Ecient approximation algorithm for minimizing makespan. on uniformly related machines. Chandra Chekuri. November 25, 1997.


 Sarah Bishop
 1 years ago
 Views:
Transcription
1 Ecient approximation algorithm for minimizing makespan on uniformly related machines Chandra Chekuri November 25, 1997 Abstract We obtain a new ecient approximation algorithm for scheduling precedence constrained jobs on machines with dierent speeds. The setting is as follows. There are n jobs, job j requires p j units of processing. The jobs are to be scheduled on a set of m machines. Machine i has a speed s i ; it takes p j =s i units of time for machine i to process job j. The precedence constraints on the jobs are given in the form of a partial order. If j k, processing of k cannot start until j is completely done. Let C j denote the completion time of job j. The objective is to minimize C max = max j C j conventionally called the makespan of the schedule. We consider nonpreemptive schedules where a job is processed on a single machine with no preemptions. Recently Chudak and Shmoys [1] gave an approximation algorithm p with a performance ratio of O(log m) improving upon the earlier ratio of O( m) due to Jae [6]. Their algorithm is based on solving a linear programming relaxation of the problem. Building on some of their ideas, we present a new ecient and combinatorial algorithm which achieves a similar approximation ratio but runs in O(n 3 ) time. In the process we also obtain a new lower bound which is of independent interest. By a general result of Shmoys, Wein, and Williamson [9] this algorithm can be extended to obtain an algorithm with an approximation ratio of O(log m) even if each job j has a release date r j before which it cannot be processed. 1 Introduction The problem of scheduling precedence constrained jobs on a set of identical parallel machines to minimize makespan is one of the oldest problems for which approximation algorithms have been devised. Graham [4] showed that a simple list scheduling gives a ratio of 2 and it is the best known algorithm till date. We consider a generalization of this model in which machines have dierent speeds which in the scheduling literature are called uniformly related. We formalize the problem below. There are n jobs 1; : : :; n, with job j requiring processing of p j units. The jobs are to be scheduled on a set of m machines. Each machine i has a speed factor s i. Job j with a processing requirement p j takes p j =s i time units to run on machine i. We restrict ourselves to nonpreemptive schedules where a job once started on a machine has to run to completion on the same machine. Our results carry over to the preemptive case as well. In the scheduling literature [5] where problems are classied in the jj notation, this problem is referred to as QjprecjC max. Supported by an IBM Cooperative Fellowship, an ARO MURI Grant DAAH and NSF Award CCR , with matching funds from IBM, Schlumberger Foundation, Shell Foundation, and erox Corporation. 1
2 Liu and Liu [8] analyzed the performance of Graham's list scheduling algorithm and showed that the approximation guarantee depends on the ratio of the largest to the smallest speed. This ratio could be arbitrarily large even for a small number of machines. The rst algorithm to have a bound independent of the speeds was given by Jae [6], who showed that list scheduling restricted to the set of machines which are within a factor of p m of the fastest machine gives an O( p m) bound. More recently, Chudak and Shymoys [1] improved the ratio considerably and gave an algorithm which has a guarantee of O(log m). At a more basic level their algorithm has a guarantee of O(K) where K is the number of distinct speeds. The above mentioned algorithm relies on solving a linear programming relaxation and uses the information obtained from the solution to allocate jobs to processors. We present a new algorithm which nds an allocation without solving a linear program. The ratio guaranteed by our algorithm is also O(log m) but it oers a couple of advantages. First, our algorithm runs in O(n 3 ) time and is combinatorial, hence is provably more ecient than the algorithm in [1]. Second, we show a new combinatorial lower bound which is natural and might be useful in other contexts. We remark here that our algorithm was inspired by, and builds upon the ideas in [1]. The rest of the paper is organized as follows. Section 2 contains some of the ideas from Chudak and Shmoy's paper [1] that are useful to us. We present the new lower bound in Section 3, and give the approximation algorithm and the analysis in Section 4. 2 Preliminaries We summarize below the basic ideas in the work of Chudak and Shmoys [1]. Their main result is an algorithm which gives a ratio of O(K) for the problem of QjprecjC max where K is the number of distinct speeds. They also show how to reduce the general case with arbitrary speeds to one in which there are only O(log m) distinct speeds as follows. Ignore all machines with speed less than 1=m times the speed of the fastest machine. Round down all speeds to the nearest power of 2. They observe that the above transformation can be done while losing only a constant factor in the approximation ratio. Therefore we will restrict ourselves to the case where we have K distinct speeds. Graham's analysis shows that any schedule produced by list scheduling has a chain of jobs j 1 j 2 : : : j r where a machine is idle only when one of the jobs in the chain is being processed. The time spent processing the above chain is a lower bound on the optimal makespan. Similarly the time spent when all machines are busy is also a lower bound. Combining these two facts give the upper bound of 2 on the performance ratio of list scheduling. One can apply a similar analysis for the multiple speed case. As observed in [1], the diculty stems from the fact that we can no longer claim that the processing time of the chain is a lower bound. All that can be said is that the processing time of any chain on the fastest machine is a lower bound. The jobs in the chain guaranteed by the list scheduling analysis do not necessarily run on the fastest machine. Based on this, the authors of [1] observed that one of the important objectives is to nd an assignment of jobs to speeds (machines) which ensures that the processing time of any chain is bounded by some factor of the optimal. We will follow the notation of [1] for sake of continuity and convenience. Let m k be the number of P machines with speed s k, k = 1; : : :; K, where s 1 > : : : > s K. Let Mu v denote v the sum m l. Let k(j) denote the speed at which job j is assigned to be processed. 2
3 The average processing allocated to a machine of a specic speed k, denoted by D k is the following. D k = 1 p j : m k s k j:k(j)=k It is also possible to compute the maximum over all chains C, of the following quantity p j s j2c k(j) which will be denoted by C. A natural variant of list scheduling called speed based list scheduling is developed in [1] which is constrained to schedule according the speed assignments of the jobs. As in the classical list scheduling, a job is scheduled as soon as a machine is free, provided the free machine matches the speed assignment of the job. The following theorem whose analysis is a simple generalization of Graham's analysis is from [1]. Theorem 1 (Chudak & Shmoys) For any job assignment k(j), j = 1; : : :; n, the speedbased list scheduling algorithm produces a schedule of length C max C + The authors of [1] use a linear programming relaxation of P the problem to obtain a job K assignment which simultaneously satises the two conditions: k=1 D k (K + p K)Cmax and C ( p K+1)Cmax where C max is the optimal makespan. Combining these with Theorem 1 gives them an O(K) approximation. We will show how to use an alternate method based on chain decompositions to obtain an assignment satisfying similar properties. 3 A new lower bound In this section we develop a simple and natural lower bound which will be used in the analysis of our algorithm. Before formally stating the lower bound we provide some intuition. The two lower bounds used in Graham's analysis for identical parallel machines are the maximum chain length and the average load. As discussed in the previous section, the maximum chain length (maximum according to processing times) has to be redened for the case when machines have dierent speeds. A naive generalization implies that the maximum chain length divided by the fastest speed is a lower bound. However it is easy to generate examples where this bound is a factor of 1=m away from the optimal. We describe the general nature of such examples to motivate the new bound. Suppose we have two speeds with s 1 = D and s 2 = 1, and l > 1 independent chains of jobs each of the same length D. Suppose m 1 = 1, and m 2 = l D. The average load can be seen to be bound by 1. Similarly the time to process any chain on the fastest processor is 1. However if D l it is easy to observe that the optimal is (l) since only l machines can be used at any time. We try to capture these types of situations in our lower bound in a simple way. We can view the precedence relations between the jobs as a weighted poset where each element of the poset has a weight associated with it that is the same as the processing time of the associated job. We will assume that we have the transitive closure of the precedence constraints. We will need a few denitions. Denition 1 A chain P is a set of jobs j 1 ; : : :; j r such that for all 1 i < r, j i j i+1. The length of a chain P denoted by j is the sum of the processing times of the jobs in P. K k=1 D k : 3
4 Denition 2 A chain decomposition P of a set of precedence constrained jobs is a partition of the poset in to a collection of chains fp 1 ; P 2 ; : : :; P r g. A maximal chain decomposition is one in which P 1 is a longest chain and fp 2 ; : : :; P r g is a maximal chain decomposition of the poset with elements of P 1 removed. Denition 3 Let P = fp 1 ; P 2 ; : : :; P r g be any maximal chain decomposition of the precedence graph of the jobs. We dene a quantity called L P associated with P as follows. L P = max 1jmin(r;m) jp ij s i With the above denitions in place we are ready to state and prove the new lower bound. Theorem 2 Let P = fp 1 ; P 2 ; : : :; P r g be any maximal chain decomposition of the precedence n graph of the jobs. Let AL = j=1 p i P m which represents the average load. Then s i C max maxfal; L Pg Moreover the lower bound is valid for the preemptive case as well. Proof: It is easy to observe that Cmax AL. We will show the following for 1 j m Cmax jp ij s i which will prove the theorem. Consider the rst j chains. Suppose our input instance was modied to have only the jobs in rst j chains with the precedence constraints induced by the original instance. It is easy to see that a lower bound for this modied instance is a lower bound for the original instance. Since it is possible to execute only one job from each chain at any time instant, only the fastest j machines are relevant for this modied instance. j The expression jp ij is nothing but the average load for the modied instance, which as s i we observed before is a lower bound. Since the average load is also a lower bound for the preemptive case, the bound is valid for that case as well. 2 Theorem 3 A maximal chain decomposition can be computed in O(n 3 ) time. If all p j the same, the running time can be improved to O(n 2p n). are Proof: It is necessary to nd the transitive closure of the given graph of precedence constraints. This can be done in O(n 3 ) time using a BFS from each vertex. From a theoretical point of view this can be improved to O(n! ) where! 2:376 using fast matrix multiplication [2]. A longest chain in a weighted DAG can be found in O(n 2 ) time using standard algorithms. Using this at most n times, a maximal chain decomposition can be obtained. If all p j are the same (without loss of generality we can assume they are all 1) the length of a chain is the same as the number of vertices in the chain. It is possible to use this additional structure to obtain a maximal chain decomposition in O(n 2p n) time. We defer the details. 2 4
5 1. compute a maximal chain decomposition P = fp 1 ; : : :; P r g of the jobs. 2. set l = 1. set B = maxfal; L P g. 3. foreach speed 1 i k do P (a) let l t r be max index such that ljt j j=(m i s i ) 4B. (b) assign jobs in chains P l ; : : :; P t to speed i. (c) set l = t + 1. If l > r return. 4. return. Figure 1: Algorithm ChainAlloc 4 The approximation algorithm The approximation algorithm we develop in this section will be based on the maximal chain decompositions dened in the previous section. As mentioned in Section 2, we will describe an algorithm to produce a job assignment where each job is assigned to a specic speed. Then we use the speed based list scheduling of Chudak and Shmoys with the job assignment produced by our algorithm. Essentially, the algorithm in Figure 1 computes a lower bound B on the optimal using a maximum chain decomposition, and allocates the chains in nonincreasing lengths to the speeds such that no speed is loaded more than four times the lower bound. We now prove several properties of the above described allocation which leads to the performance guarantee of the algorithm. Lemma 1 If chain is assigned to speed i, then jj s i 2B. Proof: Suppose some chain violates the above condition. Then let be the chain with the least index which violates it (hence longest among the violating chains) and let s u be the speed to which it is assigned. From the denition of L P and B it follows that jp 1 j=s 1 B. Therefore it must be the case that P u > 1 and that j > m 1. Let v be the index such that M v?1 1 < j M1 v (recall that M 1 v = v m l). If j > m, no such index exists and we set v to K, the slowest speed. If j m, for convenience of notation we assume that j = M1 v simply by ignoring other machines of speed P s v. From P the denition of L P, AL, j and B, we get the following. If j m then L P ( jp v ij)=( m is i ). If j > m then m i s i ). In either case we obtain the fact that j P ij v m maxfl P ; ALg = B (1) ls l AL ( jp ij)=( P K Since j j=s u > 2B, it must be the case that jp i j=s u > 2B for all M 1 < i j. This implies that j jp i j > 2B(j? M M1 1 )s u 2B <i 5 v m l s l (2)
6 We claim that each of the machines in speeds 1 to u? 1 have an average load greater than 2B. This is because for all 1 i < j we have jp i j=s k(i) 2B and, the algorithm is willing to load the machines to an average load of 4B. In addition it is true that j j=s 2B. Therefore we have jp ij P m ls l > 2B (3) From Equations 1 and 2 we obtain the following sequence of inequalities. M j jp i j B m l s l M1 jp i j j + jp i j B( m l s l + M1 <i M1 jp i v j + 2B m l s l B( m l s l + 1 jp i j + B B v v m l s l B m l s l B v m l s l From Equations 1 and 4 we obtain the following j jp i j B v m l s l B( m l s l + 2B From Equation 1 and above we conclude that j v v m l s l ) m l s l ) using Equation 2 m l s l (4) v m l s l ) m l s l using Equation 4 jp i j 2B m l s l The last inequality above contradicts Equation 3. Hence our assumption about must be incorrect which proves the lemma. 2 Lemma 2 Algorithm ChainAlloc allocates all chains to some speed. Proof: Let be the rst chain which is not allocated. As in the argument in the proof of Lemma 1, each speed is loaded to an average load greater than 2B. This is a contradiction since that would imply that AL > B where AL is the average load of each machine. 2 Lemma 3 For 1 k K, D k 4C max. 6
7 Proof: Since B C max and the algorithm never loads a speed to more than an average load of 4B the bound follows. 2 Lemma 4 For the job assignment produced C 2KC max. P Proof: Let P be any chain. We will show that p j2=s k(j) 2KCmax. Let A i be the set of jobs in P which are assigned to speed P i. Let P l be the longest chain assigned to speed i by the algorithm. We claim that jp l j j2a i p i. This is because the jobs in A i form a chain when we picked P l to be the longest chain in the max chain decomposition. From Lemma 1 we know that jp l j=s i 2B 2Cmax. Therefore it follows that k j2p p j s k(j) = ja i j s i 2KC max Theorem 4 Using speed based list scheduling on the job assignment produced by Algorithm ChainAlloc gives a 6K approximation where K is the number of distinct speeds. Furthermore the algorithm runs in O(n 3 ) time. The running time can be improved to O(n 2p n) if all p j are the same. Proof: From Lemmas 3 we obtain that D k 4Cmax for 1 k K and from Lemma 4 we obtain C 2KCmax. Putting these two facts together, for the job assignment produced by the algorithm ChainAlloc, speed based list scheduling gives the following upper bound by Theorem 1. C max C + K k=1 D k 2KC max + 4KC max 6KC max : It is easy to see that the speed based list scheduling can be implemented in O(n 2 ) time. The running time is dominated by the time to do the maximum chain decomposition. Theorem 3 gives the desired bounds. 2 Corollary 1 There is an algorithm which runs in O(n 3 ) time and gives an O(log m) approximation ratio to the problem of scheduling precedence constrained jobs on uniformly related machines. We remark here that the leading constant in the LP based algorithm in [1] is better. We also observe that the above bound is based on the lower bound which is valid for preemptive schedules as well. Hence our result is valid for preemptive schedules. In [1] it is shown that the lower bound provided by the LP relaxation is a factor of (log m= log log m) away from the optimal. Surprisingly it is easy to show using the same example as in [1] that our lower bound from Section 3 is also a factor of (log m= log log m) away from the optimal. Theorem 5 There are instances where the lower bound given in Theorem 2 is a factor of (log m= log log m) away from the optimal. Proof: The proof of Theorem 3:3 in [1] provides the instance and it is easily veried that any maximum chain decomposition of that instance is a factor of (log m= log log m) away from the optimal
8 4.1 Release Dates Now consider the scenario where each job j has a release date r j before which it cannot be processed. By a general result of Shmoys, Wein, and Williamson an approximation algorithm for the problem without release dates can be tranformed to one with release dates losing only a factor of 2 in the process. Therefore we obtain the following. Theorem 6 There is an O(log m) approximation for the problem Qjprec; r j jc max runs in time O(n 3 ). which 5 Conclusions The main contribution of this paper is a simple and ecient O(log m) approximation to the scheduling problem QjprecjC max. Chudak and Shmoys [1] provide similar approximations for the more general case when the objective function is the average weighted completion time (Qjprecj P w j C j ) using linear programming relaxations. We believe that the techniques of this paper can be extended to obtain a simpler and combinatorial algorithm for that case as well. It is known that the problem of minimizing makespan is hard to approximate to within a factor of 4=3 even if all machines have the same speed [7]. However, for the single speed case a 2 approximation is known, while the best known ratio for the multiple speed case is only O(log m). Obtaining a constant factor approximation, or improving the hardness are interesting open problems. References [1] F. Chudak and D. Shmoys. Approximation algorithms for precedenceconstrained scheduling problems on parallel machines that run at dierent speeds. Proceedings of the Eighth Annual ACMSIAM Symposium on Discrete Algorithms (SODA), [2] D. Coppersmith and S. Winograd. Matrix multiplication via arithmetic progression. Proceedings of the 19th ACM Symposium on Thoery of Computing, 1{6, [3] M.R. Garey and D.S. Johnson. Computers and Intractability: A Guide to the Theory of NPcompleteness, Freeman, San Francisco (1979). [4] R.L. Graham. Bounds for certain multiprocessor anomalies. Bell System Tech. J. 45:1563{81, [5] R.L. Graham, E.L. Lawler, J.K. Lenstra and A.H.G. Rinnooy Kan. Optimization and approximation in deterministic sequencing and scheduling: a survey. Ann. Discrete Math. 5:287{326, [6] J. Jae. Ecient scheduling of tasks without full use of processor resources. Theoretical Computer Science, 26:1{17, [7] J.K. Lenstra and A.H.G. Rinnooy Kan. Complexity of scheduling under precedence constraints. Operations Research, 26:22{35, [8] J.W.S. Lui and C.L. Lui. Bounds on scheduling algorithms for heterogeneous computing systems. In J.L Rosenfeld (ed.), Information Processing 74, NorthHolland, 349{353,
9 [9] D. Shmoys, J. Wein, and D. Williamson. Scheduling parallel machines online. SIAM Journal on Computing, vol 24, 1313{31,
Completion Time Scheduling and the WSRPT Algorithm
Completion Time Scheduling and the WSRPT Algorithm Bo Xiong, Christine Chung Department of Computer Science, Connecticut College, New London, CT {bxiong,cchung}@conncoll.edu Abstract. We consider the online
More informationScheduling Parallel Jobs with Linear Speedup
Scheduling Parallel Jobs with Linear Speedup Alexander Grigoriev and Marc Uetz Maastricht University, Quantitative Economics, P.O.Box 616, 6200 MD Maastricht, The Netherlands. Email: {a.grigoriev,m.uetz}@ke.unimaas.nl
More information1.1 Related work For nonpreemptive scheduling on uniformly related machines the rst algorithm with a constant competitive ratio was given by Aspnes e
A Lower Bound for OnLine Scheduling on Uniformly Related Machines Leah Epstein Jir Sgall September 23, 1999 lea@math.tau.ac.il, Department of Computer Science, TelAviv University, Israel; sgall@math.cas.cz,
More informationScheduling Parallel Jobs with Monotone Speedup 1
Scheduling Parallel Jobs with Monotone Speedup 1 Alexander Grigoriev, Marc Uetz Maastricht University, Quantitative Economics, P.O.Box 616, 6200 MD Maastricht, The Netherlands, {a.grigoriev@ke.unimaas.nl,
More informationAn improved online algorithm for scheduling on two unrestrictive parallel batch processing machines
This is the PrePublished Version. An improved online algorithm for scheduling on two unrestrictive parallel batch processing machines Q.Q. Nong, T.C.E. Cheng, C.T. Ng Department of Mathematics, Ocean
More informationThe power of points in preemptive single machine scheduling
JOURNAL OF SCHEDULING J. Sched. 22; 5:121 133 (DOI: 1.12/jos.93) The power of points in preemptive single machine scheduling Andreas S. Schulz 1; 2; ; and Martin Skutella 1 Massachusetts Institute of
More information2.3 Scheduling jobs on identical parallel machines
2.3 Scheduling jobs on identical parallel machines There are jobs to be processed, and there are identical machines (running in parallel) to which each job may be assigned Each job = 1,,, must be processed
More informationOptimal and nearly optimal online and semionline algorithms for some scheduling problems
Optimal and nearly optimal online and semionline algorithms for some scheduling problems Ph.D. thesis Made by: Dósa György Supervisor: Vízvári Béla University of Szeged, Faculty of Science Doctoral School
More informationAnswers to some of the exercises.
Answers to some of the exercises. Chapter 2. Ex.2.1 (a) There are several ways to do this. Here is one possibility. The idea is to apply the kcenter algorithm first to D and then for each center in D
More informationJUSTINTIME SCHEDULING WITH PERIODIC TIME SLOTS. Received December May 12, 2003; revised February 5, 2004
Scientiae Mathematicae Japonicae Online, Vol. 10, (2004), 431 437 431 JUSTINTIME SCHEDULING WITH PERIODIC TIME SLOTS Ondřej Čepeka and Shao Chin Sung b Received December May 12, 2003; revised February
More informationA(L) denotes the makespan of a schedule produced by algorithm A for scheduling the list L of jobs, and opt(l) denotes the corresponding makespan of so
Semionline scheduling with decreasing job sizes Steve Seiden Jir Sgall y Gerhard Woeginger z October 29, 1998 Abstract We investigate the problem of semionline scheduling jobs on m identical parallel
More informationMatching Nuts and Bolts Faster? MaxPlanckInstitut fur Informatik, Im Stadtwald, Saarbrucken, Germany.
Matching Nuts and Bolts Faster? Phillip G. Bradford Rudolf Fleischer MaxPlanckInstitut fur Informatik, Im Stadtwald, 6613 Saarbrucken, Germany. Email: fbradford,rudolfg@mpisb.mpg.de. Abstract. The
More informationDuplicating and its Applications in Batch Scheduling
Duplicating and its Applications in Batch Scheduling Yuzhong Zhang 1 Chunsong Bai 1 Shouyang Wang 2 1 College of Operations Research and Management Sciences Qufu Normal University, Shandong 276826, China
More informationChapter 11. 11.1 Load Balancing. Approximation Algorithms. Load Balancing. Load Balancing on 2 Machines. Load Balancing: Greedy Scheduling
Approximation Algorithms Chapter Approximation Algorithms Q. Suppose I need to solve an NPhard problem. What should I do? A. Theory says you're unlikely to find a polytime algorithm. Must sacrifice one
More information10.1 Integer Programming and LP relaxation
CS787: Advanced Algorithms Lecture 10: LP Relaxation and Rounding In this lecture we will design approximation algorithms using linear programming. The key insight behind this approach is that the closely
More informationApproximation Algorithms: LP Relaxation, Rounding, and Randomized Rounding Techniques. My T. Thai
Approximation Algorithms: LP Relaxation, Rounding, and Randomized Rounding Techniques My T. Thai 1 Overview An overview of LP relaxation and rounding method is as follows: 1. Formulate an optimization
More informationJob Scheduling Techniques for Distributed Systems with Heterogeneous Processor Cardinality
Job Scheduling Techniques for Distributed Systems with Heterogeneous Processor Cardinality HungJui Chang JanJan Wu Department of Computer Science and Information Engineering Institute of Information
More informationA class of online scheduling algorithms to minimize total completion time
A class of online scheduling algorithms to minimize total completion time X. Lu R.A. Sitters L. Stougie Abstract We consider the problem of scheduling jobs online on a single machine and on identical
More informationLecture 6: Approximation via LP Rounding
Lecture 6: Approximation via LP Rounding Let G = (V, E) be an (undirected) graph. A subset C V is called a vertex cover for G if for every edge (v i, v j ) E we have v i C or v j C (or both). In other
More information! Solve problem to optimality. ! Solve problem in polytime. ! Solve arbitrary instances of the problem. !approximation algorithm.
Approximation Algorithms Chapter Approximation Algorithms Q Suppose I need to solve an NPhard problem What should I do? A Theory says you're unlikely to find a polytime algorithm Must sacrifice one of
More information! Solve problem to optimality. ! Solve problem in polytime. ! Solve arbitrary instances of the problem. #approximation algorithm.
Approximation Algorithms 11 Approximation Algorithms Q Suppose I need to solve an NPhard problem What should I do? A Theory says you're unlikely to find a polytime algorithm Must sacrifice one of three
More informationAn Algorithm for Fractional Assignment Problems. Maiko Shigeno. Yasufumi Saruwatari y. and. Tomomi Matsui z. Department of Management Science
An Algorithm for Fractional Assignment Problems Maiko Shigeno Yasufumi Saruwatari y and Tomomi Matsui z Department of Management Science Science University of Tokyo 13 Kagurazaka, Shinjukuku, Tokyo 162,
More informationList Scheduling in Order of αpoints on a Single Machine
List Scheduling in Order of αpoints on a Single Machine Martin Skutella Fachbereich Mathematik, Universität Dortmund, D 4422 Dortmund, Germany martin.skutella@unidortmund.de http://www.mathematik.unidortmund.de/
More informationSingle machine parallel batch scheduling with unbounded capacity
Workshop on Combinatorics and Graph Theory 21th, April, 2006 Nankai University Single machine parallel batch scheduling with unbounded capacity Yuan Jinjiang Department of mathematics, Zhengzhou University
More informationOptimal Online Preemptive Scheduling
IEOR 8100: Scheduling Lecture Guest Optimal Online Preemptive Scheduling Lecturer: Jir Sgall Scribe: Michael Hamilton 1 Introduction In this lecture we ll study online preemptive scheduling on m machines
More informationApplied Algorithm Design Lecture 5
Applied Algorithm Design Lecture 5 Pietro Michiardi Eurecom Pietro Michiardi (Eurecom) Applied Algorithm Design Lecture 5 1 / 86 Approximation Algorithms Pietro Michiardi (Eurecom) Applied Algorithm Design
More informationScheduling Shop Scheduling. Tim Nieberg
Scheduling Shop Scheduling Tim Nieberg Shop models: General Introduction Remark: Consider non preemptive problems with regular objectives Notation Shop Problems: m machines, n jobs 1,..., n operations
More informationAlgorithm Design and Analysis
Algorithm Design and Analysis LECTURE 27 Approximation Algorithms Load Balancing Weighted Vertex Cover Reminder: Fill out SRTEs online Don t forget to click submit Sofya Raskhodnikova 12/6/2011 S. Raskhodnikova;
More information1 Polyhedra and Linear Programming
CS 598CSC: Combinatorial Optimization Lecture date: January 21, 2009 Instructor: Chandra Chekuri Scribe: Sungjin Im 1 Polyhedra and Linear Programming In this lecture, we will cover some basic material
More informationLecture 3: Linear Programming Relaxations and Rounding
Lecture 3: Linear Programming Relaxations and Rounding 1 Approximation Algorithms and Linear Relaxations For the time being, suppose we have a minimization problem. Many times, the problem at hand can
More informationcan be extended into NC and RNC algorithms, respectively, that solve more
Extending NC and RNC Algorithms Nimrod Megiddo A technique is presented by which NC and RNC algorithms for some problems can be extended into NC and RNC algorithms, respectively, that solve more general
More informationBatch Scheduling of Deteriorating Products
Decision Making in Manufacturing and Services Vol. 1 2007 No. 1 2 pp. 25 34 Batch Scheduling of Deteriorating Products Maksim S. Barketau, T.C. Edwin Cheng, Mikhail Y. Kovalyov, C.T. Daniel Ng Abstract.
More informationBiobjective approximation scheme for makespan and reliability optimization on uniform parallel machines
Biobjective approximation scheme for makespan and reliability optimization on uniform parallel machines Emmanuel Jeannot 1, Erik Saule 2, and Denis Trystram 2 1 INRIALorraine : emmanuel.jeannot@loria.fr
More informationApproximated Distributed Minimum Vertex Cover Algorithms for Bounded Degree Graphs
Approximated Distributed Minimum Vertex Cover Algorithms for Bounded Degree Graphs Yong Zhang 1.2, Francis Y.L. Chin 2, and HingFung Ting 2 1 College of Mathematics and Computer Science, Hebei University,
More informationAnalysis of Approximation Algorithms for kset Cover using FactorRevealing Linear Programs
Analysis of Approximation Algorithms for kset Cover using FactorRevealing Linear Programs Stavros Athanassopoulos, Ioannis Caragiannis, and Christos Kaklamanis Research Academic Computer Technology Institute
More informationThe Goldberg Rao Algorithm for the Maximum Flow Problem
The Goldberg Rao Algorithm for the Maximum Flow Problem COS 528 class notes October 18, 2006 Scribe: Dávid Papp Main idea: use of the blocking flow paradigm to achieve essentially O(min{m 2/3, n 1/2 }
More informationLinear Programming I
Linear Programming I November 30, 2003 1 Introduction In the VCR/guns/nuclear bombs/napkins/star wars/professors/butter/mice problem, the benevolent dictator, Bigus Piguinus, of south Antarctica penguins
More informationApproximation Algorithms. Scheduling. Approximation algorithms. Scheduling jobs on a single machine
Approximation algorithms Approximation Algorithms Fast. Cheap. Reliable. Choose two. NPhard problems: choose 2 of optimal polynomial time all instances Approximation algorithms. Tradeoff between time
More informationGreedy Heuristics with Regret, with Application to the Cheapest Insertion Algorithm for the TSP
Greedy Heuristics with Regret, with Application to the Cheapest Insertion Algorithm for the TSP Refael Hassin Ariel Keinan Abstract We considers greedy algorithms that allow partial regret. As an example
More informationMarkov Chains, part I
Markov Chains, part I December 8, 2010 1 Introduction A Markov Chain is a sequence of random variables X 0, X 1,, where each X i S, such that P(X i+1 = s i+1 X i = s i, X i 1 = s i 1,, X 0 = s 0 ) = P(X
More informationRonald Graham: Laying the Foundations of Online Optimization
Documenta Math. 239 Ronald Graham: Laying the Foundations of Online Optimization Susanne Albers Abstract. This chapter highlights fundamental contributions made by Ron Graham in the area of online optimization.
More informationTHEORY OF SIMPLEX METHOD
Chapter THEORY OF SIMPLEX METHOD Mathematical Programming Problems A mathematical programming problem is an optimization problem of finding the values of the unknown variables x, x,, x n that maximize
More informationSHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH
31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,
More informationmax cx s.t. Ax c where the matrix A, cost vector c and right hand side b are given and x is a vector of variables. For this example we have x
Linear Programming Linear programming refers to problems stated as maximization or minimization of a linear function subject to constraints that are linear equalities and inequalities. Although the study
More informationprinceton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora
princeton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora Scribe: One of the running themes in this course is the notion of
More information8.1 Makespan Scheduling
600.469 / 600.669 Approximation Algorithms Lecturer: Michael Dinitz Topic: Dynamic Programing: MinMakespan and Bin Packing Date: 2/19/15 Scribe: Gabriel Kaptchuk 8.1 Makespan Scheduling Consider an instance
More informationDistributed Load Balancing for Machines Fully Heterogeneous
Internship Report 2 nd of June  22 th of August 2014 Distributed Load Balancing for Machines Fully Heterogeneous Nathanaël Cheriere nathanael.cheriere@ensrennes.fr ENS Rennes Academic Year 20132014
More informationIntroduction to Scheduling Theory
Introduction to Scheduling Theory Arnaud Legrand Laboratoire Informatique et Distribution IMAG CNRS, France arnaud.legrand@imag.fr November 8, 2004 1/ 26 Outline 1 Task graphs from outer space 2 Scheduling
More informationOnline Adwords Allocation
Online Adwords Allocation Shoshana Neuburger May 6, 2009 1 Overview Many search engines auction the advertising space alongside search results. When Google interviewed Amin Saberi in 2004, their advertisement
More informationScheduling Realtime Tasks: Algorithms and Complexity
Scheduling Realtime Tasks: Algorithms and Complexity Sanjoy Baruah The University of North Carolina at Chapel Hill Email: baruah@cs.unc.edu Joël Goossens Université Libre de Bruxelles Email: joel.goossens@ulb.ac.be
More informationTHE SCHEDULING OF MAINTENANCE SERVICE
THE SCHEDULING OF MAINTENANCE SERVICE Shoshana Anily Celia A. Glass Refael Hassin Abstract We study a discrete problem of scheduling activities of several types under the constraint that at most a single
More informationOn the approximability of average completion time scheduling under precedence constraints
On the approximability of average completion time scheduling under precedence constraints Gerhard J. Woeginger Abstract We consider the scheduling problem of minimizing the average weighted job completion
More informationBest Monotone Degree Bounds for Various Graph Parameters
Best Monotone Degree Bounds for Various Graph Parameters D. Bauer Department of Mathematical Sciences Stevens Institute of Technology Hoboken, NJ 07030 S. L. Hakimi Department of Electrical and Computer
More informationOnline Scheduling with Bounded Migration
Online Scheduling with Bounded Migration Peter Sanders, Naveen Sivadasan, and Martin Skutella MaxPlanckInstitut für Informatik, Saarbrücken, Germany, {sanders,ns,skutella}@mpisb.mpg.de Abstract. Consider
More informationFairness in Routing and Load Balancing
Fairness in Routing and Load Balancing Jon Kleinberg Yuval Rabani Éva Tardos Abstract We consider the issue of network routing subject to explicit fairness conditions. The optimization of fairness criteria
More informationMATHEMATICAL BACKGROUND
Chapter 1 MATHEMATICAL BACKGROUND This chapter discusses the mathematics that is necessary for the development of the theory of linear programming. We are particularly interested in the solutions of a
More informationTopic: Greedy Approximations: Set Cover and Min Makespan Date: 1/30/06
CS880: Approximations Algorithms Scribe: Matt Elder Lecturer: Shuchi Chawla Topic: Greedy Approximations: Set Cover and Min Makespan Date: 1/30/06 3.1 Set Cover The Set Cover problem is: Given a set of
More informationPartitioning edgecoloured complete graphs into monochromatic cycles and paths
arxiv:1205.5492v1 [math.co] 24 May 2012 Partitioning edgecoloured complete graphs into monochromatic cycles and paths Alexey Pokrovskiy Departement of Mathematics, London School of Economics and Political
More informationOnline Scheduling for Cloud Computing and Different Service Levels
2012 IEEE 201226th IEEE International 26th International Parallel Parallel and Distributed and Distributed Processing Processing Symposium Symposium Workshops Workshops & PhD Forum Online Scheduling for
More informationA Lower Bound for Area{Universal Graphs. K. Mehlhorn { Abstract. We establish a lower bound on the eciency of area{universal circuits.
A Lower Bound for Area{Universal Graphs Gianfranco Bilardi y Shiva Chaudhuri z Devdatt Dubhashi x K. Mehlhorn { Abstract We establish a lower bound on the eciency of area{universal circuits. The area A
More informationA Nearlinear Time Constant Factor Algorithm for Unsplittable Flow Problem on Line with Bag Constraints
A Nearlinear Time Constant Factor Algorithm for Unsplittable Flow Problem on Line with Bag Constraints Venkatesan T. Chakaravarthy, Anamitra R. Choudhury, and Yogish Sabharwal IBM Research  India, New
More informationAdaptive Online Gradient Descent
Adaptive Online Gradient Descent Peter L Bartlett Division of Computer Science Department of Statistics UC Berkeley Berkeley, CA 94709 bartlett@csberkeleyedu Elad Hazan IBM Almaden Research Center 650
More information8.1 Min Degree Spanning Tree
CS880: Approximations Algorithms Scribe: Siddharth Barman Lecturer: Shuchi Chawla Topic: Min Degree Spanning Tree Date: 02/15/07 In this lecture we give a local search based algorithm for the Min Degree
More informationarxiv:1112.0829v1 [math.pr] 5 Dec 2011
How Not to Win a Million Dollars: A Counterexample to a Conjecture of L. Breiman Thomas P. Hayes arxiv:1112.0829v1 [math.pr] 5 Dec 2011 Abstract Consider a gambling game in which we are allowed to repeatedly
More informationSemiOnline Preemptive Scheduling: One Algorithm for All Variants
SemiOnline Preemptive Scheduling: One Algorithm for All Variants Tomáš Ebenlendr Jiří Sgall Abstract: We present a unified optimal semionline algorithm for preemptive scheduling on uniformly related
More informationTheorem 2. If x Q and y R \ Q, then. (a) x + y R \ Q, and. (b) xy Q.
Math 305 Fall 011 The Density of Q in R The following two theorems tell us what happens when we add and multiply by rational numbers. For the first one, we see that if we add or multiply two rational numbers
More informationScheduling Single Machine Scheduling. Tim Nieberg
Scheduling Single Machine Scheduling Tim Nieberg Single machine models Observation: for nonpreemptive problems and regular objectives, a sequence in which the jobs are processed is sufficient to describe
More informationMarkov Chains and Applications
Markov Chains and Applications Alexander Volfovsky August 7, 2007 Abstract In this paper I provide a quick overview of Stochastic processes and then quickly delve into a discussion of Markov Chains. There
More informationWeek 5 Integral Polyhedra
Week 5 Integral Polyhedra We have seen some examples 1 of linear programming formulation that are integral, meaning that every basic feasible solution is an integral vector. This week we develop a theory
More informationUNIVERSAL JUGGLING CYCLES. Fan Chung 1 Department of Mathematics, University of California, San Diego, La Jolla, CA Ron Graham 2.
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7(2) (2007), #A08 UNIVERSAL JUGGLING CYCLES Fan Chung 1 Department of Mathematics, University of California, San Diego, La Jolla, CA 92093 Ron
More informationPartitioned realtime scheduling on heterogeneous sharedmemory multiprocessors
Partitioned realtime scheduling on heterogeneous sharedmemory multiprocessors Martin Niemeier École Polytechnique Fédérale de Lausanne Discrete Optimization Group Lausanne, Switzerland martin.niemeier@epfl.ch
More information7.1 Introduction. CSci 335 Software Design and Analysis III Chapter 7 Sorting. Prof. Stewart Weiss
Chapter 7 Sorting 7.1 Introduction Insertion sort is the sorting algorithm that splits an array into a sorted and an unsorted region, and repeatedly picks the lowest index element of the unsorted region
More informationIntroduction to Flocking {Stochastic Matrices}
Supelec EECI Graduate School in Control Introduction to Flocking {Stochastic Matrices} A. S. Morse Yale University Gif sur  Yvette May 21, 2012 CRAIG REYNOLDS  1987 BOIDS The Lion King CRAIG REYNOLDS
More informationTwo General Methods to Reduce Delay and Change of Enumeration Algorithms
ISSN 13465597 NII Technical Report Two General Methods to Reduce Delay and Change of Enumeration Algorithms Takeaki Uno NII2003004E Apr.2003 Two General Methods to Reduce Delay and Change of Enumeration
More informationDefinition 11.1. Given a graph G on n vertices, we define the following quantities:
Lecture 11 The Lovász ϑ Function 11.1 Perfect graphs We begin with some background on perfect graphs. graphs. First, we define some quantities on Definition 11.1. Given a graph G on n vertices, we define
More informationWORSTCASE PERFORMANCE ANALYSIS OF SOME APPROXIMATION ALGORITHMS FOR MINIMIZING MAKESPAN AND FLOWTIME
WORSTCASE PERFORMANCE ANALYSIS OF SOME APPROXIMATION ALGORITHMS FOR MINIMIZING MAKESPAN AND FLOWTIME PERUVEMBA SUNDARAM RAVI, LEVENT TUNÇEL, MICHAEL HUANG Abstract. In 1976, Coffman and Sethi conjectured
More informationClassification  Examples
Lecture 2 Scheduling 1 Classification  Examples 1 r j C max given: n jobs with processing times p 1,...,p n and release dates r 1,...,r n jobs have to be scheduled without preemption on one machine taking
More informationResearch Article Batch Scheduling on TwoMachine Flowshop with MachineDependent Setup Times
Hindawi Publishing Corporation Advances in Operations Research Volume 2009, Article ID 153910, 10 pages doi:10.1155/2009/153910 Research Article Batch Scheduling on TwoMachine Flowshop with MachineDependent
More informationSchool of Computer Science, Carnegie Mellon University, Pittsburgh, PA 15213
,, 1{8 () c A Note on Learning from MultipleInstance Examples AVRIM BLUM avrim+@cs.cmu.edu ADAM KALAI akalai+@cs.cmu.edu School of Computer Science, Carnegie Mellon University, Pittsburgh, PA 1513 Abstract.
More informationPh.D. Thesis. Judit NagyGyörgy. Supervisor: Péter Hajnal Associate Professor
Online algorithms for combinatorial problems Ph.D. Thesis by Judit NagyGyörgy Supervisor: Péter Hajnal Associate Professor Doctoral School in Mathematics and Computer Science University of Szeged Bolyai
More informationMapReduce and Distributed Data Analysis. Sergei Vassilvitskii Google Research
MapReduce and Distributed Data Analysis Google Research 1 Dealing With Massive Data 2 2 Dealing With Massive Data Polynomial Memory Sublinear RAM Sketches External Memory Property Testing 3 3 Dealing With
More informationApproximation Algorithms
Approximation Algorithms or: How I Learned to Stop Worrying and Deal with NPCompleteness Ong Jit Sheng, Jonathan (A0073924B) March, 2012 Overview Key Results (I) General techniques: Greedy algorithms
More information11. APPROXIMATION ALGORITHMS
11. APPROXIMATION ALGORITHMS load balancing center selection pricing method: vertex cover LP rounding: vertex cover generalized load balancing knapsack problem Lecture slides by Kevin Wayne Copyright 2005
More informationONLINE DEGREEBOUNDED STEINER NETWORK DESIGN. Sina Dehghani Saeed Seddighin Ali Shafahi Fall 2015
ONLINE DEGREEBOUNDED STEINER NETWORK DESIGN Sina Dehghani Saeed Seddighin Ali Shafahi Fall 2015 ONLINE STEINER FOREST PROBLEM An initially given graph G. s 1 s 2 A sequence of demands (s i, t i ) arriving
More informationLecture 7: Approximation via Randomized Rounding
Lecture 7: Approximation via Randomized Rounding Often LPs return a fractional solution where the solution x, which is supposed to be in {0, } n, is in [0, ] n instead. There is a generic way of obtaining
More informationNan Kong, Andrew J. Schaefer. Department of Industrial Engineering, Univeristy of Pittsburgh, PA 15261, USA
A Factor 1 2 Approximation Algorithm for TwoStage Stochastic Matching Problems Nan Kong, Andrew J. Schaefer Department of Industrial Engineering, Univeristy of Pittsburgh, PA 15261, USA Abstract We introduce
More informationDimensioning an inbound call center using constraint programming
Dimensioning an inbound call center using constraint programming Cyril Canon 1,2, JeanCharles Billaut 2, and JeanLouis Bouquard 2 1 Vitalicom, 643 avenue du grain d or, 41350 Vineuil, France ccanon@fr.snt.com
More informationMatching Nuts and Bolts Faster? MaxPlanckInstitut fur Informatik, Im Stadtwald, Saarbrucken, Germany.
Matching Nuts and Bolts Faster? Phillip G. Bradford Rudolf Fleischer MaxPlanckInstitut fur Informatik, Im Stadtwald, 6613 Saarbrucken, Germany. Email: fbradford,rudolfg@mpisb.mpg.de. Abstract. The
More informationMinimum Makespan Scheduling
Minimum Makespan Scheduling Minimum makespan scheduling: Definition and variants Factor 2 algorithm for identical machines PTAS for identical machines Factor 2 algorithm for unrelated machines Martin Zachariasen,
More informationCS 598CSC: Combinatorial Optimization Lecture date: 2/4/2010
CS 598CSC: Combinatorial Optimization Lecture date: /4/010 Instructor: Chandra Chekuri Scribe: David Morrison GomoryHu Trees (The work in this section closely follows [3]) Let G = (V, E) be an undirected
More informationCost Model: Work, Span and Parallelism. 1 The RAM model for sequential computation:
CSE341T 08/31/2015 Lecture 3 Cost Model: Work, Span and Parallelism In this lecture, we will look at how one analyze a parallel program written using Cilk Plus. When we analyze the cost of an algorithm
More informationAll trees contain a large induced subgraph having all degrees 1 (mod k)
All trees contain a large induced subgraph having all degrees 1 (mod k) David M. Berman, A.J. Radcliffe, A.D. Scott, Hong Wang, and Larry Wargo *Department of Mathematics University of New Orleans New
More informationAn Eective Load Balancing Policy for
An Eective Load Balancing Policy for Geometric Decaying Algorithms Joseph Gil y Dept. of Computer Science The Technion, Israel Technion City, Haifa 32000 ISRAEL Yossi Matias z AT&T Bell Laboratories 600
More informationCompetitive Analysis of On line Randomized Call Control in Cellular Networks
Competitive Analysis of On line Randomized Call Control in Cellular Networks Ioannis Caragiannis Christos Kaklamanis Evi Papaioannou Abstract In this paper we address an important communication issue arising
More informationPreemptive Online Scheduling: Optimal Algorithms for All Speeds
Algorithmica (2009) 53: 504 522 DOI 10.1007/s0045300892356 Preemptive Online Scheduling: Optimal Algorithms for All Speeds Tomáš Ebenlendr Wojciech Jawor Jiří Sgall Received: 30 November 2006 / Accepted:
More informationSection Notes 4. Duality, Sensitivity, Dual Simplex, Complementary Slackness. Applied Math 121. Week of February 28, 2011
Section Notes 4 Duality, Sensitivity, Dual Simplex, Complementary Slackness Applied Math 121 Week of February 28, 2011 Goals for the week understand the relationship between primal and dual programs. know
More informationIntegrating job parallelism in realtime scheduling theory
Integrating job parallelism in realtime scheduling theory Sébastien Collette Liliana Cucu Joël Goossens Abstract We investigate the global scheduling of sporadic, implicit deadline, realtime task systems
More informationA constantfactor approximation algorithm for the kmedian problem
A constantfactor approximation algorithm for the kmedian problem Moses Charikar Sudipto Guha Éva Tardos David B. Shmoys July 23, 2002 Abstract We present the first constantfactor approximation algorithm
More informationCombinatorial PCPs with ecient veriers
Combinatorial PCPs with ecient veriers Or Meir Abstract The PCP theorem asserts the existence of proofs that can be veried by a verier that reads only a very small part of the proof. The theorem was originally
More information1 Solving LPs: The Simplex Algorithm of George Dantzig
Solving LPs: The Simplex Algorithm of George Dantzig. Simplex Pivoting: Dictionary Format We illustrate a general solution procedure, called the simplex algorithm, by implementing it on a very simple example.
More information