on the approximate tradeoff for bicriteria batching and parallel machine scheduling problems we consider multiobjective scheduling problems i e scheduling problems that are evaluated with respect to many cost criteria and we are interested in determining a trade off pareto curve among these criteria we study two types of bicriteria scheduling problems single machine batching problems and parallel machine scheduling problems instead of proceeding in a problem by problem basis we identify a class of multiobjective optimization problems possessing a fully polynomial time approximation scheme fptas for computing an approximate pareto curve this class contains a set of problems whose pareto curve can be computed via a simple pseudo polynomial dynamic program for which the objective and transition functions satisfy some easy to verify arithmetical conditions our study is based on a recent work of woeginger electronic colloquium on computational complexity report 84 short version appeared in soda 99 pp 820 829 for the single criteria optimization ex benevolent problems we show how our general result can be applied to the considered scheduling problems

using dual approximation algorithms for scheduling problems theoretical and practical results the problem of scheduling a set of n jobs on m identical machines so as to minimize the makespan time is perhaps the most well studied problem in the theory of approximation algorithms for np hard optimization problems in this paper the strongest possible type of result for this problem a polynomial approximation scheme is presented more precisely for each e an algorithm that runs in time o n e 1 e 2 and has relative error at most e is given in addition more practical algorithms for e 1 5 2 k and e 1 6 2 k which have running times o n k log n and o n km 4 log n are presented the techniques of analysis used in proving these results are extremely simple especially in comparison with the baroque weighting techniques used previously the scheme is based on a new approach to constructing approximation algorithms which is called dual approximation algorithms where the aim is to find superoptimal but infeasible solutions and the performance is measured by the degree of infeasibility allowed this notion should find wide applicability in its own right and should be considered for any optimization problem where traditional approximation algorithms have been particularly elusive

a determinant formula for a holonomic q difference system associated with jackson integrals of type bcn abstract a jackson integral of type bc n is a multisum generalization of the very well poised balanced 2 r 2 r basic hypergeometric series we state an explicit product formula for the determinant of a matrix with entries given by the bc n type jackson integrals in order to show this we treat the determinant as a solution of a holonomic q difference equation in particular we give the q difference equation explicitly as a two term recurrence relation which the determinant satisfies by introducing a set of new symmetric polynomials via the symplectic schur functions

rational function certification of multisum integral q identities the method of rational function certification for proving terminating hypergeometric identities is extended from single sums or integrals to multi integral sums and q integral sums

approximation schemes for scheduling and covering on unrelated machines we examine the problem of assigning n independent jobs to m unrelated parallel machines so that each job is processed without interruption on one of the machines and at any time every machine processes at most one job we focus on the case where m is a fixed constant and present a new rounding approach that yields approximation schemes for multi objective minimum makespan scheduling with a fixed number of linear cost constraints the same approach gives approximation schemes for covering problems like maximizing the minimum load on any machine and for assigning specific or equal loads to the machines

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

elliptic hypergeometric series on root systems abstract we derive a number of summation and transformation formulas for elliptic hypergeometric series on the root systems a n c n and d n in the special cases of classical and q series our approach leads to new elementary proofs of the corresponding identities

how asymmetry helps load balancing this article deals with randomized allocation processes placing sequentially n balls into n bins we consider multiple choice algorithms that choose d locations bins for each ball at random inspect the content of these locations and then place the ball into one of them for example in a location with minimum number of balls the goal is to achieve a good load balancing this objective is measured in terms of the maximum load that is the maximum number of balls in the same bin multiple choice algorithms have been studied extensively in the past previous analyses typically assume that the d locations for each ball are drawn uniformly and independently from the set of all bins we investigate whether a nonuniform or dependent selection of the d locations of a ball may lead to a better load balancing three types of selection resulting in three classes of algorithms are distinguished 1 uniform and independent 2 nonuniform and independent and 3 nonuniform and dependent our first result shows that the well studied uniform greedy algorithm class 1 does not obtain the smallest possible maximum load in particular we introduce a nonuniform algorithm class 2 that obtains a better load balancing surprisingly this algorithm uses an unfair tie breaking mechanism called always go left resulting in an asymmetric assignment of the balls to the bins our second result is a lower bound showing that a dependent allocation class 3 cannot yield significant further improvement our upper and lower bounds on the maximum load are tight up to additive constants proving that the always go left algorithm achieves an almost optimal load balancing among all sequential multiple choice algorithm furthermore we show that the results for the always go left algorithm can be generalized to allocation processes with more balls than bins and even to infinite processes in which balls are inserted and deleted by an oblivious adversary

optimal semi online preemptive algorithms for machine covering on two uniform machines in this paper we consider the semi online preemptive scheduling problem with decreasing job sizes on two uniform machines our goal is to maximize the continuous period of time starting from time zero when both machines are busy which is equivalent to maximizing the minimum machine completion time if idle time is not introduced before all the jobs are completed we design optimal deterministic semi online algorithms for every machine speed ratio s 1 and show that idle time is required during the assignment procedure of algorithms for any s 6 2 the competitive ratios of the algorithms match the randomized lower bound for every 1 s 3 the problem of whether randomization still does not help for the discussed preemptive scheduling problem remains open

a bilateral extension of the q selberg integral a multi dimensional bilateral q series extending the q selberg integral is studied using concepts of truncation regularization and connection formulae following aomoto s method which involves regarding theq series as a solution of aq difference equation fixed by its asymptotic behavior an infinite product evaluation is obtained the q difference equation is derived applying the shifted symmetric polynomials introduced by knop and sahi as a special case of the infinite product formula askey evans s q selberg integral evaluation and its generalization by tarasov varchenko and stokman is reclaimed and an explanation in the context of aomoto s setting is thus provided 2010 mathematical subject classification primary 33d15 33d67 secondary 39a13

bgg reciprocity for current algebras abstract we study the category i gr of graded representations with finite dimensional graded pieces for the current algebra g c t where g is a simple lie algebra this category has many similarities with the category o of modules for g and in this paper we formulate and study an analogue of the famous bgg duality we recall the definition of the projective and simple objects in i gr which are indexed by dominant integral weights the role of the verma modules is played by a family of modules called the global weyl modules we show that in the case when g is of type s l 2 the projective module admits a flag in which the successive quotients are finite direct sums of global weyl modules the multiplicity with which a particular weyl module occurs in the flag is determined by the multiplicity of a jordan holder series for a closely associated family of modules called the local weyl modules we conjecture that the result remains true for arbitrary simple lie algebras we also prove some combinatorial product sum identities involving kostka polynomials which arise as a consequence of our theorem

two semi online scheduling problems on two uniform machines this paper considers two semi online scheduling problems one with known optimal value and the other with known total sum on two uniform machines with a machine speed ratio of s 1 for the first problem we provide an optimal algorithm for s 1 3 2 1 21 4 and improved algorithms or and lower bounds for s 1 21 4 3 over which the optimal algorithm is unknown as a result the largest gap between the competitive ratio and the lower bound decreases to 0 02192 for the second problem we also present algorithms and lower bounds for s 1 the largest gap between the competitive ratio and the lower bound is 0 01762 and the length of the interval over which the optimal algorithm is unknown is 0 47382 our algorithms and lower bounds for these two problems provide insights into their differences which are unusual from the viewpoint of the known results on these two semi online scheduling problems in the literature

rarefied elliptic hypergeometric functions abstract two exact evaluation formulae for multiple rarefied elliptic beta integrals related to the simplest lens space are proved they generalize evaluations of the type i and ii elliptic beta integrals attached to the root system c n in a special n 1 case the simplest p 0 limit is shown to lead to a new class of q hypergeometric identities symmetries of a rarefied elliptic analogue of the euler gauss hypergeometric function are described and the respective generalization of the hypergeometric equation is constructed some extensions of the latter function to c n and a n root systems and corresponding symmetry transformations are considered an application of the rarefied type ii c n elliptic hypergeometric function to some eigenvalue problems is briefly discussed

hyperbolic beta integrals abstract hyperbolic beta integrals are analogues of euler s beta integral in which the role of euler s gamma function is taken over by ruijsenaars hyperbolic gamma function they may be viewed as q q bibasic analogues of the beta integral in which the two bases q and q are interrelated by modular inversion and they entail q analogues of the beta integral for q 1 the integrals under consideration are the hyperbolic analogues of the ramanujan integral the askey wilson integral and the nassrallah rahman integral we show that the hyperbolic nassrallah rahman integral is a formal limit case of spiridonov s elliptic nassrallah rahman integral

some q beta and mellin barnes integrals on compact lie groups and lie algebras multidimensional generalizations of beta type integrals of bames ramanujan askey wilson and others are evaluated these integrals are analogues of the summation theorems for multilateral hypergeometric series associated to the simple lie algebras of classical type and type g2 many of these integrals can also be written as group integrals over a compact lie group or conjugation invariant integrals over the corresponding lie algebra

an algebraic approach to macdonald koornwinder polynomials rodrigues type formula and inner product identity we study macdonald koornwinder polynomials in the context of double affine hecke algebras nonsymmetric macdonald koornwinder polynomials are constructed by use of raising operators provided by a representation theory of the double affine hecke algebra associated with a2l 2 type affine root system this enables us to evaluate diagonal terms of scalar products of the nonsymmetric polynomials algebraically the macdonald koornwinder polynomials are expressed by linear combinations of the nonsymmetric counterparts we show a new proof of the inner product identity of the macdonald koornwinder polynomials without opdam cherednik s shift operators

an algorithmic proof theory for hypergeometric ordinary and q multisum integral identities it is shown that every proper hypergeometric multisum integral identity orq identity with a fixed number of summations and or integration signs possesses a short computer constructible proof we give a fast algorithm for finding such proofs most of the identities that involve the classical special functions of mathematical physics are readily reducible to the kind of identities treated here we give many examples of the method including computer generated proofs of identities of mehta dyson selberg hille hardy q saalschutz and others the prospect of using the method for proving multivariate identities that involve an arbitrary number of summations integrations is discussed

some multidimensional integrals related to many body systems with the 1 r2 potential an n dimensional integral evaluated by k aomoto 1988 is shown to represent the density matrix for an impurity particle in the 1 r2 quantum many body problem on a line the value of the n dimensional integral representing the same density matrix in periodic boundary conditions is conjectured as too is the value of an n dimensional integral which represents a two point correlation function in the system also the partition function of a related classical hamiltonian is evaluated by formulating a conjecture which asserts that the sum of jacobians of a certain change of variables in n dimensions is a constant

a calculus for su 3 leading to an algebraic formula for the clebsch gordan coefficients we develop a simple computational tool for su 3 analogous to bargmann s calculus for su 2 crucial new inputs are i explicit representation of the gelfand zetlin basis in terms of polynomials in four variables and positive or negative integral powers of a fifth variable ii an auxiliary gaussian measure with respect to which the gelfand zetlin states are orthogonal but not normalized iii simple generating functions for generating all basis states and also all invariants as an illustration of our techniques an algebraic formula for the clebsch gordan coefficients is obtained for the first time this involves only gaussian integrations thus su 3 is made as accessible for computations as su 2 is

a generalization of the sears slater transformation and elliptic lagrange interpolation of type bcn abstract the connection formula for the jackson integral of type b c n is obtained in the form of a sears slater type expansion of a bilateral multiple basic hypergeometric series as a linear combination of several specific bilateral multiple series the coefficients of this expansion are expressed by certain elliptic lagrange interpolation functions analyzing basic properties of the elliptic lagrange interpolation functions an explicit determinant formula is provided for a fundamental solution matrix of the associated system of q difference equations

online bin stretching with bunch techniques we are given a sequence of items that can be packed into m unit size bins and the goal is to assign these items online to m bins while minimizing the stretching factor bins have infinite capacities and the stretching factor is the size of the largest bin we present an algorithm with stretching factor 26 17 1 5294 improving the best known algorithm by kellerer and kotov 2013 1 with a stretching factor 11 7 1 5714 our algorithm has 2 stages and uses bunch techniques we aggregate bins into batches sharing a common purpose

on line bin stretching we are given a sequence of items that can be packed into m unit size bins in the classical bin packing problem we fix the size of the bins and try to pack the items in the minimum number of such bins in contrast in the bin stretching problem we fix the number of bins and try to pack the items while stretching the size of the bins as least as possible we present two on line algorithms for the bin stretching problem that guarantee a stretching factor of 5 3 for any number m of bins we then combine the two algorithms and design an algorithm whose stretching factor is 1 625 for any m the analysis for the performance of this algorithm is tight the best lower bound for any algorithm is 4 3 for any m 2 we note that the bin stretching problem is also equivalent to the classical scheduling load balancing problem in which the value of the makespan maximum load is known in advance copyright 2001 elsevier science b v

q difference shift for a bcn type jackson integral arising from elementary symmetric polynomials we study a q difference equation of a bcn type jackson integral introduced by van diejen the bcn type jackson integral is a multiple q series generalized from a q analogue of selberg s integral the equation is characterized by some new symmetric polynomials defined via symplectic schur functions as an application we give another proof of a product formula for the bcn type jackson integral which is equivalent to the so called q macdonald morris identity for the root system bcn first obtained by gustafson

macdonald s constant term conjectures for exceptional root systems we announce proofs of macdonald s constant term conjectures for the affine root systems s f 4 and s f 4 v we also give an algorithm for deciding the conjectures for the remaining root systems s e 6 s e 7 and s e 8 and prove that the constant term in question can indeed be expressed in closed form combined with previous work of zeilberger bressoud kadell and gustafson our results imply that macdonald s conjectures are true in form for any root system

a new boson calculus for su 3 it is shown that the gelfand weyl pattern for su 3 can be uniquely expressed in terms of four non negative and one positive or negative free integers this provides an optimal albeit non linear boson calculus for su 3 in terms of four harmonic oscillators and one planar rotor by optimal it is meant that every irreducible representation is obtained once and once only our techniques can be generalized to other groups

properties of some families of hypergeometric orthogonal polynomials in several variables limiting cases are studied of the koornwinder macdonald multivariable generalization of the askey wilson polynomials we recover recently and not so recently introduced families of hypergeometric orthogonal polynomials in several variables consisting of multivariable wilson continuous hahn and jacobi type polynomials respectively for each class of polynomials we provide systems of difference or differential equations recurrence relations and expressions for the norms of the polynomials in terms of the norm of the constant polynomial

inversions of integral operators and elliptic beta integrals on root systems we prove a novel type of inversion formula for elliptic hypergeometric integrals associated to a pair of root systems using the a c inversion formula to invert one of the known c type elliptic beta integrals we obtain a new elliptic beta integral for the root system of type a validity of this integral is established by a different method as well

the importance of the selberg integral it has been remarked that a fair measure of the impact of atle selberg s work is the number of mathematical terms that bear his name one of these is the selberg integral an n dimensional generalization of the euler beta integral we trace its sudden rise to prominence initiated by a question to selberg from enrico bombieri more than thirty years after its initial publication in quick succession the selberg integral was used to prove an outstanding conjecture in random matrix theory and cases of the macdonald conjectures it further initiated the study of q analogues which in turn enriched the macdonald conjectures we review these developments and proceed to exhibit the sustained prominence of the selberg integral as evidenced by its central role in random matrix theory calogero sutherland quantum many body systems knizhnik zamolodchikov equations and multivariable orthogonal polynomial theory

a polynomial approximation scheme for scheduling on uniform processors using the dual approximation approach in this paper we present a polynomial approximation scheme for the minimum makespan problem on uniform parallel processors more specifically the problem is to find a schedule for a set of independent jobs on a collection of machines of different speeds so that the last job to finish is completed as quickly as possible we give a family of polynomial time algorithms a such that a delivers a solution that is within a relative error of e of the optimum the technique employed is the dual approximation approach where infeasible but superoptimal solutions for a related dual problem are converted to the desired feasible but possibly suboptimal solution

complementary group resolution of the su n outer multiplicity problem i the littlewood rules and a complementary u 2n 2 group structure a complementary group to su n is found that realizes all features of the littlewood rules for kronecker products of su n representations this is accomplished by considering a state of su n to be a special gel fand state of the complementary group u 2n 2 with labels of the latter used to distinguish multiple occurrences of irreducible representations of su n irreps in the su n su n su n decomposition that is obtained from the littlewood rules furthermore this realization also helps us to determine su n su n 1 u 1 reduced wigner coefficients rwcs frequently called isoscalar factors and clebsch gordan coefficients cgcs or full nonreduced wigner coefficients of su n using algebraic or numeric methods in either the canonical or a noncanonical basis new explicit formulas for the su 3 and su 4 multiplicities are obtained by using this technique

a determinant formula associated with the elliptic hypergeometric integrals of type bcn we establish a determinant formula for the bilinear form associated with the elliptic hypergeometric integrals of type bcn by studying the structure of q difference equations to be satisfied by them the determinant formula is proved by combining the q difference equations of the determinant and its asymptotic analysis along the singularities the elliptic interpolation functions of type bcn are essentially used in the study of the q difference equations we establish a determinant formula for the bilinear form associated with the elliptic hypergeometric integrals of type bcn by studying the structure of q difference equations to be satisfied by them the determinant formula is proved by combining the q difference equations of the determinant and its asymptotic analysis along the singularities the elliptic interpolation functions of type bcn are essentially used in the study of the q difference equations

askey wilson polynomials by means of a q selberg type integral abstract an integral representation of the askey wilson polynomials is presented in terms of a q selberg type integral our motivation consists in the study of q selberg type integrals from the viewpoint of de rham theory or holonomic systems

analysis of greedy solutions for a replacement part sequencing problem this paper considers the problem of sequencing the installation of replacement parts for the repeated repair of a machine consisting of two working parts a finite inventory of spares is initially available for each type of part each spare has a known deterministic field life and spares of each type may have different field lives the inventories are eventually depleted since no replacement or repairs of parts occur the primary objective is to obtain a sequencing policy which minimizes the total number of installations replacements this problem is a special case of many opportunistic replacement problems and is called a build problem the motivation for this work is based on the maintenance of modular gas turbine aircraft engines used by domestic airlines and the u s air force we show this build problem is np hard which indicates the necessity of efficient heuristic solution procedures this paper considers several greedy type algorithms based on grouping parts according to exact matching of field lives then a generalization is presented which allows approximate matches the focus of the analysis is to obtain tight worst case bounds on the number of replacements obtained by each of the greedy algorithms

twisted strong macdonald theorems and adjoint orbits the strong macdonald theorems state that for l reductive and s an odd variable the cohomology algebras h l z z n and h l z s are freely generated and describe the cohomological s and z degrees of the generators the resulting identity for the z weighted euler characteristic is equivalent to macdonald s constant term identity for a finite root system we calculate h mathfrak p z n mathfrak p and h mathfrak p s for mathfrak p a standard parahoric in a twisted loop algebra giving strong macdonald theorems that take into account both a parabolic component and a possible diagram automorphism twist in particular we show that h mathfrak p z n mathfrak p contains a parabolic subalgebra of the coinvariant algebra of the fixed point subgroup of the weyl group of l and thus is no longer free we also prove a strong macdonald theorem for h mathfrak b s mathfrak n and h mathfrak b z n mathfrak n when mathfrak b and mathfrak n are iwahori and nilpotent subalgebras respectively of a twisted loop algebra for each strong macdonald theorem proved taking z weighted euler characteristics gives an identity equivalent to macdonald s constant term identity for the corresponding affine root system as part of the proof we study the regular adjoint orbits for the adjoint action of the twisted arc group associated to l proving an analogue of the kostant slice theorem

an approximation algorithm for the generalized assignment problem the generalized assignment problem can be viewed as the following problem of scheduling parallel machines with costs each job is to be processed by exactly one machine processing jobj on machinei requires timep ij and incurs a cost ofc ij each machinei is available fort i time units and the objective is to minimize the total cost incurred our main result is as follows there is a polynomial time algorithm that given a valuec either proves that no feasible schedule of costc exists or else finds a schedule of cost at mostc where each machinei is used for at most 2t i time units we also extend this result to a variant of the problem where instead of a fixed processing timep ij there is a range of possible processing times for each machine job pair and the cost linearly increases as the processing time decreases we show that these results imply a polynomial time 2 approximation algorithm to minimize a weighted sum of the cost and the makespan i e the maximum job completion time we also consider the objective of minimizing the mean job completion time we show that there is a polynomial time algorithm that given valuesm andt either proves that no schedule of mean job completion timem and makespant exists or else finds a schedule of mean job completion time at mostm and makespan at most 2t

a multidimensional generalization of the barnes integral and the continuous hahn polynomial employing a multidimensional generalization of the beta integral of barnes we derive an expression of the continuous hahn polynomials motivation of the present work is to study such integrals from the viewpoint of a finite difference version of the de rham theory

the missing link operators for labelling multiplicity in the clebsch gordan series it is indicated that there are invariant operators on the direct product space of irreducible representations of lie groups which distinguish between repeating representations they complete the labelling of the clebsch gordan series

a unified approach to approximation algorithms for bottleneck problems in this paper a powerful and yet simple technique for devising approximation algorithms for a wide variety of np complete problems in routing location and communication network design is investigated each of the algorithms presented here delivers an approximate solution guaranteed to be within a constant factor of the optimal solution in addition for several of these problems we can show that unless p np there does not exist a polynomial time algorithm that has a better performance guarantee

branching formula for macdonald koornwinder polynomials abstract we present an explicit branching formula for the six parameter macdonald koornwinder polynomials with hyperoctahedral symmetry

on line load balancing the setup for our problem consists of n servers that must complete a set of tasks each task can be handled only by a subset of the servers requires a different level of service and once assigned cannot be reassigned we make the natural assumption that the level of service is known at arrival time but that the duration of service is not the on line load balancing problem is to assign each task to an appropriate server in such a way that the maximum load on the servers is minimized in this paper we derive matching upper and lower bounds for the competitive ratio of the on line greedy algorithm for this problem namely 3n 23 2 1 o 1 and derive a lower bound n12 for any other deterministic or randomized on line algorithm

the q dixon anderson integral and multi dimensional 1 psi 1 summations the dixon anderson integral is a multi dimensional integral evaluation fundamental to the theory of the selberg integral the 11 summation is a bilateral generalization of the q binomial theorem it is shown that a q generalization of the dixon anderson integral due to evans and multi dimensional generalizations of the 11 summation due to milne and gustafson can be viewed as having a common origin in the theory of q difference equations as expounded by aomoto each is shown to be determined by a q difference equation of rank 1 and a certain asymptotic behavior in calculating the latter essential use is made of the concepts of truncation regularization and connection formulae