the asymptotic optimality of the lpt rule for the problem of minimizing makespan on parallel machines of different speed the behaviour of list scheduling rules is subjected to a probabilistic analysis under the assumption that the processing requirements of the jobs are independent identically distributed nonnegative random variables under mild conditions on the probability distribution we obtain strong asymptotic optimality results for the lpt longest processing time rule in which the jobs are assigned to the machines in order of nonincreasing processing requirements

stochastic analysis of a modified first fit decreasing packing we make a stochastic analysis of a modified version mffd of first fit decreasing in which each bin is closed after it receives its first fallback item consider a probability measure on 0 1 and independent random variables x1 xn distributed according to let rn rx1 xn be the number of unit size bins that mffd needs to pack items of size x1 xn we prove that c limn ern n exists and that the random variable rn nc n converges in distribution the main tools are deterministic inequalities concerning mffd that might be of independent interest

optimal multi way number partitioning the np hard number partitioning problem is to separate a multiset s of n positive integers into k subsets such that the largest sum of the integers assigned to any subset is minimized the classic application is scheduling a set of n jobs with different runtimes on k identical machines such that the makespan the elapsed time to complete the schedule is minimized the two way number partitioning decision problem is one of the original 21 problems that richard karp proved np complete it is also one of garey and johnson s six fundamental np complete problems and the only one based on numbers this article explores algorithms for solving multi way number partitioning problems optimally we explore previous algorithms as well as our own algorithms which fall into three categories sequential number partitioning snp a branch and bound algorithm binary search improved bin completion bsibc a bin packing algorithm and cached iterative weakening ciw an iterative weakening algorithm we show experimentally that for large random numbers snp and ciw are state of the art algorithms depending on the values of n and k both algorithms outperform the previous state of the art by up to seven orders of magnitude in terms of runtime

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

the price of envy freeness in machine scheduling we consider k envy free assignments for scheduling problems in which the completion time of each machine is not k times larger than the one she could achieve by getting the jobs of another machine for a given factor k 1 we introduce and investigate the notion of price of k envy freeness defined as the ratio between the makespan of the best k envy free assignment and that of an optimal allocation achievable without envy freeness constraints we provide exact or asymptotically tight bounds on the price of k envy freeness for all the basic scheduling models that is unrelated related and identical machines moreover we show how to efficiently compute such allocations with a worsening multiplicative factor being at most the best approximation ratio for the minimum makespan problem guaranteed by a polynomial time algorithm for each specific model finally we extend our results to the case of restricted assignments and to the objective of minimizing the sum of the completion times of all the machines

the universal r matrix for u_q rm sl 3 and beyond ther matrices for the quantised lie algebrasa n are constructed through the quantum double procedure given by drinfel d 6 the case ofu q sl 3 is thoroughly analysed initially to demonstrate the more subtle points of the calculation the ease of the calculation fora n is very dependent on a choice of generators for the borel subalgebrau q b and its dual and a certain ordering imposed on these generators which is related to the length of a certain word in the weyl group

a polynomial time approximation scheme for the two stage multiprocessor flow shop problem in this paper we investigate the two stage multiprocessor flow shop scheduling problem f2 p cmax where the numbers m1 and m2 of machines available in the two stages are part of the input we demonstrate the existence of a polynomial time approximation scheme for this problem this result solves the simplest case of an open problem that has been posed by leslie hall in a recent paper hall 1995 an extension of our algorithm yields an approximation scheme for the closely related two stage multiprocessor job shop problem

finite dimensional representations of the quantum analog of the enveloping algebra of a complex simple lie algebra let g be a complex simple lie algebra we show that whent is not a root of 1 all finite dimensional representations of the quantum analogu t g are completely reducible and we classify the irreducible ones in terms of highest weights in particular they can be seen as deformations of the representations of the classical u g

scheduling and fixed parameter tractability fixed parameter tractability analysis and scheduling are two core domains of combinatorial optimization which led to deep understanding of many important algorithmic questions however even though fixed parameter algorithms are appealing for many reasons no such algorithms are known for many fundamental scheduling problems in this paper we present the first fixed parameter algorithms for classical scheduling problems such as makespan minimization scheduling with job dependent cost functions one important example being weighted flow time and scheduling with rejection to this end we identify crucial parameters that determine the problems complexity in particular we manage to cope with the problem complexity stemming from numeric input values such as job processing times which is usually a core bottleneck in the design of fixed parameter algorithms we complement our algorithms with mathsf w 1 w 1 hardness results showing that for smaller sets of parameters the respective problems do not allow fixed parameter algorithms in particular our positive and negative results for scheduling with rejection explore a research direction proposed by daniel marx

strongly polynomial algorithms for the high multiplicity scheduling problem a high multiplicity scheduling problem consists of many jobs which can be partitioned into relatively few groups where all the jobs within each group are identical polynomial and even strongly polynomial algorithms for the standard scheduling problem in which all jobs are assumed to be distinct become exponential for the corresponding high multiplicity problem in this paper we study various high multiplicity problems of scheduling unit time jobs on a single machine we provide strongly polynomial algorithms for constructing optimal schedules with respect to several measures of efficiency completion time lateness tardiness the number of tardy jobs and their weighted counterparts the algorithms require a number of operations that are polynomial in the number of groups rather than in the total number of jobs as a by product we identify a new family of nxn transportation problems which are solvable in o n log n time by a simple greedy algorithm

optimal bin packing with items of random sizes consider a probability on 0 1 and i i d random variables x1 x2 xn distributed like let qn denote the optimal minimum number of unit size bins needed to pack items of size x1 x2 xn we characterize the class of which have the property that limn qn n e x1 a s or equivalently that the expected level of occupancy of bins converges to one

scheduling to minimize average completion time off line and on line approximation algorithms in this paper we introduce two general techniques for the design and analysis of approximation algorithms for np hard scheduling problems in which the objective is to minimize the weighted sum of the job completion times for a variety of scheduling models these techniques yield the first algorithms that are guaranteed to find schedules that have objective function value within a constant factor of the optimum in the first approach we use an optimal solution to a linear programming relaxation in order to guide a simple list scheduling rule consequently we also obtain results about the strength of the relaxation our second approach yields on line algorithms for these problems in this setting we are scheduling jobs that continually arrive to be processed and for each time t we must construct the schedule until time t without any knowledge of the jobs that will arrive afterwards our on line technique yields constant performance guarantees for a variety of scheduling environments and in some cases essentially matches the performance of our off line lp based algorithms

probabilistic analysis of a generalized bin packing problem and applications we give a unified probabilistic analysis for a general class of bin packing problems by directly analyzing corresponding mathematical programs in this general class of packing problems objects are described by a given number of attribute values some attributes may be discrete others may be continuous bins are sets of objects and the collection of feasible bins is merely required to satisfy some general consistency properties we characterize the asymptotic optimal value as the value of an easily specified linear program whose size is independent of the number of objects to be packed or as the limit of a sequence of such linear program values we also provide bounds for the rate of convergence of the average cost to its asymptotic value the analysis suggests an a s asymptotically optimal heuristic that runs in linear time the heuristics can be designed to be asymptotically optimal while still running in polynomial time we also show that in several important cases the algorithm has both po

multidimensional optimal bin packing with items of random size consider a probability measure on 0 1 d and independent random variables x 1 x n distributed according to let vn vnx 1 x n resp rn rnx 1 x n be the minimal number of unit size d dimensional bins needed to pack items of size x 1 x n in the vector packing resp rectangular packing problem for a large class of distributions we characterize v limn vn n and r limn rn n

tight bounds and probabilistic analysis of two heuristics for parallel processor scheduling we study the partitioning problem consisting in partitioning a sublist of n positive numbers into m disjoint sublists such that the maximum sublist is minimized this is equivalent to minimizing the completion time of n jobs on m parallel identical processors we establish upper bounds on the deviation from optimality of two heuristics the well known lpt heuristic and the on line rlp heuristic these bounds serve to establish a probabilistic analysis of these heuristics for both of them the absolute deviation from optimality remains finite when the size of the list of numbers becomes infinite this is a stronger result than previous convergence theorems and it is valid whenever the processing times are iid random variables with finite mean and arbitrary distributions

approximation algorithms for single source unsplittable flow in the single source unsplittable flow problem we are given a network g a source vertex s and k commodities with sinks ti and real valued demands rho i 1 leq i leq k we seek to route the demand rho i of each commodity i along a single s ti flow path so that the total flow routed across any edge e is bounded by the edge capacity ue the conceptual difficulty of this np hard problem arises from combining packing constraints due to the existence of capacities with path selection in a graph of arbitrary topology in this paper we give a generic framework which yields approximation algorithms that are simpler than those previously known and achieve significant improvements upon the approximation ratios our framework with appropriate subroutines applies to all optimization versions previously considered and unlike previous work treats in a unified manner directed and undirected graphs we provide extensions of our algorithms which yield the best possible approximation guarantees for restricted sets of demand values and an associated scheduling problem

approximation algorithms for scheduling unrelated parallel machines we consider the following scheduling problem there arem parallel machines andn independent jobs each job is to be assigned to one of the machines the processing of jobj on machinei requires timep ij the objective is to find a schedule that minimizes the makespan our main result is a polynomial algorithm which constructs a schedule that is guaranteed to be no longer than twice the optimum we also present a polynomial approximation scheme for the case that the number of machines is fixed both approximation results are corollaries of a theorem about the relationship of a class of integer programming problems and their linear programming relaxations in particular we give a polynomial method to round the fractional extreme points of the linear program to integral points that nearly satisfy the constraints in contrast to our main result we prove that no polynomial algorithm can achieve a worst case ratio less than 3 2 unlessp np we finally obtain a complexity classification for all special cases with a fixed number of processing times

property testing and its connection to learning and approximation in this paper we consider the question of determining whether a function f has property p or is e far from any function with property p a property testing algorithm is given a sample of the value of f on instances drawn according to some distribution in some cases it is also allowed to query f on instances of its choice we study this question for different properties and establish some connections to problems in learning theory and approximation in particular we focus our attention on testing graph properties given access to a graph g in the form of being able to query whether an edge exists or not between a pair of vertices we devise algorithms to test whether the underlying graph has properties such as being bipartite k colorable or having a p clique clique of density p with respect to the vertex set our graph property testing algorithms are probabilistic and make assertions that are correct with high probability while making a number of queries that is independent of the size of the graph moreover the property testing algorithms can be used to efficiently i e in time linear in the number of vertices construct partitions of the graph that correspond to the property being tested if it holds for the input graph

representations of quantum matrix algebra mq 2 and its q boson realization in this paper it is proved that the nontrivial finite dimensional irreducible representations of the quantum matrix algebra mq 2 or the coordinate ring of the quantum group glq 2 exist only when q is a root of unity and thereby construct such representations for mq 2 some finite dimensional indecomposable representations of mq 2 are also explicitly presented finally the q boson realization of mq 2 is discussed and it is shown that it works in this case as well as in the quantum algebra case

performance guarantees of local search for minsum scheduling problems we study the worst case performance guarantee of locally optimal solutions for the problem of minimizing the total weighted and unweighted completion time on parallel machine environments our method makes use of a mapping that maps a schedule into an inner product space so that the norm of the mapping is closely related to the cost of the schedule we apply the method to study the most basic local search heuristics for scheduling namely jump and swap and establish their worst case performance in the case of unrelated restricted related and restricted identical machines

probabilistic analysis of bin packing heuristics in this paper we present both a probabilistic and statistical analysis of several bin packing heuristics with mild conditions on the distribution of problem data we show that the ratio of the expected number of bins packed for these heuristics to the number of elements converges to some constant this constant can either be derived analytically or estimated through simulation we analyze the variance in performance of these heuristics and we further show that as the number of elements n increases the probability that the number of bins required for these heuristics exceeds any fixed percentage of their expected performance goes to zero

online multiple strip packing we study an online multiple strip packing problem whose goal is to pack the given rectangles into m vertical strips of unit width such that the maximum height used among the strips is minimized rectangles arrive one by one the decision of delivering the rectangles to a strip as well as packing them into the strip must be done immediately and irrevocably without any information on the next rectangles both randomized and deterministic online algorithms are investigated all of which are guaranteed a constant competitive ratio

completeness in approximation classes we introduce a formal framework for studying approximation properties of np optimization npo problems the classes we consider are those appearing in the literature namely the class of approximable problems within a constant e apx the class of problems having a polynomial time approximation scheme pas and the class of problems having a fully polynomial time approximation scheme fpas we define natural approximation preserving reductions and obtain completeness results for these classes a complete problem in a class can not have stronger approximation properties unless p np we also show that the degree structure of npo allows intermediate degrees that is if p np there are problems which are neither complete nor belong to a lower class

scheduling under linear constraints abstract we introduce a parallel machine scheduling problem in which the processing times of jobs are not given in advance but are determined by a system of linear constraints the objective is to minimize the makespan i e the maximum job completion time among all feasible choices this novel problem is motivated by various real world application scenarios we discuss the computational complexity and algorithms for various settings of this problem in particular we show that if there is only one machine with an arbitrary number of linear constraints or there is an arbitrary number of machines with no more than two linear constraints or both the number of machines and the number of linear constraints are fixed constants then the problem is polynomial time solvable via solving a series of linear programming problems if both the number of machines and the number of constraints are inputs of the problem instance then the problem is np hard we further propose several approximation algorithms for the latter case

a general lower bound for the makespan problem abstract the problem of scheduling jobs on identical parallel processors to minimize makespan is np hard we introduce a general lower bound which includes as special cases bounds that either dominate or are equivalent to lower bounds previously proposed in the literature the general lower bound unifies earlier results and can be used to identify new bounds improve previously proposed bounds and extend bounds for the problem with identical processor ready times to the problem with variable processor ready times

the quantum group suq 2 and a q analogue of the boson operators a new realisation of the quantum group suq 2 is constructed by means of a q analogue to the jordan schwinger mapping determining thereby both the complete representation structure and q analogues to the wigner and racah operators to achieve this realisation a new elementary object is defined a q analogue to the harmonic oscillator the uncertainty relation for position and momentum in a q harmonic oscillator is quite unusual

lower bounds and heuristic algorithms for the ki partitioning problem we consider the problem of partitioning a set of positive integers values into a given number of subsets each having an associated cardinality limit so that the maximum sum of values in a subset is minimized and the number of values in each subset does not exceed the corresponding limit the problem is related to scheduling and bin packing problems we give combinatorial lower bounds reduction criteria constructive heuristics a scatter search approach and a lower bound based on column generation the outcome of extensive computational experiments is presented

a realization of quantum algebras some applications some applications of a new realization of quantum algebras are given finite dimensional modules are constructed in a new basis with explicit matrix elements for q a root of 1 the structure of weyl modules for is analysed in detail as a corollary a partial result on tensor product structure is obtained generalizations to other quantum algebras are briefly discussed realizations of in terms of difference operators and some unusual representations are also presented

an analysis of the lpt algorithm for the max min and the min ratio partition problems given a set of positive numbers the max min partition problem asks for a k partition such that the minimum part is maximized the min ratio partition problem has the similar definition but the objective is to minimize the ratio of the maximum to the minimum parts in this paper we analyze the performances of the longest processing time lpt algorithm for the two problems we show that the tight bounds of the lpt are respectively 4k 2 3k 1 and 7 5

approximation algorithms for scheduling jobs with release times and arbitrary sizes on batch machines with non identical capacities abstract we consider the problem of scheduling jobs with release times and arbitrary sizes on batch machines with non identical capacities to minimize makespan a job can only be assigned to a machine whose capacity is not less than the size of the job a machine can process a group of jobs as a batch simultaneously as long as the total size of the jobs in this batch does not exceed the capacity of the machine the processing time of a batch is equal to the longest processing time of all the jobs in the batch for the case of equal release times we obtain a fast 5 approximation algorithm as well as an algorithm with approximation ratio arbitrarily close to 2 for the case of unequal release times and a fixed number of different batch capacities we obtain an algorithm with approximation ratio arbitrarily close to 2 we also study the problem of scheduling jobs with equal release times arbitrary sizes inclusive processing set restrictions and incompatible families on batch machines with identical capacities to minimize makespan we obtain a 8 3 approximation algorithm as well as an algorithm with approximation ratio arbitrarily close to 2 if only a single family is considered a 2 approximation algorithm is presented it is likely that these results cannot be too substantially improved upon since all of the problems studied cannot be approximated to within a ratio smaller than 2 unless p np

turnpike optimality of smith s rule in parallel machines stochastic scheduling consider scheduling a batch of jobs with stochastic processing times on parallel machines with minimization of expected weighted flowtime as objective smith s rule which orders job starts by decreasing ratio of weight to expected processing time provides a natural heuristic for this problem in a previous paper we have found an upper bound for the difference between the expected objective under smith s rule and under the optimal strategy in this paper we find an upper bound on the expected number of times that smith s rule differs from the optimal decision under some mild and reasonable assumptions these bounds remain constant when the number of jobs increases hence under these conditions smith s rule is asymptotically optimal and it has a turnpike optimality property that is smith s rule is optimal most of the time

fast approximation algorithms for job scheduling with processing set restrictions we consider the problem of scheduling n independent jobs on m parallel machines where the machines differ in their functionality but not in their processing speeds each job has a restricted set of machines to which it can be assigned called its processing set preemption is not allowed our goal is to minimize the makespan of the schedule we study two variants of this problem 1 the case of tree hierarchical processing set and 2 the case of nested processing set we first give a fast algorithm for the case of tree hierarchical processing set with a worst case bound of 4 3 which is better than the best known algorithm whose worst case bound is 2 we then give a more complicated algorithm for the case of nested processing set with a worst case bound of 5 3 which is better than the best known algorithm whose worst case bound is 7 4 in both cases we will give examples achieving the worst case bounds

makespan minimization in open shops a polynomial time approximation scheme in this paper we demonstrate the existence of a polynomial time approximation scheme for makespan minimization in the open shop scheduling problem with an arbitrary fixed numberm of machines for the variant of the problem where the number of machines is part of the input it is known that the existence of an approximation scheme would implyp np hence our result draws a precise separating line between approximable cases i e withm fixed and non approximable cases i e withm part of the input of this shop problem 1998 the mathematical programming society inc published by elsevier science b v

minmax scheduling with job classes and earliness tardiness costs abstract we address scheduling problems with job dependent due dates and general possibly nonlinear and asymmetric earliness and tardiness costs the number of distinct due dates is substantially smaller than the number of jobs thus jobs are partitioned to classes where all jobs of a given class share a common due date we consider the settings of a single machine and parallel identical machines our objective is of a minmax type i e we seek a schedule that minimizes the maximum earliness tardiness cost among all jobs we introduce a nonlinear program that needs to be solved in order to obtain an optimal solution for the single machine case the case of parallel identical machines is np hard even for two machines linear costs and a single common due date we introduce an lpt based largest processing time first heuristic and a simple lower bound on the optimal cost both the heuristic and the lower bound are asymptotically accurate under very general conditions our numerical tests indicate that the heuristic produces very close to optimal schedules in all settings

semidefinite and linear programming integrality gaps for scheduling identical machines sherali and adams siam j discrete math 3 411 430 1990 and lovasz and schrijver siam j optim 1 166 190 1991 developed systematic procedures to construct the hierarchies of relaxations known as lift and project methods they have been proven to be a strong tool for developing approximation algorithms matching the best relaxations known for problems like max cut and sparsest cut in this work we provide lower bounds for these hierarchies when applied over the configuration lp for the problem of scheduling identical machines to minimize the makespan first we show that the configuration lp has an integrality gap of at least 1024 1023 by providing a family of instances with 15 different job sizes then we show that for any integer n there is an instance with n jobs in this family such that after varomega n rounds of the sherali adams text sa or the lovasz schrijver text ls hierarchy the integrality gap remains at least 1024 1023

a unified approach to truthful scheduling on related machines we present a unified framework for designing deterministic monotone polynomial time approximation schemes ptass for a wide class of scheduling problems on uniformly related machines this class includes among others minimizing the makespan maximizing the minimum load and minimizing the p norm of the machine loads vector previously this kind of result was only known for the makespan objective monotone algorithms have the property that an increase in the speed of a machine cannot decrease the amount of work assigned to it our results imply the existence of a truthful mechanism that can be implemented in polynomial time where the social goal is approximated within arbitrary precision

some distributions that allow perfect packing a probability distribution on 0 1 allows perfect packing if n items of size x 1 x n independent and identically distributed according to can be packed in unit size bins in such a way that the expected wasted space is o n a large class of distributions that allow perfect packing is exhibited as a corollary the intervals a b for which the uniform distribution on a b allows perfect packing are determined

a note on expected makespans for largest first sequences of independent tasks on two processors we consider the scheduling of sets of n simultaneously available tasks on two identical processors the task execution times are assumed to be independent samples from the uniform distribution on 0 1 we analyze the expected makespan schedule length performance of two well known largest task first nonpreemptive approximation rules with the largest first lf rule tasks are assigned in nonincreasing order of execution time to the processors as they become available with the restricted largest first rlf rule tasks are assigned in pairs one to a processor the larger task of a pair is the first to be scheduled if n is odd the last task is simply assigned to the earner finishing processor in the schedule for the first n 1 tasks we prove that the expected makespan for lf is bounded by frac n 4 frac e 2n 1 and for rlf it is precisely frac n 4 frac 1 2n 1 from these results simple bounds are derived on the ratio of the expected performance of lf and rlf to the expected performance of an optimization rule finally a lower bound on expected lf makespans is shown to be frac n 4 frac 1 4n 1

structure and complexity of extreme nash equilibria we study extreme nash equilibria in the context of a selfish routing game specifically we assume a collection of n users each employing a mixed strategy which is a probability distribution over m parallel identical links to control the routing of its own assigned traffic in a nash equilibrium each user selfishly routes its traffic on those links that minimize its expected latency cost the social cost of a nash equilibrium is the expectation over all random choices of the users of the maximum over all links latency through a link we provide substantial evidence for the fully mixed nash equilibrium conjecture which states that the worst nash equilibrium is the fully mixed nash equilibrium where each user chooses each link with positive probability specifically we prove that the fully mixed nash equilibrium conjecture is valid for pure nash equilibria furthermore we show that under a certain condition the social cost of any nash equilibrium is within a factor of 2h 1 e of that of the fully mixed nash equilibrium where h is the factor by which the largest user traffic deviates from the average user traffic considering pure nash equilibria we provide a ptas to approximate the best social cost we give an upper bound on the worst social cost and we show that it is n p hard to approximate the worst social cost within a multiplicative factor better than 2 2 m 1

weight conservation and quantum group construction of the braid group representation it is proved that the braid group representation constructed in terms of the quantum group can be covered by that obtained from the proposal of weight conservation by the extended kauffman diagram technique the non standard braid group representations associated with su 2 are obtained by adding the terms of weight conservation to the standard universal r matrix

an fptas for the parallel two stage flowshop problem we consider the np hard m parallel two stage flowshop problem abbreviated as the m 2 pfs problem where we need to schedule n jobs to m parallel identical two stage flowshops in order to minimize the makespan i e the maximum completion time of all the jobs on the m flowshops the m 2 pfs problem can be decomposed into two subproblems to assign the n jobs to the m parallel flowshops and for each flowshop to schedule the jobs assigned to the flowshop we first present a pseudo polynomial time dynamic programming algorithm to solve the m 2 pfs problem optimally for any fixed m based on an earlier idea for solving the 2 2 pfs problem using the dynamic programming algorithm as a subroutine we design a fully polynomial time approximation scheme fptas for the m 2 pfs problem

quantum deformations of certain simple modules over enveloping algebras soit f un corps soit f 0 un sous corps de f soit u l algebre enveloppante universelle sur f 0 de l algebre de lie de kac moody correspondant a la matrice a d entiers on montre qu un module integrable de poids le plus eleve v de u admet une deformation quantique

relations for clebsch gordan and racah coefficients in suq 2 and yang baxter equations relations are exploited among clebsch gordan cg and racah coefficients in the algebra suq 2 known as a deformation of su 2 these are used to show that the yang baxter yb relation for the irf interaction round a face model results from one of the symmetry relations for the 9 j symbol specific to suq 2 and that in an asymptotic limit this yb relation becomes the yb relation for the two dimensional vertex model the racah coefficient which has a particularly simple dependence on q is efficiently used such that an asymptotic limit of the racah coefficient is the cg coefficient and another limit gives the factorized s matrix of the vertex model

on symmetric numerical semigroups a numerical semigroup is a subset s of n such that is closed under sums contains the zero and generates z as a group from this definition w x one obtains see for example 2 that s has a conductor c i e the maximum among all the natural numbers not belonging to s a numerical semigroup s is called symmetric if for every integer z f s c y z g s the study of the subsemigroups of n is a classical problem which is equivalent to study of the sets of natural solutions of linear equations with w x coefficients in n from 1970 and after the works 5 6 and others the study of the subsemigroups of n is also motivated by its applications to algebraic geometry an example arises when we consider an analytically irreducible one dimensional noetherian local domain r the integral closure v of r in the quotient field of r is a discrete valuation ring if we assume that r and v have the same residue class field and we denote by the valuation attached to v then s s r is a numerical semigroup r then properties on the semigroup s imply properties of the ring r for r w x w x example 6 proves that r is gorenstein see 1 if and only if s is r symmetric the close relation between numerical semigroups and monomial curves has produced the usage of algebraic geometry terminology in the field of numerical semigroups for example the smallest integer m greater than zero in s is usually called the multiplicity of s the cardinal e of a minimal system of generators for s is called the embedding dimension of s and a finite presentation for s is called a relation for s

a non standard quantum double as the l state boson algebra and its r matrices for the quantum yang baxter equation in this paper we construct a new quantum double by endowing the l state boson algebra with a non trivial hopf algebra structure which is not a q deformation of the lie algebra or superalgebra the universal r matrix for the yang baxter equation associated with this new quantum group structure is obtained explicitly by building the representations of this quantum double we obtain some r matrices which can result in new representations of the braid group

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

a general framework for designing approximation schemes for combinatorial optimization problems with many objectives combined into one in this paper we present a general framework for designing approximation schemes for combinatorial optimization problems in which the objective function is a combination of more than one function examples of such problems include those in which the objective function is a product or ratio of two linear functions parallel machine scheduling problems with the makespan objective robust versions of weighted multiobjective optimization problems and assortment optimization problems with logit choice models the main idea behind our approximation schemes is the construction of an approximate pareto optimal frontier of the functions that constitute the given objective using this idea we give the first fully polynomial time approximation schemes for the max min resource allocation problem with a fixed number of agents combinatorial optimization problems in which the objective function is the sum of a fixed number of ratios of linear functions or the product of a fixed number of linear functions and assor

quantifiers and approximation we investigate the relationship between logical expressibility of np optimization problems and their approximation properties first such attempt was made by papadimitrou and yannakakis 1988 who defined the class of npo problems max np we show that many important optimization problems do not belong to max np and that in fact there are problems in p which are not in max np the problems that we consider fit naturally in a new complexity class that we call max 1 we prove that several natural optimization problems are complete for max 1 under approximation preserving reductions all these complete problems are not approximable unless p np this motivates the definition of subclasses of max 1 that only contain problems which are presumably eaiser with respect to approximation in particular the class that we call rmax 2 contains approximable problems and problems like max clique that are not known to be nonapproximable we prove the max clique and several other optimization problems are complete for rmax 2 all the complete problems in rmax 2 share the interesting property that they either are nonapproximable or are approximable to any degree of accuracy

optimal cost sharing for resource selection games joint use of resources with usage dependent cost raises the question who pays how much we study cost sharing in resource selection games where the strategy spaces are either singletons or bases of a matroid defined on the ground set of resources our goal is to design cost sharing protocols so as to minimize the resulting price of anarchy and price of stability we investigate three classes of protocols basic protocols guarantee the existence of at least one pure nash equilibrium separable protocols additionally require that the resulting cost shares only depend on the set of players on a resource uniform protocols are separable and require that the cost shares on a resource may not depend on the instance that is they remain the same even if new resources are added to or removed from the instance we find optimal basic and separable protocols that guarantee the price of stability and price of anarchy to grow logarithmically in the number of players except for the case of matroid games induced by separable protocols where the price of anarchy grows linearly with the number of players for uniform protocols we show that the price of anarchy is unbounded even for singleton games

an approximation scheme for minimizing the makespan of the parallel identical multi stage flow shops in the parallel k stage flow shops problem we are given m identical k stage flow shops and a set of jobs each job can be processed by any one of the flow shops but switching between flow shops is not allowed the objective is to minimize the makespan which is the finishing time of the last job this problem generalizes the classical parallel identical machine scheduling where k 1 k 1 and the classical flow shop scheduling where m 1 m 1 problems and thus it is np hard we present a polynomial time approximation scheme ptas for the problem when m and k are fixed constants the key technique is to partition the jobs into big jobs and small jobs enumerate over all feasible schedules for the big jobs and handle the small jobs by solving a linear program and employing a sliding method such a technique has been used in the design of ptas for several flow shop scheduling variants our main contributions are the non trivial application of this technique and a valid answer to the open question in the literature

average case analysis of cutting and packing in two dimensions abstract stock cutting and packing in two dimensions is a source of problems of great practical significance much has been written on the subject including several surveys but the focus of the mathematically oriented research has been largely on combinatorial optimization and worst case analysis recently substantial progress has been made in the average case analysis of algorithms for two dimensional packing this paper reviews selected illustrative results in this area of research and discusses desirable extensions and open problems several cutting and packing algorithms are defined and the results of their probabilistic analysis reported from the presentation of this material it is clear that the field is still in early stages of development the algorithms and probability models tend to be simplistic and stimates of performance are far more common that exact measures in spite of these limitations valuable and in some cases unexpected insights have emerged

the differencing algorithm ldm for partitioning a proof of a conjecture of karmarkar and karp the algorithm ldm largest differencing method divides a list of n random items into two blocks the parameter of interest is the expected difference between the two block sums it is shown that if the items are i i d and uniform then the rate of convergence of this parameter to zero is n log n an algorithm for balanced partitioning is constructed with the same rate of convergence to zero

martingale inequalities and np complete problems we use martingale inequalities in the probabilistic analysis of stochastic models of np complete problems we give examples of applications to the bin packing and traveling salesman problems in both cases we obtain sharp bounds on pr un eun t where un is the objective function value of the problem with n random data

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

a sublinear time approximation scheme for bin packing the bin packing problem is defined as follows given a set of n items with sizes 00 we present an algorithm a e that has sampling access to the input instance and outputs a value k such that c o p t k 1 e c o p t 1 where c o p t is the cost of an optimal solution it is clear that uniform sampling by itself will not allow a sublinear time algorithm in this setting a small number of items might constitute most of the total weight and uniform samples will not hit them in this work we use weighted samples where item i is sampled with probability proportional to its weight that is with probability w i iw i in the presence of weighted samples the approximation algorithm runs in o n poly 1 e g 1 e time where g x is an exponential function of x when both weighted sampling and uniform sampling are allowed o n 1 3 poly 1 e g 1 e time suffices in addition to an approximate value to c o p t our algorithm can also output a constant size template of a packing that can later be used to find a near optimal packing in linear time

average performance of greedy heuristics for the integer knapsack problem abstract this paper derives a lower bound on the average performance of a total value greedy heuristic for the integer knapsack problem this heuristic selects items in order of their maximum possible contribution to the solution value at each stage we show that as for the worst case bound the average performance bound for the total value heuristic dominates the corresponding bound for the density ordered greedy heuristic

an efficient polynomial time approximation scheme for load balancing on uniformly related machines we consider basic problems of non preemptive scheduling on uniformly related machines for a given schedule defined by a partition of the jobs into m subsets corresponding to the m machines c i c i denotes the completion time of machine i our goal is to find a schedule that minimizes or maximizes sum i 1 m c i p i 1 m c i p for a fixed value of p such that 0 1 p 1 the minimization problem is equivalent to the well known problem of minimizing the ell p l p norm of the vector of the completion times of the machines and for 0 p 1 0 p 1 the maximization problem is of interest our main result is an efficient polynomial time approximation scheme eptas for each one of these problems our schemes use a non standard application of the so called shifting technique we focus on the work total size of jobs assigned to each machine and introduce intervals of work that are forbidden these intervals are defined so that the resulting effect on the goal function is sufficiently small this allows the partition of the problem into sub problems with subsets of machines and jobs whose solutions are combined into the final solution using dynamic programming our results are the first eptas s for this natural class of load balancing problems

structure of regular representation of quantum universal enveloping algebra slq 2 with qp 1 the authors construct the regular representation quantum lie algebra of slq 2 and its quotient representations and then analyse the reducible structures of these representations when qp 1 for an integer p they show how the indecomposable representations of slq 2 are obtained by a purely algebraic method

average case performance analysis of an approximation algorithm for maximum subset sum using recurrence relations given a positive integer m and a set s x1 x2 xn of positive integers the maximum subset sum problem is to find a subset s of s such that x s x is maximized under the constraint that the summation is no larger than m in addition the cardinality of s is also a maximum among all subsets of s which achieve the maximum subset sum this problem is known to be np hard we analyze the average case performance of a simple on line approximation algorithm assuming that all integers in s are independent and have the same probability distribution we develop a general methodology i e using recurrence relations to evaluate the expected values of the content and the cardinality of s for any distribution the maximum subset sum problem has important applications especially in static job scheduling in multiprogrammed parallel systems the algorithm studied can also be easily adapted for dynamic job scheduling in such systems

asymptotic methods in the probabilistic analysis of sequencing and packing heuristics recently there has been considerable interest in the average case performance of heuristics this paper pursues that interest where it concerns sequencing and packing problems in particular we survey the methods that have been used to obtain formal probabilistic analyses of heuristics for makespan scheduling and one dimensional bin packing in so doing we present many of the key results in these research areas

optimal bin packing with items of random sizes iii consider a probability measure mu on 0 1 and independent random variables x 1 cdots x n distributed according to mu let q n q n x 1 cdots x n be the minimum number of unit size bins needed to pack items of size x 1 cdots x n let c mu lim n to infty e q n n in this paper it is proved that the random variable q n nc mu sqrt n converges in distribution the limit is identified as a distribution of the supremum of a certain gaussian process canonically attached to mu

combination of parallel machine scheduling and vertex cover this paper studies a combination of parallel machine scheduling and the vertex cover problem given some weighted vertices in an undirected graph a set of vertices is called a vertex cover if for each edge at least one endpoint belongs to this set our problem is to schedule a set of weighted vertices on m identical parallel machines such that the set of vertices is a vertex cover and the makespan is minimized we develop an approximation algorithm based on the local ratio method and the lpt rule and prove that it is a 3 2m 1 approximation algorithm

random walk and the area below its path a simple random walk with reflected origin is considered the walk starts at the origin and it must return to the origin at time 2n we show that the expected area below the path of this walk is n22n 1 cn2n if however the walk is required to return to the origin for the first time at time 2n then the expected area below the path of this bernoulli excursion is 2n 1 22n 1 cn2n we also show that if v1 vn is the order statistics based on a sample of size n from a uniform distribution over 0 1 and that if u1 un is another independent set of order statistics from the same distribution then e i 1n vi ui n22n 1 2n 1 cn2n we use this result to find an average case performance of the earliest due date edd heuristic for one machine scheduling problem with earliness and tardiness penalties we also apply some of the results to larson s queue inference engine 1990

afptas results for common variants of bin packing a new method for handling the small items we consider two well known natural variants of bin packing and show that these packing problems admit asymptotic fully polynomial time approximation schemes afptass in bin packing problems a set of one dimensional items of size at most 1 is to be assigned packed to subsets of sum at most 1 bins it has been known for a while that the most basic problem admits an afptas in this paper we develop new methods that allow us to extend this result to other variants of bin packing consisting of a family of two parameter bin packing problems we demonstrate the use of our methods by designing afptass for the following problems the first problem is bin packing with cardinality constraints where a parameter k is given such that a bin may contain up to k items the goal is to minimize the number of bins used the second problem is bin packing with rejection where every item has a rejection penalty associated with it an item needs to be either packed to a bin or rejected and the goal is to minimize the number of bins used and the total rejection penalty of unpacked items this resolves the complexity of two important variants of the bin packing problem our approximation schemes use a novel method for packing the small items this new method is the core of the improved running times of our schemes over the running times of the previous results which are only asymptotic polynomial time approximationschemes

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

on multidimensional packing problems we study the approximability of multidimensional generalizations of three classical packing problems multiprocessor scheduling bin packing and the knapsack problem specifically we study the vector scheduling problem its dual problem namely the vector bin packing problem and a class of packing integer programs the vector scheduling problem is to schedule n d dimensional tasks on m machines such that the maximum load over all dimensions and all machines is minimized the vector bin packing problem on the other hand seeks to minimize the number of bins needed to schedule all n tasks such that the maximum load on any dimension across all bins is bounded by a fixed quantity say 1 such problems naturally arise when scheduling tasks that have multiple resource requirements finally packing integer programs capture a core problem that directly relates to both vector scheduling and vector bin packing namely the problem of packing a maximum number of vectors in a single bin of unit height we obtain a variety of new algorithmic as well as inapproximability results for these three problems

minimizing machine assignment costs over δ approximate solutions of the scheduling problem p cmax abstract bi criteria lexicographical minimization problems with the makespan as the primary objective and the total machine assignment costs as the secondary objective have been recently introduced to the scheduling research and polynomial time r 1 1 approximation algorithms have been suggested for their solution where 1 r 2 is the performance ratio of an approximation algorithm for p c max we improve these results by presenting a polynomial time 1 5 r 1 1 approximation algorithm for the additive cost type then we introduce a problem of minimizing the total machine assignment cost over the approximate solutions of the makespan minimization problem we prove that this new problem is strongly np hard and pseudo polynomially non approximable in general a polynomial time approximation algorithm with a guaranteed approximation ratio is presented for the additive cost type and bounded ratio between the maximal and minimal machine costs an o m n 2 k time dynamic programming algorithm is also presented where k is the fixed number of distinct job processing times

q analogues of verma representations of quantum algebras for q a root of unity the case of uq sl 3 verma representation theory for classical lie algebra is extended to study the representation of quantum universal enveloping algebra quantum algebra for the nongeneric case that q is a root of unity on certain subspaces and quotient spaces of the verma space finite and infinite dimensional irreducible or indecomposable representations of slq 3 uq sl 3 are obtained in explicit matrix forms