p2 free let lp be the orbit space of the unit

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

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

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

algorithms for the relaxed online bin packing model the typical online bin packing problem requires the fitting of a sequence of rationals in 0 1 into a minimum number of bins of unit capacity by packing the ith input element without any knowledge of the sizes or the number of input elements that follow moreover unlike typical online problems this one issue does not admit any data reorganization i e no element can be moved from one bin to another in this paper first of all the relaxed online bin packing model will be formalized this model allows a constant number of elements to move from one bin to another as a consequence of the arrival of a new input element then in the context of this new model two online algorithms will be described the first presents linear time and space complexities with a 1 5 approximation ratio and moves at most once only small elements the second instead is an o n log n time and linear space algorithm with a 1 33 approximation ratio and moves each element a constant number of times in the worst case as a result of the arrival of a new input element the first algorithm moves no more than three elements while the second moves as many as seven elements please note that the number of movements performed is explicitly considered in the complexity analysis both algorithms are below the theoretical 1 536 lower bound effective for the online bin packing algorithms without the movement of elements moreover our algorithms are more online than any other linear space online bin packing algorithm because unlike the algorithms already known they allow the return of a possibly relevant fraction of bins before the work is carried out

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

bin packing problems with rejection penalties and their dual problems in this paper we consider the following problems we are given a set of n items u1 un and a number of unit capacity bins each item ui has a size wi 0 1 and a penalty pi 0 an item can be either rejected in which case we pay its penalty or put into one bin under the constraint that the total size of the items in the bin is no greater than 1 no item can be spread into more than one bin the objective is to minimize the total number of used bins plus the total penalty paid for the rejected items we call the problem bin packing with rejection penalties and denote it as bpr for the on line bpr problem we present an algorithm with an absolute competitive ratio of 2 618 while the lower bound is 2 343 and an algorithm with an asymptotic competitive ratio arbitrarily close to 1 75 while the lower bound is 1 540 for the off line bpr problem we present an algorithm with an absolute worst case ratio of 2 while the lower bound is 1 5 and an algorithm with an asymptotic worst case ratio of 1 5 we also study a closely related bin covering version of the problem in this case pi means some amount of profit if an item is rejected we get its profit or it can be put into a bin in such a way that the total size of the items in the bin is no smaller than 1 the objective is to maximize the number of covered bins plus the total profit of all rejected items we call this problem bin covering with rejection bcr for the on line bcr problem we show that no algorithm can have absolute competitive ratio greater than 0 and present an algorithm with asymptotic competitive ratio 1 2 which is the best possible for the off line bcr problem we also present an algorithm with an absolute worst case ratio of 1 2 which matches the lower bound

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

separating vectors and reflexivity abstract there is a close connection between separating vectors and reflexivity but the existence of a separating vector does not guarantee reflexivity this paper clarifies the relation we first find that every subspace of linear transformations whose dimension is less than the cardinality of the scalar field and having a disjoint pair of separating vectors is algebraically reflexive then we refine the result and conclude that a finite dimensional subspace of operators having a two dimensional subspace of separating vectors is algebraically reflexive if its scalar field is algebraically closed examples are provided to show that these results are sharp an application of the first result shows that finite dimensional spaces of operators of large rank are reflexive

hyperreflexivity and operator ideals abstract suppose b is an operator ideal and a is a linear space of operators between banach spaces x and y modifying the classical notion of hyperreflexivity we say that a is called b hyperreflexive if there exists a constant c such that for any t b x y with sup q t i the supremum runs over all isometric embeddings i into x and all quotient maps of y satisfying q a i 0 there exists a a for which t a c in this paper we give examples of b hyperreflexive spaces as well as of spaces failing this property in the last section we apply s e hyperreflexivity of operator algebras s e is a regular symmetrically normed operator ideal to constructing operator spaces with prescribed families of completely bounded maps

rigidity and crystallographic groups i this is the first in a series of articles about rigidity theorems for crystallographic groups the main theorem is 1 1 it is an h cobordism rigidity statement it implies see 1 6 that any crystallographic manifold with odd order holonomy which is h cobordant and simply homotopically equivalent to the standard one is actually homeomorphic to it in the second part of this work 9 we will exploit this result to prove that one can remove the word h cobordant from the previous sentence the main theorem concerns an involution on the group wh p mr this is the group of g h cobordisms of the flat torus mr of a crystallographic group f with holonomy group g definitions of these terms appear in 1 4 we prove a vanishing result of the tate cohomology

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

approximation by equivariant homeomorphisms i locally linear locally smoothable actions of finite groups on finite dimensional manifolds are considered in which two incident components of fixed point sets of subgroups either coincide or one has codimension at least three in the other for these actions an equivariant o approximation theorem is proved using engulfing techniques as corollaries are obtained equivariant fibrations are bundles and controlled h cobordism theorems as well as an equivariant version of edwards cell like mapping theorem and the vanishing of the set of transfer invariant g homotopy topological structures rel bound ary on tn x dp when tn is the n torus with trivial g action and dp is a representation disc here we consider locally linear locally smooth br pl and topological actions of a finite group g on n manifolds in addition we require that a g manifold m have gaps of codimension 3 i e for h c g if mh is a component of the h fixed point set mh and m h c mh is the subspace of points x for which the isotropy subgroup gx strictly contains h then either m h mh or m h has codimension 3 in mh we shall use the term not necessarily locally linear g manifold when we wish to drop the local linearity hypothesis but retain the property that if mh c m but mh m then mh is a locally flat inequivariantly submanifold of mk of codimension at least three this while self contained is the second in a series of papers sw2 8 s in which we analyze the extent of failure of the topological invariance of equivari ant whitehead torsion and the consequent failure of subgroups of the equivariant whitehead group whgl m of illman ill cf rothenberg r to classify pl or smooth g h cobordisms up to topological equivalence under our gap hypotheses the arguments of browder and quinn bq and rothenberg r show that g h cobordisms are classified up to pl or smooth equivalence by such subgroups we eventually conclude in sw7 s cf sw2 that the topological equivalence classes of g h cobordisms on a g manifold m that are products over the union of fixed point components of m having dimension 5 are classised by a group whg p m to which whgl m maps homomorphically the image in whg p m of the per tinent subgroup of whgl m classifies up to equivariant homeomorphism rel m those g h cobordisms on m that admit equivariant handlebody decompositions relative to m this homomorphism is not generally surjective and there are g h cobordisms that do not admit such handle decompositions cf sw5 s in order

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

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

on the relative reflexivity of finitely generated modules of operators let s be a von neumann algebra on a hilbert space z with commutant and centre w for each subspace e of s let ref r be the space of all b e w such that xby 0 for all x y e s satisfying xyy o if ref y the space y is called s reflexive if s z and 5 is an algebra containing the identity operator s reflexivity reduces to the usual reflexivity in operator theory the main result of the paper is the following if y is one dimensional or if r is arbitrary finitedimensional but s has no central portions of type in for n 1 then the space is s reflexive and the space is z reflexive where the bar denotes the closure in the ultraweak operator topology if s is a factor then is closed in the weak operator topology for each finite dimensional subspace 5 of s

locally linearly dependent operators and reflexivity of operator spaces we obtain two results on the existence of large subspaces of operators of small rank in locally linearly dependent spaces of operators as a consequence we obtain an upper bound for the rank of operators belonging to a minimal locally linearly dependent space of operators it has been known that the only obstruction to the reflexivity of a finite dimensional operator space comes from the operators with small ranks our results improve known bounds on the minimal rank that guarantees the reflexivity

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

an asymptotic fully polynomial time approximation scheme for bin covering in the bin covering problem there is a group l a1 an of items with sizes s ai 0 1 and the goal is to find a packing of the items into bins to maximize the number of bins that receive items of total size at least 1 this is a dual problem to the classical bin packing problem in this paper we present the first asymptotic fully polynomial time approximation scheme for the problem

on bin packing with conflicts we consider the offline and online versions of a bin packing problem called bin packing with conflicts given a set of items v 1 2 dots n with sizes s 1 s 2 dots s n in 0 1 and a conflict graph g v e the goal is to find a partition of the items into independent sets of g where the total size of items in each independent set is at most 1 so that the number of independent sets in the partition is minimized this problem is clearly a generalization of both the classical one dimensional bin packing problem where e emptyset and of the graph coloring problem where s i 0 for all i 1 2 dots n since coloring problems on general graphs are hard to approximate following previous work we study the problem on specific graph classes for the offline version we design improved approximation algorithms for perfect graphs and other special classes of graphs these are a frac52 2 5 approximation algorithm for perfect graphs a frac73 approx2 33333 approximation algorithm for a subclass of perfect graphs which contains interval graphs and chordal graphs and a frac74 1 75 approximation for algorithm bipartite graphs for the online problem on interval graphs we design a 4 7 competitive algorithm and show a lower bound of frac 155 36 approx4 30556 on the competitive ratio of any algorithm to derive the last lower bound we introduce the first lower bound on the asymptotic competitive ratio of any online bin packing algorithm with known optimal value which is frac 47 36 approx1 30556

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

the maximum resource bin packing problem usually for bin packing problems we try to minimize the number of bins used or in the case of the dual bin packing problem maximize the number or total size of accepted items this paper presents results for the opposite problems where we would like to maximize the number of bins used or minimize the number or total size of accepted items we consider off line and on line variants of the problems for the off line variant we require that there be an ordering of the bins so that no item in a later bin fits in an earlier bin we find the approximation ratios of two natural approximation algorithms first fit increasing and first fit decreasing for the maximum resource variant of classical bin packing for the on line variant we define maximum resource variants of classical and dual bin packing for dual bin packing no on line algorithm is competitive for classical bin packing we find the competitive ratio of various natural algorithms we study the general versions of the problems as well as the parameterized versions where there is an upper bound of 1 k on the item sizes for some integer k

bounded reflexivity of kernels of elementary operators abstract kernels of elementary operators of length two give rise to many interesting classes of operators in this paper we study bounded reflexivity of these kernels for general banach spaces a sufficient condition is given for the kernels to be boundedly reflexive in particular this assures that the commutant of any banach space operator is boundedly reflexive we also show that the bounded reflexive cover of the set of all toeplitz matrices in m m n is the entire m m n for finite dimensional banach spaces using kronecker canonical form for matrix pencils we obtain some necessary and sufficient conditions for the kernels to be boundedly reflexive these conditions enable us to easily determine bounded reflexivity by simply computing rank or nullity of matrices

the lazy bureaucrat scheduling problem we introduce a new class of scheduling problems in which the optimization is performed by the worker single machine who performs the tasks a typical worker s objective is to minimize the amount of work he does he is lazy or more generally to schedule as inefficiently in some sense as possible the worker is subject to the constraint that he must be busy when there is work that he can do we make this notion precise both in the preemptive and nonpreemptive settings the resulting class of perverse scheduling preblems which we denote lazy bureaucrat problems gives rise to a rich set of new questions that explore the distinction between maximization and minimization in computing optimal schedules

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

semiclassical non concentration near hyperbolic orbits for a large class of semiclassical pseudodifferential operators including schrodinger operators p h h2 g v x on compact riemannian manifolds we give logarithmic lower bounds on the mass of eigenfunctions outside neighbourhoods of generic closed hyperbolic orbits more precisely we show that if a is a pseudodifferential operator which is microlocally equal to the identity near the hyperbolic orbit and microlocally zero away from the orbit then u c log 1 h h p h u clog 1 h i a u this generalizes earlier estimates of colin de verdiere and parisse y colin de verdiere b parisse equilibre instable en regime semi classique i concentration microlocale comm partial differential equations 19 1994 1535 1563 equilibre instable en regime semi classique ii conditions de bohr sommerfeld ann inst h poincare phys theor 61 1994 347 367 obtained for a special case and of burq and zworski n burq m zworski geometric control in the presence of a black box j amer math soc 17 2004 443 471 for real hyperbolic orbits

selfish bin packing with cardinality constraints bin packing with cardinality constraints is a variant of bin packing in this problem items with sizes of at most 1 are to be partitioned or packed into subsets called bins such that the total size of items packed into a bin would not exceed 1 and each bin would contain at most k items for a given integer parameter k 1 we consider a class of games resulting from this problem by seeing the items as selfish players trying to be packed into a valid bin where the total size of items is maximized such games always admit pure nash equilibria we analyze the price of anarchy poa of such games defined as the asymptotic worst case ratio between the maximum number of bins in a pure nash equilibrium ne and the minimum number of bins in any valid solution that is the number of bins in a socially optimal solution we provide a complete analysis of the poa as a function of k we prove that for k 2 any ne is an optimal solution for k 4 the poa is exactly 2 1k and for k 3 the poa is exactly 117

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

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

online bin packing with arbitrary release times we study a new variant of the online bin packing problem in which each item a i is associated with a size a i and also a release time r i so that it must be placed at least r i above the bottom of a bin items arrive in turn and must be assigned without any knowledge of subsequent items the goal is to pack all items into unit size bins using the minimum number of bins we study the problem with all items have equal size first we show that the any fit algorithm cannot be approximated within any constant then we present a best possible online algorithm with asymptotic competitive ratio of two

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

compressions graphs and hyperreflexivity abstract in this note we extend the ideas in 11 to prove reflexivity and especially hyperreflexivity results that pertain to compressions rather than summands these results in turn are relevant for some remarkable classes of operators

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

new lower bounds for certain classes of bin packing algorithms on line algorithms have been extensively studied for the one dimensional bin packing problem in this paper we investigate two classes of one dimensional bin packing algorithms and we give better lower bounds for their asymptotic worst case behavior for on line algorithms so far the best lower bound was given by van vliet in 1992 12 he proved that there is no on line bin packing algorithm with better asymptotic performance ratio than 1 54014 in this paper we give an improvement on this bound to 248161 1 54037 and we investigate the parametric case as well for those lists where the elements are preprocessed according to their sizes in non increasing order csirik et al 1983 1 proved that no on line algorithm can have an asymptotic performance ratio smaller than 87 we improve this result to 5447

on the minimal rank in non reflexive operator spaces over finite fields let u and v be vector spaces over a field k and s be an n dimensional linear subspace of l u v the space s is called algebraically reflexive whenever it contains every linear map g u v such that for all x u there exists f s with g x f x a theorem of meshulam and semrl states that if s is not algebraically reflexive then it contains a non zero operator of rank at most 2n 2 provided that k has more than n 2 elements in the present article we prove that the provision on the cardinality of the underlying field is unnecessary to do so we demonstrate that the above result holds for all finite fields

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

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

reflexivity algebraic reflexivity and linear interpolation introduction this paper was motivated by a discovery that most finite dimensional subalgebras of b h for h an infinite dimensional hilbert space are reflexive with the only obstructions to reflexivity being finite rank considerations proofs section 2 do not depend on multiplicative structure nor on topology so extend to linear subspaces of transformations in an abstract setting abstract reflexivity has been studied in 3 4 7 primarily for singly generated algebras reflexivity properties can be interpreted as linear interpolation properties and we shall adopt this point of view the results of section 2 extend in section 3 to algebraic reflexivity counterparts for countably generated algebraically linear subspaces of bounded linear transformations acting on a banach space as consequences in section 4 we obtain generalizations of two single operator results one is a multivariate version of the result due to kaplansky 10 theorem 15 that a bounded locally algebraic operator acting on a banach space is algebraic the other is a multivariate version of the result due to douglas and foias 5 for hilbert space and to hadwin 7 for a general banach space extending work of fillmore 6 that a bounded non algebraic operator acting on a banach space is topologically algebraically reflexive in section 5 we give an application kindly suggested by d hadwin concerning joint strong similarity orbits of n tupules of operators we wish to thank e azoff k davidson j erdos d hadwin j kraus c lance and a sourour for conversations and some correspon

on line sequential bin packing we consider a sequential version of the classical bin packing problem in which items are received one by one before the size of the next item is revealed the decision maker needs to decide whether the next item is packed in the currently open bin or the bin is closed and a new bin is opened if the new item does not fit it is lost if a bin is closed the remaining free space in the bin accounts for a loss the goal of the decision maker is to minimize the loss accumulated over n periods we present an algorithm that has a cumulative loss not much larger than any strategy in a finite class of reference strategies for any sequence of items special attention is payed to reference strategies that use a fixed threshold at each step to decide whether a new bin is opened some positive and negative results are presented for this case

reflexivity and hyperreflexivity of operator spaces abstract in this paper we show that if s is an n dimensional subspace of l v such that every nonzero transformation of s has rank greater than or equal to 2 n 1 then s is algebraically reflexive if s is an n dimensional subspace of b h such that every nonzero transformation of s has rank greater than or equal to 2 n 1 then s is hyperreflexive we also consider how to construct some new hyperreflexive subspaces

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