heuristics for the container loading problem abstract the knapsack container loading problem is the problem of loading a subset of rectangular boxes into a rectangular container of fixed dimensions such that the volume of the packed boxes is maximized a new heuristic based on the wall building approach is proposed which decomposes the problem into a number of layers which again are split into a number of strips the packing of a strip may be formulated and solved optimally as a knapsack problem with capacity equal to the width or height of the container the depth of a layer as well as the thickness of each strip is decided through a branch and bound approach where at each node only a subset of branches is explored several ranking rules for the selection of the most promising layer depths and strip widths are presented and the performance of the corresponding algorithms is experimentally compared for homogeneous and heterogeneous instances the best ranking rule is then used in a comprehensive computational study involving large sized instances these computational results show that instances with a total box volume up to 90 easily may be solved to optimality and that average fillings of the container volume exceeding 95 may be obtained for large sized instances

p torsion monodromy representations of elliptic curves over geometric function fields given a complex quasiprojective curve b and a nonisotrivial familye of elliptic curves over b the p torsione p yields a monodromy representation e p 1 b gl2 fp we prove that if e p e0 p thene ande 0 are isogenous provided p is larger than a constant depending only on the gonality of b this can be viewed as a function eld analog of the frey mazur conjecture which states that an elliptic curve over q is determined up to isogeny by its p torsion galois representation for p 17 the proof relies on hyperbolic geometry and is therefore only applicable in characteristic 0

measured early lateral energy fractions in concert halls and opera houses in the 30 years since early lateral reflections were first suggested as important for concert halls spatial impression and source broadening have become almost universally accepted as essential characteristics of halls with good acoustics two objective measures of source broadening have been proposed measured values of the best defined of these measures the early lateral energy fraction lf are considered here results from two independent measurement surveys are discussed comparisons of lf values by hall show a significant link between hall mean lf and hall width there is however considerable overlap between measured lf values in different halls so the relevance of describing halls by their mean early lateral energy fraction values is questionable the behaviour of lf values within auditoria is discussed for different concert hall plan forms and within opera houses a measure of source broadening including sound level is proposed and results considered in the context of auditorium design

new lower bounds for bin packing problems with conflicts abstract the bin packing problem with conflicts bpc consists of minimizing the number of bins used to pack a set of items where some items cannot be packed together in the same bin due to compatibility restrictions the concepts of dual feasible functions dff and data dependent dual feasible functions ddff have been used in the literature to improve the resolution of several cutting and packing problems in this paper we propose a general framework for deriving new ddff as well as a new concept of generalized data dependent dual feasible functions gddff a conflict generalization of ddff the gddff take into account the structure of the conflict graph using the techniques of graph triangulation and tree decomposition then we show how these techniques can be used in order to improve the existing lower bounds

approximation of the k batch consolidation problem we consider a problem of minimizing the number of batches of a fixed capacity processing the orders of various sizes on a finite set of items this batch consolidation problem is motivated by the production system typical in raw material industries in which multiple items can be processed in the same batch if they share sufficiently close production parameters if the number of items processed in a batch is restricted to being up to some fixed integer k the problem is referred to as the k batch consolidation problem we will show that the k batch consolidation problem admits an approximation whose factor is twice that of the k set cover problem in particular this implies an upper bound on the approximation factor 2hk 1 where hk 1 1 k

bounded coloring of co comparability graphs and the pickup and delivery tour combination problem the double traveling salesman problem with multiple stacks is a vehicle routing problem in which pickups and deliveries must be performed in two independent networks the items are stored in stacks and repacking is not allowed given a pickup and a delivery tour the problem of checking if there exists a valid distribution of items into s stacks of size h that is consistent with the given tours is known as pickup and delivery tour combination pdtc problem in the paper we show that the pdtc problem can be solved in polynomial time when the number of stacks s is fixed but the size of each stack is not we build upon the equivalence between the pdtc problem and the bounded coloring bc problem on permutation graphs for the latter problem s is the number of colors and h is the number of vertices that can get a same color we show that the bc problem can be solved in polynomial time when s is a fixed constant on co comparability graphs a superclass of permutation graphs to the contrary the bc problem is known to be hard on permutation graphs when h 6 is a fixed constant but s is unbounded

approximation algorithms for np complete problems on planar graphs this paper describes a general technique that can be used to obtain approximation schemes for various np complete problems on planar graphs the strategy depends on decompos ing a planar graph into subgraphs of a form we call k outerplanar for fixed k the problems of interest are solvable optimally in linear time on k outerplanar graphs by dynamic programming for general planar graphs if the problem is a maximization problem such as maximum independent set this technique gives for each k a linear time algorithm that produces a solution whose size is at least k k 1 optimal if the problem is a minimization problem such as minimum vertex cover it gives for each k a linear time algorithm that produces a solution whose size is at most k 1 k optimal taking k c log log nl or k c log nl where n is the number of nodes and c is some constant we get polynomial time approximation algorithms whose solution sizes converge toward optimal as n increases the class of problems for which this approach provides approximation schemes includes maximum independent set maximum tile salvage partition into triangles maximum h matching minimum vertex cover minimum dominat ing set and minimum edge dominating set for these and certain other problems the proof of solvability on k outerplanar graphs also enlarges the class of planar gmphs for which the problems are known to be solvable in polynomial time

the bin packing problem with precedence constraints given a set of identical capacitated bins a set of weighted items and a set of precedences among such items we are interested in determining the minimum number of bins that can accommodate all items and can be ordered in such a way that all precedences are satisfied the problem denoted as the bin packing problem with precedence constraints bpp p has a very intriguing combinatorial structure and models many assembly and scheduling issues according to our knowledge the bpp p has received little attention in the literature and in this paper we address it for the first time with exact solution methods in particular we develop reduction criteria a large set of lower bounds a variable neighborhood search upper bounding technique and a branch and bound algorithm we show the effectiveness of the proposed algorithms by means of extensive computational tests on benchmark instances and comparison with standard integer linear programming techniques

weighted and locally bounded list colorings in split graphs cographs and partial k trees abstract for a fixed number of colors we show that in node weighted split graphs cographs and graphs of bounded tree width one can determine in polynomial time whether a proper list coloring of the vertices of a graph such that the total weight of vertices of each color equals a given value in each part of a fixed partition of the vertices exists we also show that this result is tight in some sense by providing hardness results for the cases where any one of the assumptions does not hold the edge coloring variant is also studied as well as special cases of cographs and split graphs

new classes of fast lower bounds for bin packing problems the bin packing problem is one of the classical np hard optimization problems in this paper we present a simple generic approach for obtaining new fast lower bounds based on dual feasible functions worst case analysis as well as computational results show that one of our classes clearly outperforms the previous best economical lower bound for the bin packing problem by martello and toth which can be understood as a special case in particular we prove an asymptotic worst case performance of 3 4 for a bound that can be computed in linear time for items sorted by size in addition our approach provides a general framework for establishing new bounds

restrictions of graph partition problems let j be a set of n jobs all with job processing time or job length one and let m be a set of k machines incompatibilities between jobs are described by an undirected graph g j e with vertex set j if g contains an edge between two jobs j and j j j we demand that these two jobs can not be executed by the same machine an assignment of the jobs to the machines which satisfies the incompatibility relation is called a schedule the processing time of a machine is given by the number of assigned jobs since each job has processing time one

algorithms for two dimensional cutting stock and strip packing problems using dynamic programming and column generation abstract we investigate several two dimensional guillotine cutting stock problems and their variants in which orthogonal rotations are allowed we first present two dynamic programming based algorithms for the rectangular knapsack rk problem and its variants in which the patterns must be staged the first algorithm solves the recurrence formula proposed by beasley the second algorithm for staged patterns also uses a recurrence formula we show that if the items are not so small compared to the dimensions of the bin then these algorithms require polynomial time using these algorithms we solved all instances of the rk problem found at the or library including one for which no optimal solution was known we also consider the two dimensional cutting stock problem we present a column generation based algorithm for this problem that uses the first algorithm above mentioned to generate the columns we propose two strategies to tackle the residual instances we also investigate a variant of this problem where the bins have different sizes at last we study the two dimensional strip packing problem we also present a column generation based algorithm for this problem that uses the second algorithm above mentioned where staged patterns are imposed in this case we solve instances for two three and four staged patterns we report on some computational experiments with the various algorithms we propose in this paper the results indicate that these algorithms seem to be suitable for solving real world instances we give a detailed description a pseudo code of all the algorithms presented here so that the reader may easily implement these algorithms

objective measures of listener envelopment this paper reports the results of subjective studies to determine objective predictors of perceived listener envelopment in concert halls subjects seated in an anechoic room were exposed to simulated sound fields that were expected to have varied listener envelopment as independent variables the reverberation time the early to late sound ratio the overall sound level and the angular distribution of the late arriving sound levels were varied all of these factors had statistically significant effects on perceived listener envelopment the results indicate however that the angular distribution of the late arriving sound and the overall level would have the largest effects in real concert halls thus listener envelopment depends on having strong lateral reflections arriving at the listener 80 ms or more after the direct sound several objective measures correlated significantly with listener envelopment however a new measure the late lateral sound level as measured using a figure of eight microphone was found to be both conceptually simple and a very good predictor of the perceived listener envelopment of the sound fields in this experiment

volumes of complex analytic subvarieties of hermitian symmetric spaces we give lower bounds of volumes of k dimensional complex analytic subvarieties of certain naturally defined domains in n dimensional complex space forms of constant positive zero or negative holomorphic sectional curvature for each 1 k n the lower bounds are sharp in the sense that these bounds are attained by k dimensional complete totally geodesic complex submanifolds such lower bounds are obtained by constructing singular potential functions corresponding to blow ups of the kahler metrics involved similar lower bounds are also obtained in the case of hermitian symmetric spaces of noncompact type in this case the lower bounds are sharp for those values of k at which the hermitian symmetric space contains k dimensional complete totally geodesic complex submanifolds which are complex hyperbolic spaces of minimum holomorphic sectional curvature

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

characterization of jacobian varieties in terms of soliton equations pour dans le demi espace superieur de siegel h g soit x c g z g z g la variete abelienne principalement polarisee correspondante et soit z z m e z gexp 2 i t mz i t m m la fonction theta de riemann pour x les deux conditions a et b pour x sont equivalentes a 1 il existe des vecteurs a 1 a 2 a 3 c g a 1 0 et une forme quadratique q t i j 1 3 q ij t i t j q ij c tel que pour c g t t exp q t t 1 a 1 t 2 a 2 t 3 a 3 donne une solution a l equation de kadomtsev petviashvili sous la forme d hirota d 1 4 3d 2 2 4d 1 d 3 0 2 le diviseur de x est irreductible b x est isomorphe a la variete jacobienne d une courbe lisse complete de genre g sur c

complexity of approximating bounded variants of optimization problems we study low degree graph problems such as maximum independent set and minimum vertex cover the goal is to improve approximation lower bounds for them and for a number of related problems like max b set packing min b set cover and max b dimensional matching b 3 we prove for example that it is np hard to achieve an approximation factor of 95 94 for max 3 dm and a factor of 48 47 for max 4 dm in both cases the hardness result applies even to instances with exactly two occurrences of each element

on the period matrix of a riemann surface of large genus with an appendix by j h conway and n j a sloane riemann showed that a period matrix of a compact riemann surface of genusg 1 satisfies certain relations we give a further simple combinatorial property related to the length of the shortest non zero lattice vector satisfied by such a period matrix see 1 13 in particular it is shown that for large genus the entire locus of jacobians lies in a very small neighborhood of the boundary of the space of principally polarized abelian varieties

scheduling agreement graph vs resource constraints abstract we investigate two scheduling problems the first is scheduling with agreements swa that consists in scheduling a set of jobs non preemptively on identical machines in a minimum time subject to constraints that only some specific jobs can be scheduled concurrently these constraints are represented by an agreement graph we extend the np hardness of swa with three distinct values of processing times to only two values and this definitely closes the complexity status of swa on two machines with two fixed processing times the second problem is the so called resource constrained scheduling we prove that swa is polynomially equivalent to a special case of the resource constrained scheduling and deduce new complexity results of the latter

on the hardness of approximating minimization problems we prove results indicating that it is hard to compute efficiently good approximate solutions to the graph coloring set covering and other related minimization problems specifically there is an e 0 such that graph coloring cannot be approximated with ratio n e unless p np set covering cannot be approximated with ratio c log n for any c 1 4 unless np is contained in dtime n poly log n similar results follow for related problems such as clique cover fractional chromatic number dominating set and others

a numerical criterion for very ample line bundles let x be a projective algebraic manifold of dimension n and let l be an ample line bundle over x we give a numerical criterion ensuring that the adjoint bundle kx l is very ample the sufficient conditions are expressed in terms of lower bounds for the intersection numbers l y over subvarieties y of x in the case of surfaces our criterion gives universal bounds and is only slightly weaker than i reider s criterion when dimx 3 and codimy 2 the lower bounds for l y involve a numerical constant which depends on the geometry of x by means of an iteration process it is finally shown that 2kx ml is very ample form 12n our approach is mostly analytic and based on a combination of hormander s l estimates for the operator lelong number theory and the aubin calabi yau theorem

a spectral technique for coloring random 3 colorable graphs let g3n p 3 be a random 3 colorable graph on a set of 3n vertices generated as follows first split the vertices arbitrarily into three equal color classes and then choose every pair of vertices of distinct color classes randomly and independently to be edges with probability p we describe a polynomial time algorithm that finds a proper 3 coloring of g3n p 3 with high probability whenever p geq c n where c is a sufficiently large absolute constant this settles a problem of blum and spencer who asked if an algorithm can be designed that works almost surely for p geq polylog n n j algorithms 19 1995 pp 204 234 the algorithm can be extended to produce optimal k colorings of random k colorable graphs in a similar model as well as in various related models implementation results show that the algorithm performs very well in practice even for moderate values of c

divisors on symmetric products of curves for a curve with general moduli the neron severi group of its symmetric products is generated by the classes of two divisors x and in this paper we give bounds for the cones of effective and ample divisors in the x plane

a new conformal invariant and its applications to the willmore conjecture and the first eigenvalue of compact surfaces let m be a compact riemannian manifold with a fixed conformal structure then we introduce the concept of conformal volume of m in the following manner for each branched conformal immersion q9 of m into the unit sphere s n we consider the set of all branched conformal immersions obtained by composi t ion of qo with the conformal automorphisms of s we let vc n qg be the max imum volume of these branched immersions the conformal volume of m is defined to be the infimum of v n q0 where qo ranges over all branched conformal immersions of m into the unit sphere s in this paper we study the case when m is a compact surface and we call the conformal volume of m to be the conformal area of m we demonst ra te that this conformal invariant is non trivial in fact we prove that if there exists a minimal immersion of m into s where coordinate functions are first eigenfunctions then the conformal area of m is given by the area of m with respect to the induced metric this enables us to compute the conformal area for several surfaces for example the conformal area of r p z is 6n and the conformal area of the square torus is 27c 2 we believe that the computa t ion of the conformal area for general surfaces will be very impor tant in studying the geometry of compact surfaces we demonstrate this claim by applying the concept of conformal area to two different branches of surface theory the first application is to study the total curvature of a compact surface in r this problem has a long history fenchel and fary 9 proved that for a closed curve o in r slk 27c where k is its curvature then milnor 12

exact solution of the two dimensional finite bon packing problem given a set of rectangular pieces to be cut from an unlimited number of standardized stock pieces bins the two dimensional finite bin packing problem is to determine the minimum number of stock pieces that provide all the pieces the problem is np hard in the strong sense and finds many practical applications in the cutting and packing area we analyze a well known lower bound and determine its worst case performance we propose not lower bounds which are used within a branch and bound algorithm for the exact solution of the problem extensive computational testing on problem instances from the literature involving up to 120 pieces shows the effectiveness of the proposed approach

simple linear time algorithms to test chordality of graphs test acyclicity of hypergraphs and selectively reduce acyclic hypergraphs chordal graphs arise naturally in the study of gaussian elimination on sparse symmetric matrices acyclic hypergraphs arise in the study of relational data bases rose tarjan and lueker siam j c

exact algorithms for the two dimensional strip packing problem with and without rotations we propose exact algorithms for the two dimensional strip packing problem 2sp with and without 90 rotations we first focus on the perfect packing problem pp which is a special case of 2sp wherein all given rectangles are required to be packed without wasted space and design branch and bound algorithms introducing several branching rules and bounding operations a combination of these rules yields an algorithm that is especially efficient for feasible instances of pp we then propose several methods of applying the pp algorithms to 2sp our algorithms succeed in efficiently solving benchmark instances of pp with up to 500 rectangles and those of 2sp with up to 200 rectangles they are often faster than existing exact algorithms specially tailored for problems without rotations

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

approximation algorithms for time constrained scheduling in this paper we consider the following time constrained scheduling problem given a set of jobsjwith execution timese j 0 1 and an undirected graphg j e we consider the problem to find a schedule for the jobs such that adjacent jobs j j eare assigned to different machines and that the total execution time for each machine is at most 1 the goal is to find a minimum number of machines to execute all jobs under this time constraint this scheduling problem is a natural generalization of the classical bin packing problem we propose and analyse several approximation algorithms with constant absolute worst case ratio for graphs that can be colored in polynomial time

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

the mutual exclusion scheduling problem for permutation and comparability graphs in this paper we consider the mutual exclusion scheduling problem for comparability graphs given an undirected graph g and a fixed constant m the problem is to find a minimum coloring of g such that each color is used at most m times the complexity of this problem for comparability graphs was mentioned as an open problem by mohring problem 9 10 in i rival ed graphs and orders reidel dordrecht 1985 p 583 and for permutation graphs a subclass of comparability graphs as an open problem by lonc on complexity of some chain and antichain partition problem in g schmidt r berghammer eds graph theoretical concepts in computer science wg 91 lecture notes in computer science vol 570 1999 pp 97 104 we prove that this problem is already np complete for permutation graphs and for each fixed constant m 6

distribution of lateral acoustic energy in mudejar gothic churches abstract in this work the physical measures of spatial impression are considered in 12 mudejar gothic churches in the city of seville in the south of spain this study describes the spatial distribution of the early and late lateral acoustic energy through monaural parameters derived from impulse response analysis using a maximum length sequence measurement system in each church in the first time analysis the two early lateral energy measures early lateral fraction lf and early lateral fraction cosine lfc are taken in order to assess apparent source width asw and the late lateral level gll in the second to assess listener envelopment lev are conducted parameters have been studied spectrally in each temple and were averaged at low and mid frequency values in their different naves in order to study how these two attributes of sound perception vary with source receiver distance experimental results have been compared with the theoretical early lateral energy fractions and late lateral level both of which are derived by assuming that reflected energy in these places of worship is solely dependent on source receiver distance this comparison is carried out in accordance with the model proposed by the authors in an earlier paper in order to describe the dependence of acoustic monaural omnidirectional energy parameters on source receiver distance thus it is supposed that the directional distribution of reflections is similar to a diffuse distribution to conclude these spatially averaged monoaural parameters have been correlated with geometric variables by using linear regression and only weak correlations with the mean width of the churches and with the height width ratio have been found

comparison of concert hall measurements of spatial impression two different types of measures of spatial impression lateral energy fractions and interaural cross correlations were measured at a number of locations in 14 different concert halls the mean and range of the hall average values of these quantities give an indication of typical conditions to be expected in other larger concert halls comparisons of hall average values of the two types of quantities showed them to be significantly related at medium and lower frequencies in good agreement with theoretical expectations individual position measurements were less well related and showed that other factors had a major influence on the individual position measurements in some halls the two types of quantities varied quite differently with increasing source receiver distance at a particular seat location the two types of measures cannot always give similar indications of expected spatial impression calculating hall average values from a small number of measurement locations is shown to introduce significant errors

mutual exclusion scheduling abstract mutual exclusion scheduling is the problem of scheduling unit time tasks non preemptively on m processors subject to constraints represented by a graph g such that tasks represented by adjacent vertices in g must run in disjoint time intervals this problem arises in load balancing the parallel solution of partial differential equations by domain decomposition minimizing the completion time is np hard even if either the number of processors or the completion time is fixed but greater than two however polynomial time is sufficient to produce optimal schedules for forests and simple heuristics perform well on certain classes of graphs for graphs derived from the two dimensional domain decomposition problem heuristics yield solutions within 4 c 7 time units of optimal where c is the maximal number of regions that touch each other at a single point in the domain decomposition these solutions are within a constant factor of optimal

restrictions of graph partition problems part i in this paper partition problems into k independent sets or cliques of bounded size k are analyzed for several classes of graphs we prove the computational complexity of both problems restricted to cographs split graphs bipartite graphs and interval graphs given general or constant k and k it is shown that the assignment problem for operations in a branching flow graph to processors each with a limit on the number of executable operations equals the first problem restricted to cographs in addition a job assignment problem given intervals for each job and k machines each executing at most k jobs equals the first problem restricted to interval graphs it is shown that both problem are np complete

equitable colorings of bounded treewidth graphs a proper coloring of a graph g is equitable if the sizes of any two color classes differ by at most one a proper coloring is l bounded when each color class has size at most l we consider the problems to determine for a given graph g and a given integer l whether g has an equitable l bounded k coloring we prove that both problems can be solved in polynomial time on graphs of bounded treewidth and show that a precolored version remains np complete on trees

a new exact method for the two dimensional orthogonal packing problem abstract the two dimensional orthogonal packing problem 2 opp consists in determining if a set of rectangles items can be packed into one rectangle of fixed size bin in this paper we propose two exact algorithms for solving this problem the first algorithm is an improvement on a classical branch bound method whereas the second algorithm is based on a new relaxation of the problem we also describe reduction procedures and lower bounds which can be used within enumerative methods we report computational experiments for randomly generated benchmarks which demonstrate the efficiency of both methods the second method is competitive compared to the best previous methods it can be seen that our new relaxation allows an efficient detection of non feasible instances

the influence of late arriving energy on spatial impression new subjective experiments using sound fields simulated in an anechoic room confirm that spatial impression in concert halls is composed of at least two distinct aspects apparent source width and listener envelopment previously published research has shown that apparent source width is related to the relative level of early lateral reflections this new work demonstrates that listener envelopment is related to the level direction of arrival and temporal distribution of late arriving reflections it is further shown that increased amounts of listener envelopment decrease the subject s sensitivity to changes in the apparent source width of the sound field listener envelopment is related to objective acoustical measures and the implications for concert hall design are discussed

knapsack problems with setups knapsack problems with setups find their application in many concrete industrial and financial problems moreover they also arise as subproblems in a dantzig wolfe decomposition approach to more complex combinatorial optimization problems where they need to be solved repeatedly and therefore efficiently here we consider the multiple class integer knapsack problem with setups items are partitioned into classes whose use implies a setup cost and associated capacity consumption item weights are assumed to be a multiple of their class weight the total weight of selected items and setups is bounded the objective is to maximize the difference between the profits of selected items and the fixed costs incurred for setting up classes a special case is the bounded integer knapsack problem with setups where each class holds a single item and its continuous version where a fraction of an item can be selected while incurring a full setup the paper shows the extent to which classical results for the knapsack problem can be generalized to these variants with setups in particular an extension of the branch and bound algorithm of horowitz and sahni is developed for problems with positive setup costs our direct approach is compared experimentally with the approach proposed in the literature consisting in converting the problem into a multiple choice knapsack with pseudo polynomial size

new lower bounds for the three dimensional orthogonal bin packing problem abstract in this paper we consider the three dimensional orthogonal bin packing problem which is a generalization of the well known bin packing problem we present new lower bounds for the problem from a combinatorial point of view and demonstrate that they theoretically dominate all previous results from the literature the comparison is also done concerning asymptotic worst case performance ratios the new lower bounds can be more efficiently computed in polynomial time in addition we study the non oriented model which allows items to be rotated

injectivity radius and gonality of a compact riemann surface we obtain a sharp lower bound for the volumes of purely 1 dimensional complex analytic subvarieties in a geodesic tubular neighborhood of the diagonal of the cartesian product of a compact riemann surface with itself this leads to a lower bound of the seshadri number of the canonical line bundle of the cartesian product with respect to the diagonal as a consequence we obtain an upper bound for the hyperbolic injectivity radii of compact riemann surfaces of a fixed gonality in particular we obtain the limiting behavior of the gonalities of a tower of compact riemann surfaces we also give an application of our results to an invariant related to the ample cone of the symmetric product of a riemann surface