a polynomial time approximation scheme for the multiple knapsack problem the multiple knapsack problem mkp is a natural and well known generalization of the single knapsack problem and is defined as follows we are given a set of n items and m bins knapsacks such that each item i has a profit p i and a size s i and each bin j has a capacity c j the goal is to find a subset of items of maximum profit such that they have a feasible packing in the bins mkp is a special case of the generalized assignment problem gap where the profit and the size of an item can vary based on the specific bin that it is assigned to gap is apx hard and a 2 approximation for it is implicit in the work of shmoys and tardos math program a 62 1993 pp 461 474 and thus far this was also the best known approximation for mkp the main result of this paper is a polynomial time approximation scheme ptas for mkp apart from its inherent theoretical interest as a common generalization of the well studied knapsack and bin packing problems it appears to be the strongest special case of gap that is not apx hard we substantiate this by showing that slight generalizations of mkp are apx hard thus our results help demarcate the boundary at which instances of gap become apx hard an interesting aspect of our approach is a ptas preserving reduction from an arbitrary instance of mkp to an instance with o log n distinct sizes and profits

the equilibrium generalized assignment problem and genetic algorithm abstract the well known generalized assignment problem gap is to minimize the costs of assigning n jobs to m capacity constrained agents or machines such that each job is assigned to exactly one agent this problem is known to be np hard and it is hard from a computational point of view as well in this paper follows from practical point of view in real systems the gap is extended to the equilibrium generalized assignment problem egap and the equilibrium constrained generalized assignment problem ecgap a heuristic equilibrium strategy based genetic algorithm ga is designed for solving the proposed egap finally to verify the computational efficiency of the designed ga some numerical experiments are performed on some known benchmarks the test results show that the designed ga is very valid for solving egap

a new rounding procedure for the assignment problem with applications to dense graph arrangement problems we present a randomized procedure for rounding fractional perfect matchings to integral matchings if the original fractional matching satisfies any linear inequality then with high probability the new matching satisfies that linear inequality in an approximate sense this extends the well known lp rounding procedure of raghavan and thompson which is usually used to round fractional solutions of linear programs we use our rounding procedure to design an additive approximation algorithm to the quadratic assignment problem the approximation error of the algorithm is en 2 and it runs in n o log n e2 time we also describe polynomial time approximation schemes ptass for dense subcases of many well known np hard arrangement problems including minimum linear arrangement minimum cut linear arrangement maximum acyclic subgraph and betweenness

an evaluation of semidefinite programming based approaches for discrete lot sizing problems abstract the present work is intended as a first step towards applying semidefinite programming models and tools to discrete lot sizing problems including sequence dependent changeover costs and times such problems can be formulated as quadratically constrained quadratic binary programs we investigate several semidefinite relaxations by combining known reformulation techniques recently proposed for generic quadratic binary problems with problem specific strengthening procedures developed for lot sizing problems our computational results show that the semidefinite relaxations consistently provide lower bounds of significantly improved quality as compared with those provided by the best previously published linear relaxations in particular the gap between the semidefinite relaxation and the optimal integer solution value can be closed for a significant proportion of the small size instances thus avoiding to resort to a tree search procedure the reported computation times are significant however improvements in sdp technology can still be expected in the future making sdp based approaches to discrete lot sizing more competitive

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

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

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

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

certain hypergeometric series related to the root system bc we show that the generalized hypergeometric function 2 f 1 of matrix argument is the series expansion at the origin of a special case of the hypergeometric function associated with the root system of type bc in addition we prove that the jacobi polynomials of matrix argument correspond to the jacobi polynomials associated with the root system of type bc we also give a precise relation between jack polynomials and the jacobi polynomials associated with the root system of type a as a side result one obtains generalized hook length formulas which are related to harish chandra s c function and one can prove a conjecture due to macdonald relating two inner products on a space of symmetric functions

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

stochastic depletion problems effective myopic policies for a class of dynamic optimization problems this paper presents a general class of dynamic stochastic optimization problems we refer to as stochastic depletion problems a number of challenging dynamic optimization problems of practical interest are stochastic depletion problems optimal solutions for such problems are difficult to obtain both from a pragmatic computational perspective as well as from a theoretical perspective as such simple heuristics are desirable we isolate two simple properties that if satisfied by a problem within this class guarantee that a myopic policy incurs a performance loss of at most 50 relative to the optimal adaptive control policy for that problem we are able to verify that these two properties are satisfied for several interesting families of stochastic depletion problems and as a consequence we identify computationally efficient approximations to optimal control policies for a number of interesting dynamic stochastic optimization problems

improved combinatorial algorithms for facility location problems we present improved combinatorial approximation algorithms for the uncapacitated facility location problem two central ideas in most of our results are cost scaling and greedy improvement we present a simple greedy local search algorithm which achieves an approximation ratio of 2 414 epsilon in tilde o n 2 epsilon time this also yields a bicriteria approximation tradeoff of 1 gamma 1 2 gamma for facility cost versus service cost which is better than previously known tradeoffs and close to the best possible combining greedy improvement and cost scaling with a recent primal dual algorithm for facility location due to jain and vazirani we get an approximation ratio of 1 853 in tilde o n 3 time this is very close to the approximation guarantee of the best known algorithm which is linear programming lp based further combined with the best known lp based algorithm for facility location we get a very slight improvement in the approximation factor for facility location achieving 1 728 we also consider a variant of the capacitated facility location problem and present improved approximation algorithms for this

source identities and kernel functions for the deformed koornwinder van diejen models we consider generalizations of the bc type relativistic calogero moser sutherland models comprising of the rational trigonometric hyperbolic and elliptic cases due to koornwinder and van diejen and construct an explicit eigenfunction for these generalizations in special cases we find the various kernel function identities and also a chalykh feigin sergeev veselov type deformation of these operators and their corresponding kernel functions which generalize the known kernel functions for the koornwinder van diejen models

a faster combinatorial approximation algorithm for scheduling unrelated parallel machines we consider the problem of scheduling n independent jobs on m unrelated parallel machines without preemption job i takes processing time p i j on machine j and the total time used by a machine is the sum of the processing times for the jobs assigned to it the objective is to minimize makespan the best known approximation algorithms for this problem compute an optimum fractional solution and then use rounding techniques to get an integral 2 approximation in this paper we present a combinatorial approximation algorithm that matches this approximation quality it is much simpler than the previously known algorithms and its running time is better this is the first time that a combinatorial algorithm always beats the interior point approach for this problem our algorithm is a generic minimum cost flow algorithm without any complex enhancements tailored to handle unsplittable flow it pushes unsplittable jobs through a two layered bipartite generalized network defined by the scheduling problem in our analysis we take advantage from addressing the approximation problem directly in particular we replace the classical technique of solving the lp relaxation and rounding afterwards by a completely integral approach we feel that this approach will be helpful also for other applications

truthful generalized assignments via stable matching in the generalized assignment problem gap a set of jobs seek to be assigned to a set of machines for every job machine pair there are a specific value and an accommodated capacity for the assignment the objective is to find an assignment that maximizes the total sum of values given that the capacity constraint of every machine is satisfied the gap is a classic optimization problem and has been studied extensively from the algorithmic perspective dughmi and ghosh dughmi s ghosh a 2010 truthful assignment without money acm conf electronic commerce acm new york 325 334 proposed the game theoretical framework in which every job is owned by a selfish agent who aims to maximize the value of his own assignment they gave a logarithmic approximation truthful in expectation mechanism and left open the problem whether there exists a truthful mechanism with a constant approximation factor in this paper we give an affirmative answer to this question and provide a constant approximation mechanism

semidefinite programming in combinatorial optimization we discuss the use of semidefinite programming for combinatorial optimization problems the main topics covered include i the lovasz theta function and its applications to stable sets perfect graphs and coding theory ii the automatic generation of strong valid inequalities iii the maximum cut problem and related problems and iv the embedding of finite metric spaces and its relationship to the sparsest cut problem

approximating semidefinite packing programs in this paper we define semidefinite packing programs and describe an algorithm to approximately solve these problems semidefinite packing programs arise in many applications such as semidefinite programming relaxations for combinatorial optimization problems sparse principal component analysis and sparse variance unfolding techniques for dimension reduction our algorithm exploits the structural similarity between semidefinite packing programs and linear packing programs

worst case bound of an lrf schedule for the mean weighted flow time problem this paper studies the problem of scheduling a set of n independent tasks on m identical processors so as to minimize mean weighted flow time the problem is known to be np complete for m geqq 2 and to be np complete in the strong sense for m arbitrary the worst case behavior of a heuristic algorithm which requires time o n log n is investigated and it is shown that the mean weighted flow time obtained by the algorithm does not exceed sqrt 2 1 2 cong 1 0207 times that of an optimal schedule moreover the bound sqrt 2 1 2 is best possible

coupled and k sided placements generalizing generalized assignment in modern data centers and cloud computing systems jobs often require resources distributed across nodes providing a wide variety of services motivated by this we study the coupled placement problem in which we place jobs into computation and storage nodes with capacity constraints so as to optimize some costs or profits associated with the placement the coupled placement problem is a natural generalization of the widely studied generalized assignment problem which concerns the placement of jobs into single nodes providing one kind of service we also study a further generalization the k sided placement problem in which we place jobs into k tuples of nodes each node in a tuple offering one of k services for both the coupled and k sided placement problems we consider minimization and maximization versions in the minimization versions mincpand min k ksp the goal is to achieve minimum placement cost while incurring a minimum blowup in the capacity of the individual nodes our first main result is an algorithm for min k ksp that achieves optimal cost while increasing capacities by at most a factor of k 1 k 1 also yielding the first constant factor approximation for mincp in the maximization versions maxcp and max k ksp the goal is to maximize the total weight of the jobs that are placed under hard capacity constraints max k ksp can be expressed as a k column sparse integer program and can be approximated to within a factor of o k factor using randomized rounding of a linear program relaxation we consider alternative combinatorial algorithms that are much more efficient in practice our second main result is a local search based combinatorial algorithm that yields a 15 approximation and o k 2 o k2 approximation for maxcp and max k ksp respectively finally we consider an online version of max k ksp and present algorithms that achieve logarithmic competitive ratio under certain necessary technical assumptions

solving optimization problems with diseconomies of scale via decoupling we present a new framework for solving optimization problems with a diseconomy of scale in such problems our goal is to minimize the cost of resources used to perform a certain task the cost of resources grows superlinearly as xq q 1 with the amount x of resources used we define a novel linear programming relaxation for such problems and then show that the integrality gap of the relaxation is aq where aq is the q th moment of the poisson random variable with parameter 1 using our framework we obtain approximation algorithms for the minimum energy efficient routing minimum degree balanced spanning tree load balancing on unrelated parallel machines and unrelated parallel machine scheduling with nonlinear functions of completion times problems our analysis relies on the decoupling inequality for nonnegative random variables the inequality states that ie1nxi q cq ie1nyi q where xi are independent nonnegative random variables yi are possibly dependent nonnegative random variables and each yi has the same distribution as xi the inequality was proved by de la pena in 1990 de la pena ibragimov and sharakhmetov showed that cq 2 for qi 1 2 and cq aq1 q for q 2 we show that the optimal constant is cqe aq1 q for any q 1 we then prove a more general inequality for every convex function v elv ie1nxi r elv p ie1n yi r and for every concave function el ie1n xi r el p ie1n yi r where p is a poisson random variable with parameter 1 independent of the random variables yi

scheduling to minimize average completion time 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 t

approximability of average completion time scheduling on unrelated machines we show that minimizing the average job completion time on unrelated machines is mathcal apx apx hard if preemption of jobs is allowed this provides one of the last missing pieces in the complexity classification of machine scheduling with weighted sum of completion times objective the proof is based on a mixed integer linear program this means that verification of the reduction is partly done by an ilp solver this gives a concise proof which is easy to verify in addition we give a deterministic 1 698 approximation algorithm for the weighted version of the problem the improvement is made by modifying and combining known algorithms and by the use of new lower bounds these results improve on the known mathcal np np hardness and 2 approximability

approximation schemes for job shop scheduling problems with controllable processing times in this paper we study the job shop scheduling problem under the assumption that the jobs have controllable processing times the fact that the jobs have controllable processing times means that it is possible to reduce the processing time of the jobs by paying a certain cost we consider two models of controllable processing times continuous and discrete for both models we present polynomial time approximation schemes when the number of machines and the number of operations per job are fixed

convex quadratic and semidefinite programming relaxations in scheduling we consider the problem of scheduling unrelated parallel machines subject to release dates so as to minimize the total weighted completion time of jobs the main contribution of this paper is a provably good convex quadratic programming relaxation of strongly polynomial size for this problem the best previously known approximation algorithms are based on lp relaxations in time or interval indexed variables those lp relaxations however suffer from a huge number of variables as a result of the convex quadratic programming approach we can give a very simple and easy to analyze 2 approximation algorithm which can be further improved to performance guarantee 3 2 in the absence of release dates we also consider preemptive scheduling problems and derive approximation algorithms and results on the power of preemption which improve upon the best previously known results for these settings finally for the special case of two machines we introduce a more sophisticated semidefinite programming relaxation and apply the random hyperplane technique introduced by goemans and williamson for the maxcut problem this leads to an improved 1 2752 approximation

parameterized approximation scheme for the multiple knapsack problem the multiple knapsack problem mkp is a well known generalization of the classical knapsack problem we are given a set a of n items and set b of m bins knapsacks such that each item a in a has a size size a and a profit value profit a and each bin b in b has a capacity c b the goal is to find a subset u subset a of maximum total profit such that u can be packed into b without exceeding the capacities the decision version of mkp is strongly np complete since it is a generalization of the classical knapsack and bin packing problem furthermore mkp does not admit a fully time polynomial time approximation scheme fptas even if the number m of bins is two kellerer gave a polynomial time approximation scheme ptas for mkp with identical capacities and chekuri and khanna presented a ptas for mkp with general capacities with running time n o log 1 epsilon epsilon 8 in this paper we propose an efficient ptas eptas with parameterized running time 2 o log 1 epsilon epsilon 5 cdot poly n o m for mkp this also solves an open question by chekuri and khanna

truthful mechanism design via correlated tree rounding a powerful algorithmic technique for truthful mechanism design is the maximal in distributional range midr paradigm unfortunately many such algorithms use heavy algorithmic machinery e g the ellipsoid method and approximate solution of convex programs in this paper we present a correlated rounding technique for designing mechanisms that are truthful in expectation it is elementary and can be implemented quickly the main property we rely on is that the domain offers fractional optimum solutions with a tree structure in auctions based on the generalized assignment problem each bidder has a publicly known knapsack constraint that captures the subsets of items that are of value to him he has a private valuation for each item and strives to maximize the value of assigned items minus payment for this domain we design a truthful 2 approximate midr mechanism for social welfare maximization it avoids using the ellipsoid method or convex programming in contrast to some previous work our mechanism achieves exact truthfulness in restricted related scheduling with selfish machines each job comes with a public weight and it must be assigned to a machine from a public job specific subset each machine has a private speed and strives to maximize payments minus workload of jobs assigned to it here we design a mechanism for makespan minimization this is a single parameter domain but the approximation status of the optimization problem is similar to unrelated machine scheduling the best known algorithm obtains a non truthful 2 approximation for unrelated machines and there is 1 5 hardness our mechanism matches this bound with a truthful 2 approximation

nonsymmetric koornwinder polynomials and duality in the fundamental work of lusztig l on affine hecke algebras a special role is played by the root system of type cn the affine hecke algebra is a deformation of the group algebra of an affine weyl group which usually depends on as many parameters as there are distinct root lengths i e one or two for an irreducible root system however in the cn case the hecke algebra h has three parameters corresponding to the fact that there is a simple coroot which is divisible by 2 recently cherednik c1 c3 has introduced the notion of a double affine hecke algebra and has used it to prove several conjectures on macdonald polynomials these polynomials and cherednik s double affine hecke algebra involve two or three parameters i e one more than the number of root lengths

penalty cost constrained identical parallel machine scheduling problem we consider a version of parallel machine scheduling with rejection an instance of the problem is given by m identical parallel machines and a set of n independent jobs with each job having a processing time and a penalty a job may be accepted to be processed or be rejected at its penalty the objective of the problem is to partition the set of jobs into two subsets the subset of accepted and the subset of rejected jobs and to schedule the set of accepted jobs such that the makespan is minimized under the constraint that the total penalty of the rejected jobs is no more than a given bound in this paper we present a 2 approximation algorithm within strongly polynomial time for the problem we also present a polynomial time approximation scheme whose running time is o n m o 1 2 mn 2 for the problem moreover for the case where the number of machines is a fixed constant m our results lead to a fully polynomial time approximation scheme for the problem our result is fairly good in the sense that in a reasonable size of jobs our fptas improves previous best running time from o n m 2 m to o 1 2 m 3 mn 2

an asymptotic formula for the koornwinder polynomials a formula for the large degree asymptotics of koornwinder s multivariate askey wilson polynomials is presented in the special case of a single variable this asymptotic formula agrees with the known leading asymptotics of the askey wilson polynomials determined by ismail and wilson

the constrained minimum weighted sum of job completion times problem we consider the problem of minimizing the weighted sum of job completion times on a single machine subject to certain job weights with an additional side constraint on the weighted sum of job completion times with respect to different job weights this problem is np hard and we provide a polynomial time approximation scheme for this problem our method is based on lagrangian relaxation mixed with carefully guessing the positions of certain jobs in the schedule

approximation algorithms for data placement problems we develop approximation algorithms for the problem of placing replicated data in arbitrary networks where the nodes may both issue requests for data objects and have capacity for storing data objects so as to minimize the average data access cost we introduce the data placement problem to model this problem we have a set of caches mathcal f a set of clients mathcal d and a set of data objects mathcal o each cache i can store at most u i data objects each client j in mathcal d has demand d j for a specific data object o j in mathcal o and has to be assigned to a cache that stores that object storing an object o in cache i incurs a storage cost of f i o and assigning client j to cache i incurs an access cost of d jc ij the goal is to find a placement of the data objects to caches respecting the capacity constraints and an assignment of clients to caches so as to minimize the total storage and client access costs we present a 10 approximation algorithm for this problem our algorithm is based on rounding an optimal solution to a natural linear programming relaxation of the problem one of the main technical challenges encountered during rounding is to preserve the cache capacities while incurring only a constant factor increase in the solution cost we also introduce the connected data placement problem to capture settings where write requests are also issued for data objects so that one requires a mechanism to maintain consistency of data we model this by requiring that all caches containing a given object be connected by a steiner tree to a root for that object which issues a multicast message upon a write to any copy of that object the total cost now includes the cost of these steiner trees we devise a 14 approximation algorithm for this problem we show that our algorithms can be adapted to handle two variants of the problem a a k median variant where there is a specified bound on the number of caches that may contain a given object and b a generalization where objects have lengths and the total length of the objects stored in any cache must not exceed its capacity

a ptas for minimizing the total weighted completion time on identical parallel machines we consider the problem of scheduling a set of n jobs on m identical parallel machines so as to minimize the weighted sum of job completion times this problem is np hard in the strong sense the best approximation result known so far was a documentclass aastex usepackage amsbsy usepackage amsfonts usepackage amssymb usepackage bm usepackage mathrsfs usepackage pifont usepackage stmaryrd usepackage textcomp usepackage portland xspace usepackage amsmath amsxtra pagestyle empty declaremathsizes 10 9 7 6 begin document 1 over 2 1 sqrt 2 end document approximation algorithm that has been derived by kawaguchi and kyan back in 1986 the contribution of this paper is a polynomial time approximation scheme for this setting which settles a problem that was open for a long time moreover our result constitutes the first known approximation scheme for a strongly np hard scheduling problem with minsum objective

moments of characteristic polynomials for compact symmetric spaces and jack polynomials we express the averages of products of characteristic polynomials for random matrix ensembles associated with compact symmetric spaces in terms of jack polynomials or heckman and opdam s jacobi polynomials depending on the root system of the space we also give explicit expressions for the asymptotic behavior of these averages in the limit as the matrix size goes to infinity

the generalized assignment problem with minimum quantities abstract we consider a variant of the generalized assignment problem gap where the amount of space used in each bin is restricted to be either zero if the bin is not opened or above a given lower bound a minimum quantity we provide several complexity results for different versions of the problem and give polynomial time exact algorithms and approximation algorithms for restricted cases for the most general version of the problem we show that it does not admit a polynomial time approximation algorithm unless p np even for the case of a single bin this motivates to study dual approximation algorithms that compute solutions violating the bin capacities and minimum quantities by a constant factor when the number of bins is fixed and the minimum quantity of each bin is at least a factor 1 larger than the largest size of an item in the bin we show how to obtain a polynomial time dual approximation algorithm that computes a solution violating the minimum quantities and bin capacities by at most a factor 1 1 and 1 1 respectively and whose profit is at least as large as the profit of the best solution that satisfies the minimum quantities and bin capacities strictly in particular for 2 we obtain a polynomial time 1 2 approximation algorithm

quantum knizhnik zamolodchikov equations and affine root systems quantum difference knizhnik zamolodchikov equations s1 fr are generalized for ther matrices from ch1 with the arguments in arbitrary root systems and their formal counterparts in particular qkz equations with certain boundary conditions are introducted the self consistency of the equations from fr and the cross derivative integrability conditions for ther matrix kz equations from ch2 are obtained as corollaries a difference counterpart of the quantum many body problem connected with macdonald s operators is defined as an application

the demand matching problem we examine formulations for the well known b matching problem in the presence of integer demands on the edges a subset m of edges is feasible if for each node v the total demand of edges in m incident to v is at most bv we examine the system of star inequalities for this problem this system yields an exact linear description for b matchings in bipartite graphs for the demand version we show that the integrality gap for this system is at least 2 5 and at most 2 764 for general graphs the gap lies between 3 and 3 264 we also describe a 3 approximation algorithm 2 5 for bipartite graphs for the cardinality version of the problem a fully polynomial approximation scheme is also presented for the problem on a tree thus generalising a well known result for the knapsack problem recently the notion of demand matching has arisen in the design of clos network switches

a finite branch and bound algorithm for nonconvex quadratic programming via semidefinite relaxations existing global optimization techniques for nonconvex quadratic programming qp branch by recursively partitioning the convex feasible set and thus generate an infinite number of branch and bound nodes an open question of theoretical interest is how to develop a finite branch and bound algorithm for nonconvex qp one idea which guarantees a finite number of branching decisions is to enforce the first order karush kuhn tucker kkt conditions through branching in addition such an approach naturally yields linear programming lp relaxations at each node however the lp relaxations are unbounded a fact that precludes their use in this paper we propose and study semidefinite programming relaxations which are bounded and hence suitable for use with finite kkt branching computational results demonstrate the practical effectiveness of the method with a particular highlight being that only a small number of nodes are required

improved approximation schemes for scheduling unrelated parallel machines we consider the problem of schedulingn independent jobs onm unrelated parallel machines where each job has to be processed by exactly one machine processing jobj on machinei requiresp ijtime units and the objective is to minimize the makespan i e the maximum job completion time focusing on the case whenm is fixed we present for both preemptive and nonpreemptive variants of the problem fully polynomial approximation schemes whose running times depend only linearly onn we also study an extension of the problem where processing jobj on machinei incurs a cost ofc ij and thus there are two optimization criteria makespan and cost we show that for any fixedm there is a fully polynomial approximation scheme that given valuest andc computes for any fixed e 0 a schedule in0 n time with makespan at most 1 e t and cost at most 1 e c if there exists a schedule of makespant and costc

optimal solutions for unrelated parallel machines scheduling problems using convex quadratic reformulations abstract in this work we take advantage of the powerful quadratic programming theory to obtain optimal solutions of scheduling problems we apply a methodology that starts in contrast to more classical approaches by formulating three unrelated parallel machine scheduling problems as 0 1 quadratic programs under linear constraints by construction these quadratic programs are non convex therefore before submitting them to a branch and bound procedure we reformulate them in such a way that we can ensure convexity and a high quality continuous lower bound experimental results show that this methodology is interesting by obtaining the best results in literature for two of the three studied scheduling problems

minimizing average completion time in the presence of release dates a natural and basic problem in scheduling theory is to provide good average quality of service to a stream of jobs that arrive over time in this paper we consider the problem of schedulingn jobs that are released over time in order to minimize the average completion time of the set of jobs in contrast to the problem of minimizing average completion time when all jobs are available at time 0 all the problems that we consider are np hard and essentially nothing was known about constructing good approximations in polynomial time we give the first constant factor approximation algorithms for several variants of the single and parallel machine models many of the algorithms are based on interesting algorithmic and structural relationships between preemptive and nonpreemptive schedules and linear programming relaxations of both many of the algorithms generalize to the minimization of averageweighted completion time as well 1998 the mathematical programming society inc published by elsevier science b v

assigning sporadic tasks to unrelated machines we study the problem of assigning sporadic tasks to unrelated machines such that the tasks on each machine can be feasibly scheduled despite its importance for modern real time systems this problem has not been studied before we present a polynomial time algorithm which approximates the problem with a constant speedup factor of 8 2 sqrt 6 approx 12 9 8 26 12 9 and show that any polynomial time algorithm needs a speedup factor of at least 2 2 unless p np in the case of a constant number of machines we give a polynomial time approximation scheme key to these results are two new relaxations of the demand bound function the function that yields a sufficient and necessary condition for a task system on a single machine to be feasible in particular we present new methods to approximate this function to obtain useful structural properties while incurring only bounded loss in the approximation quality for the constant speedup result we employ a very general rounding procedure for linear programs lps which model assignment problems with capacity type constraints it ensures that the cost of the rounded integral solution is no more than the cost of the optimal fractional lp solution and the capacity constraints are violated only by a bounded factor depending on the structure of the matrix that defines the lp in fact our rounding scheme generalizes the well known 2 approximation algorithm for the generalized assignment problem due to shmoys and tardos

lp based approximation algorithms for capacitated facility location in the capacitated facility location problem with hard capacities we are given a set of facilities mathcal f and a set of clients mathcal d in a common metric space each facility i has a facility opening cost f i and capacity u i that specifies the maximum number of clients that may be assigned to this facility we want to open some facilities from the set mathcal f and assign each client to an open facility so that at most u i clients are assigned to any open facility i the cost of assigning client j to facility i is given by the distance c ij and our goal is to minimize the sum of the facility opening costs and the client assignment costs the only known approximation algorithms that deliver solutions within a constant factor of optimal for this np hard problem are based on local search techniques it is an open problem to devise an approximation algorithm for this problem based on a linear programming lower bound or indeed to prove a constant integrality gap for any lp relaxation we make progress on this question by giving a 5 approximation algorithm for the special case in which all of the facility costs are equal by rounding the optimal solution to the standard lp relaxation one notable aspect of our algorithm is that it relies on partitioning the input into a collection of single demand capacitated facility location problems approximately solving them and then combining these solutions in a natural way

cooperative catalysis and chemical chaos a chemical model for the lorenz equations abstract a chemical reaction mechanism is presented that gives the lorenz equations after the axes are translated so that the range of all the variables now representing concentrations is in the positive quadrant the resulting set of nonlinear equations can be interpreted in terms of straightforward chemical reactions the mechanism requires some species to be in excess constant concentration and for some steps to be a slow rate determining reaction followed by a very fast instantaneous reaction the mechanism does not involve autocatalysis but rather cooperative catalysis where the reactants mutually catalyze appropriate reactions

a new average case analysis for completion time scheduling we present a new average case analysis for the problem of scheduling n jobs on m machines so that the sum of job completion times is minimized our goal is to use the concept of competitive ratio which is a typical worst case notion also within an average case analysis we show that the classic sept scheduling strategy with n worst case competitive ratio achieves an average of o 1 under several natural distributions among them the exponential distribution our analysis technique allows to also roughly estimate the probability distribution of the competitive ratio thus our result bridges the gap between worst case and average case performance guarantee

approximate graph coloring by semidefinite programming we consider the problem of coloring k colorable graphs with the fewest possible colors we present a randomized polynomial time algorithm that colors a 3 colorable graph on n vertices with min o 1 3 log 1 2 log n o n 1 4 log 1 2 n colors where is the maximum degree of any vertex besides giving the best known approximation ratio in terms of n this marks the first nontrivial approximation result as a function of the maximum degree this result can be generalized to k colorable graphs to obtain a coloring using min o 1 2 k log 1 2 log n o n 1 3 k 1 log 1 2 n colors our results are inspired by the recent work of goemans and williamson who used an algorithm for semidefinite optimization problems which generalize linear programs to obtain improved approximations for the max cut and max 2 sat problems an intriguing outcome of our work is a duality relationship established between the value of the optimum solution to our semidefinite program and the lovasz t function we show lower bounds on the gap between the optimum solution of our semidefinite program and the actual chromatic number by duality this also demonstrates interesting new facts about the t function

domain synchronization of a cyclic lotka volterra reaction diffusion mechanism abstract the domain synchronization of a cyclic lotka volterra reaction diffusion mechanism is studied in a dilute solution this is done by performing a number of molecular dynamics simulations and by setting up a mathematical model that can account for the dynamics seen in the simulations the simulations show that below a certain critical length the domain is synchronized but for larger domains the system quickly becomes unsynchronized a mathematical model based on the law of mass action and fick s law will not reveal this critical length however if an ad hoc gaussian noise term is added to the model it resemble the synchronization phenomena seen in the molecular dynamics simulations

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

the effect of clustering on the lotka volterra model abstract using the lotka volterra model as a reference for an oscillating system the effects of nonideality excluded volume and clustering are investigated it is found that these two different types of interaction have opposite effects on the stability of the system and that for a range of parameters stable limit cycles can result even limited clustering such as dimer formation is sufficient to give rise to limit cycle behavior lattice models are also investigated where clustering is treated using the bethe approximation

scheduling multiple variable speed machines we examine scheduling problems where we control not only the assignment of jobs to machines but also the time used by the job on the machine for instance many tooling machines allow control of the speed at which a job is run increasing the speed incurs costs due to machine wear but also increases throughput we discuss some fundamental scheduling problems in this environment and give algorithms for some interesting cases some cases are inherently difficult so for these we give heuristics our approach illustrates the exploitation of underlying network structure in combinatorial optimization problems we create heuristics that optimally schedule a large portion of the jobs and then attempt to fit in the remainder this also gives a method for quickly finding valid inequalities violated by the linear relaxation solution for the problem of minimizing the sum of makespan and production costs a rounding heuristic is within a constant factor of optimal our heuristics are compared to other classical heu

dependent rounding and its applications to approximation algorithms we develop a new randomized rounding approach for fractional vectors defined on the edge sets of bipartite graphs we show various ways of combining this technique with other ideas leading to improved approximation algorithms for various problems these include low congestion multi path routing richer random graph models for graphs with a given degree sequence improved approximation algorithms for i throughput maximization in broadcast scheduling ii delay minimization in broadcast scheduling as well as iii capacitated vertex cover and fair scheduling of jobs on unrelated parallel machines

fast approximation schemes for boolean programming and scheduling problems related to positive convex half product we address a version of the half product problem and its restricted variant with a linear knapsack constraint for these minimization problems of boolean programming we focus on the development of fully polynomial time approximation schemes with running times that depend quadratically on the number of variables applications to various single machine scheduling problems are reported minimizing the total weighted flow time with controllable processing times minimizing the makespan with controllable release dates minimizing the total weighted flow time for two models of scheduling with rejection

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

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

derandomizing approximation algorithms based on semidefinite programming remarkable breakthroughs have been made recently in obtaining approximate solutions to some fundamental np hard problems namely max cut max k cut max sat max dicut max bisection k vertex coloring maximum independent set etc all these breakthroughs involve polynomial time randomized algorithms based upon semidefinite programming a technique pioneered by goemans and williamson in this paper we give techniques to derandomize the above class of randomized algorithms thus obtaining polynomial time deterministic algorithms with the same approximation ratios for the above problems at the heart of our technique is the use of spherical symmetry to convert a nested sequence of n integrations which cannot be approximated sufficiently well in polynomial time to a nested sequence of just a constant number of integrations which can be approximated sufficiently well in polynomial time

refined cauchy littlewood identities and six vertex model partition functions iii deformed bosons abstract we study hall littlewood polynomials using an integrable lattice model of t deformed bosons working with row to row transfer matrices we review the construction of hall littlewood polynomials of the a n root system within the framework of this model introducing appropriate double row transfer matrices we extend this formalism to hall littlewood polynomials based on the b c n root system and obtain a new combinatorial formula for them we then apply our methods to prove a series of refined cauchy and littlewood identities involving hall littlewood polynomials the last two of these identities are new and relate infinite sums over hyperoctahedrally symmetric hall littlewood polynomials with partition functions of the six vertex model on finite domains

approximating the single source unsplittable min cost flow problem in the single source unsplittable min cost flow problem commodities must be routed simultaneously from a common source vertex to certain destination vertices in a given graph with edge capacities and costs the demand of each commodity must be routed along a single path so that the total flow through any edge is at most its capacity moreover the total cost must not exceed a given budget this problem has been introduced by kleinberg 7 and generalizes several np complete problems from various areas in combinatorial optimization such as packing partitioning scheduling load balancing and virtual circuit routing kolliopoulos and stein 9 and dinitz garg and goemans 4 developed algorithms improving the first approximation results of kleinberg for the problem of minimizing the violation of edge capacities and for other variants however known techniques do not seem to be capable of providing solutions without also violating the cost constraint we give the first approximation results with hard cost constraints moreover all our results dominate the best known bicriteria approximations finally we provide results on the hardness of approximation for several variants of the problem

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

tight approximation algorithms for maximum separable assignment problems a separable assignment problem sap is defined by a set of bins and a set of items to pack in each bin a value fij for assigning item j to bin i and a separate packing constraint for each bin i e for each bin a family of subsets of items that fit in to that bin the goal is to pack items into bins to maximize the aggregate value this class of problems includes the maximum generalized assignment problem gap 1 and a distributed caching problem dcp described in this paper given a approximation algorithm for finding the highest value packing of a single bin we give a polynomial time lp rounding based 1 1 e approximation algorithm a simple polynomial time local search 1 e approximation algorithm for any e 0 therefore for all examples of sap that admit an approximation scheme for the single bin problem we obtain an lp based algorithm with 1 1 e e approximation and a local search algorithm with e approximation guarantee furthermore for cases in which the subproblem admits a fully polynomial approximation scheme such as for gap the lp based algorithm analysis can be strengthened to give a guarantee of 1 1 e the best previously known approximation algorithm for gap is a approximation by shmoys and tardos and chekuri and khanna our lp algorithm is based on rounding a new linear programming relaxation with a provably better integrality gap to complement these results we show that sap and dcp cannot be approximated within a factor better than 1 1 e unless np dtime no log log n even if there exists a polynomial time exact algorithm for the single bin problem we extend the 1 1 e approximation algorithm to a constant factor approximation algorithms for a nonseparable assignment problem with applications in maximizing revenue for budget constrained combinatorial auctions and the adwords assignment problem we generalize the local search algorithm to yield a e approximation algorithm for the maximum k median problem with hard capacities

iterated greedy local search methods for unrelated parallel machine scheduling abstract this work deals with the parallel machine scheduling problem which consists in the assignment of n jobs on m parallel machines the most general variant of this problem is when the processing time depends on the machine to which each job is assigned to this case is known as the unrelated parallel machine problem similarly to most of the literature this paper deals with the minimization of the maximum completion time of the jobs commonly referred to as makespan c max many algorithms and methods have been proposed for this hard combinatorial problem including several highly sophisticated procedures by contrast in this paper we propose a set of simple iterated greedy local search based metaheuristics that produce solutions of very good quality in a very short amount of time extensive computational campaigns show that these solutions are most of the time better than the current state of the art methodologies by a statistically significant margin

double affine hecke algebras and macdonald s conjectures in this paper we prove the main results about the structure of double affine hecke algebras announced in c1 c2 the technique is based on the realization of these algebras in terms of demazure lusztig operators bgg d l2 ls c3 and rather standard facts from the theory of affine weyl groups in particular it completes the proof partially published in c2 of the macdonald scalar product conjecture see ml 12 6 including the famous macdonald constant term conjecture the q t case we mainly follow the opdam paper 0 where the macdonald mehta conjectures in the degenerate differential case were deduced from certain properties of the heckman opdam operators ho and the existence of the shift operators heckman s interpretation of these operators via the so called dunkl operators see he and also c5 was important to our approach we note that the ho operators are closely related to the so called quantum many body problem calogero sutherland moser olshanetsky perelomov the conformal field theory knizhnik zamolodchikov equations the harmonic analysis on symmetric spaces harish chandra helgason etc and last but not the least the classic theory of the hypergeometric functions establishing the connection between the difference counterparts of heckman opdam operators introduced in c4 and the macdonald theory ml m2 including the construction of the difference shift operators is the main thrust of this paper once the connection is established it is not very difficult to calculate the scalar squares of the macdonald polynomials and to prove the constant term conjecture from his fundamental paper m3 to simplify the exposition we discuss the reduced root systems only and impose the relation q tk for k e z to avoid infinite products in the definition of macdonald s pairing the purpose of this work is to present a concrete application of the new technique arbitrary q t can be handled in a

a 699 approximation algorithm for max bisection we present a 699 approximation algorithm for max bisection i e partitioning the nodes of a weighted graph into two blocks of equal cardinality so as to maximize the weights of crossing edges this is an improved result from the 651 approximation algorithm of frieze and jerrum and the semidefinite programming relaxation of goemans and williamson

a unified approach to scheduling on unrelated parallel machines we develop a single rounding algorithm for scheduling on unrelated parallel machines this algorithm works well with the known linear programming quadratic programming and convex programming relaxations for scheduling to minimize completion time makespan and other well studied objective functions this algorithm leads to the following applications for the general setting of unrelated parallel machines i a bicriteria algorithm for a schedule whose weighted completion time and makespan simultaneously exhibit the current best individual approximations for these criteria ii better than two approximation guarantees for scheduling to minimize the lp norm of the vector of machine loads for all 1

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

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