moderately exponential approximation for makespan minimization on related machines we consider in this paper the classical q c m a x scheduling problem the objective is to minimize the maximum completion time called makespan while scheduling independent jobs in parallel on machines that have different speeds while several approximation schemes has been proposed and in particular a recent eptas jansen 2010 12 the current best direct algorithm i e an algorithm specifically designed for reaching a given approximation ratio is still due to chen 1991 4 with a 1 382 ratio our objective in this work is not to provide yet another improvement of the asymptotic dependencies in email protected ensuring a email protected ratio but to design faster direct algorithms by targeting respectively 43 and 54 ratios indeed instantiating any of the existing approximation scheme for e 13 respectively 14 leads to polynomial complexities but not to practical algorithms because of the hidden large constants in the computational complexity thus our approach focuses on a moderately exponential algorithm and provides a email protected dual approximation algorithm running in o m 1 1 3 e l o g 3 b 1 m n where m is the number of machines n the number of jobs b an integer lower than m depending on the instance this result is obtained through an oracle framework where the algorithm guesses possible answers from an oracle the terseness of the answers points out the critical information needed while solving any instance such an approach leads to a better comprehension of the problem similarly we obtain the same kind of results for a email protected ratio moreover the proposed techniques seem promising for tackling classical specific cases like scheduling on identical machines as the complexity becomes a low degree polynomial when the speeds are non arbitrary

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 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

a truthful constant approximation for maximizing the minimum load on related machines abstract designing truthful mechanisms for scheduling on related machines is a very important problem in single parameter mechanism design we consider the covering objective that is we are interested in maximizing the minimum completion time of a machine this problem falls into the class of problems where the optimal allocation can be truthfully implemented a major open issue for this class is whether truthfulness affects the polynomial time implementation we provide the first constant factor approximation for deterministic truthful mechanisms in particular we come up with an approximation guarantee of 2 e significantly improving on the previous upper bound of min m 2 e s m s 1

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

approximation schemes for the restricted shortest path problem this note contains two fully polynomial approximation schemes for the shortest path problem with an additional constraint the main difficulty in constructing such algorithms arises since no trivial lower and upper bounds on the solution value whose ratio is polynomially bounded are known in spite of this difficulty one of the algorithms presented here is strongly polynomial applications to other problems are also discussed

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

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

smoothed performance guarantees for local search we study popular local search and greedy algorithms for standard machine scheduling problems the performance guarantee of these algorithms is well understood but the worst case lower bounds seem somewhat contrived and it is questionable whether they arise in practical applications to find out how robust these bounds are we study the algorithms in the framework of smoothed analysis in which instances are subject to some degree of random noise while the lower bounds for all scheduling variants with restricted machines are rather robust we find out that the bounds are fragile for unrestricted machines in particular we show that the smoothed performance guarantee of the jump and the lex jump algorithm are in contrast to the worst case independent of the number of machines they are theta phi and theta log phi respectively where 1 phi is a parameter measuring the magnitude of the perturbation the latter immediately implies that also the smoothed price of anarchy is theta log phi for routing games on parallel links additionally we show that for unrestricted machines also the greedy list scheduling algorithm has an approximation guarantee of theta log phi

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

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

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

improved approximation algorithms for scheduling parallel jobs on identical clusters the multiple cluster scheduling problem corresponds to minimizing the maximum completion time makespan of a set of n parallel rigid and non preemptive jobs submitted to n identical clusters it cannot be approximated with a ratio better than 2 unless p np we present in this paper the methodology that encompasses several existing results 1 2 we detail first how to apply it for obtaining a 5 2 approximation then we use it to provide a new 7 3 approximation running in o log n h max n n log n where h max is the processing time of the longest job finally we apply it to a restriction of the problem to jobs of limited size leading to a 2 approximation which is the best possible ratio since the restriction remains 2 inapproximable

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

periodic solutions of a discrete hamiltonian system with a change of sign in the potential in this paper an existence theorem is obtained for periodic solutions of a second order discrete hamiltonian system with a change of sign in the potential by the minimax methods in the critical point theory

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

trudinger moser inequalities on complete noncompact riemannian manifolds abstract let m g be a complete noncompact riemannian n manifold n 2 if there exist positive constants and such that sup u w 1 n m u 1 1 m e u n n 1 k 0 n 2 k u n k n 1 k d v g where u 1 g u l n m u l n m then we say that the trudinger moser inequality holds suppose the trudinger moser inequality holds we prove that there exists some positive constant such that vol g b x 1 for all x m also we give a sufficient condition under which the trudinger moser inequality holds say the ricci curvature of m g has lower bound and its injectivity radius is positive moreover the adams inequality is discussed in this paper for application of the trudinger moser inequality we obtain existence results for some quasilinear equations with nonlinearity of exponential growth

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

an aptas for generalized cost variable sized bin packing bin packing is a well known problem which has a large number of applications classical bin packing is a simple model in which all bins are identical in the bin packing problem with variable sized bins we are given a supply of a variety of sizes this latter model assumes however that the cost of a bin is always defined to be its exact size in this paper we study the more general problem where an available bin size is associated with a fixed cost which may be smaller or larger than its size the costs of different bin sizes are unrelated this generalized problem has various applications in storage and scheduling in order to generalize previous work we design new rounding and allocation methods our main result is an asymptotic polynomial time approximation scheme for the generalized problem

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