Parallel machine scheduling under a grade of service provision
|
|
- Paula Shelton
- 7 years ago
- Views:
Transcription
1 Computers & Operations Research 31 (200) Parallel machine scheduling under a grade of service provision Hark-Chin Hwang a;1, Soo Y. Chang b;, Kangbok Lee b a Department of Industrial Engineering, Chosun University, 375 Susuk-Dong, Dong-Gu, Gwangju , South Korea b Industrial Engineering Department, Division of Electrical and Computer Engineering, Pohang University of Science and Technology, San 31 Hyoja-Dong, Pohang, Kyungbook , South Korea Abstract We consider the problem of scheduling parallel machines that process service requests from various customers who are entitled to many dierent grade of service (GoS) levels. We propose and analyze one simple way to ensure such dierentiated service. In particular, we investigate how the longest processing time rst algorithm (LPT) would perform in the worst case and show that a slight modication of LPT could signicantly improve its worst-case performance.? 2003 Elsevier Ltd. All rights reserved. Keywords: Parallel machine scheduling; Longest processing time rst; Eligibility; Grade of service 1. Introduction In service industry, the service providers often have special customers, such as, silver, gold, or platinum members who are more valued than the regular members. Those special members are usually entitled to premium services, and, hence, some kind of dierentiated service policy must be implemented by the service provider. One simple scheme for providing dierentiated service is to label machines and jobs with prespecied grade of service (or GoS) levels and allow each job to be processed by a particular machine only when the GoS level of the job is no less than the GoS level of the machine. In eect, the processing capability of the machines labelled with high GoS levels tends to be reserved for the jobs with high GoS levels. Hence, if we label relatively higher GoS levels on the jobs from valued Corresponding author. address: syc@postech.ac.kr (S.Y. Chang). 1 Supported by research funds from Chosun University /$ - see front matter? 2003 Elsevier Ltd. All rights reserved. doi: /s (03)0016-3
2 2056 H.-C. Hwang et al. / Computers & Operations Research 31 (200) customers, we can ensure better service to more valued customers. In such situation, assigning jobs to the machines becomes a parallel machine scheduling problem with a special eligibility constraint. Although the makespan is not always the best choice for measuring the goodness of the schedule, it is chosen as the objective function to be minimized in this paper for the following reason. We are looking at the particular case where we have a few clusters of the same number of identical parallel processors and one special processor called load balancer. The load balancer receives and collects customer service requests over a short period of time and assigns the batch of collected jobs to one of the idle clusters. The load balancer recognizes each cluster to be idle only when each cluster completes the processing of all the jobs in the batch under processing and register its idle state to the load balancer. In order to have each cluster get ready as soon as possible for the next batch, the makespan of the schedule for each batch is certainly the most appropriate measure to be minimized. For more formal denition of our problem instance, we dene J ={a 1 ;:::;a n } and M ={ ;:::; } to be the sets of jobs and machines, respectively. Each job a j requires the processing time of p(a j ) and is labelled with the GoS level of g(a j ). Each machine b i is also labelled with the GoS level g(b i ) and b i is allowed to process job a j only when g(b i ) 6 g(a j ). A schedule is, then, a partition of J into m disjoint sets, S = S 1 ;:::;S m, such that S i is allowed to include a j only if g(b i ) 6 g(a j ). The makespan of a schedule S can be dened as max 16i6m t(s i ), where t(s i )= a j S i p(a j )ifs i, and 0, otherwise. Then, an instance of our scheduling problem can be denoted by (J; M; p; g) or (J; M) for short, which we call an instance of the parallel machine scheduling with GoS eligibility. When all the jobs have the same GoS level, say, we may delete all the machines with their GoS level greater than, and only have to deal with the machines having their GoS level less than or equal to. In such special case, if the number of machines that we need to deal with is m, we see that our problem is the ordinary identical parallel machine scheduling problem of minimizing the makespan which is often denoted by P m C max, following the notational convention developed in [1]. The decision version of this problem is NP-complete but pseudo-polynomially solvable for m = 2. It is NP-complete in the strong sense for m 3, since the 3-PARTITION problem, which is known to be NP-complete in the strong sense, can be reduced to the decision version of P 3 C max [2]. To tackle NP-hard problems, theoretical research has been focused on analyzing the worst-case performance of approximation algorithms. In the case of P m C max without eligibility constraints, LPT (longest processing times rst) and MULTIFIT [3] yield schedules with makespan no more than m [] and times the optimum [5], respectively In the case of the problem with eligibility, Azar et al. [6] presented a kind of list scheduling algorithm, AW, and proved that it always generates a schedule with makespan no greater than log 2 2m times optimum. However, this bound is not a constant but an increasing function of m. Lenstra et al. [7] proposed a binary search algorithm based on linear programming and proved the constant worst-case bound of two for the most general problem of unrelated parallel machine scheduling which certainly covers the case of the problem with eligibility. However, their algorithm must employ linear programming solution module whose computational complexity is not yet proven to be strongly polynomial. In other words, its complexity is polynomially bounded with respect not to the number of constraints and variables but to the actual magnitudes of the values of data dening the particular linear programming problem to be solved. In this paper, we propose a simple algorithm, that we call lowest grade-longest processing times rst (LG-LPT) whose computational complexity is strongly polynomial and the worst-case makespan
3 H.-C. Hwang et al. / Computers & Operations Research 31 (200) of the generated schedule is no more than twice the optimum. In particular, the makespan of the schedule generated by the proposed algorithm is proven to be no more than 5 for m = 2 and 2 1=(m 1) for m 3 times the optimum. In Section 2, we demonstrate the importance of taking the eligibility into account and establish a basic property of our proposed algorithm. In Section 3, we present the result of our worst-case analysis. 2. Algorithm AW The algorithm AW, proposed by Azar et al. in [6], simply picks and assigns an arbitrary unassigned job to one of the eligible machines that has the least load, while breaking ties by choosing the machine with the lowest index. We note that AW does not take the eligibility into account in deciding the order of jobs to be assigned. Due to this, for some of our problem instances, AW may yield a schedule with theoretically proven worst-case makespan of log 2 2m times the optimum, as demonstrated in the following example. Example 1. Consider a problem with J and M, where m = n =2 k for some positive integer k. Every job has the same processing time of 1 and the GoS levels of machines and jobs are set as; g(a 1 )=m; g(a 2 )=m 1;:::;g(a m )=1 and g( )=1;g( )=2;:::;g( )=m. Then, AW may yield a schedule that the rst m=2 jobs, a 1 ;a 2 ;:::;a m=2, are assigned to the machines ; ;:::;=2, next m= jobs are assigned to the machines ; ;:::;= and so on, until the last two jobs a m and a m 1 are both assigned to the machine yielding the makespan of log 2m as shown in Fig. 1. However, an optimal schedule with makespan of 1 can be constructed by assigning job a j to machine j+1 for all j, 16 j 6 m. A worthwhile thing to be noted at this point is the fact that even LPT could also yield exactly the same schedule as AW, since every job has identical processing time in this particular example. This clearly suggests a better approach to the problem in which not only the processing time but also the eligibility of each job is taken into account. In fact, as demonstrated below, we can even get an optimum solution for this problem, if we assign jobs to their eligible machines in a proper order obtainable by the use of the eligibility information. a 1 a 2 a m/2+1 a m-3 a m/2+2 a m-1 a m-2 a m a m a m-1 / a m/ / a m/-1 /2 a m/2 /2 /2+1 /2+1 a m/2+1 a m/2 (a) (b) a 1 Fig. 1. (a) AW and (b) optimal schedule.
4 2058 H.-C. Hwang et al. / Computers & Operations Research 31 (200) Instead of an arbitrary ordering, consider the ordering of jobs where the job a j precedes a k when g(a j ) g(a k ), or if g(a j )=g(a k ), p(a j ) p(a k ) holds. And we denote such precedence by a j a k, and call it LG-LPT order. In the case of the above example, one can easily observe that an optimum solution can be obtained, if AW picks and assigns jobs in the order of LG-LPT instead of choosing jobs arbitrarily. The development of our algorithm is motivated by this simple observation. Throughout the remaining of this paper, we assume that a j 1 a j, j =2; 3;:::;n and g(b i 1 ) 6 g(b i ), i=2; 3;:::;m, for notational simplicity. Also, we assume that the optimum makespan is always 1, since we can redene p(a j )asp(a j ) divided by the optimum makespan, without loss of generality. For further notational convenience, we introduce the concept of the range of each job, denoted by (a j ) which is simply the number of machines that are eligible to process a j, namely the number of machines with g(b i ) 6 g(a j ). Then, due to the nature of (a j ) and the assumed ordering of machines, we observe that the job a j can be assigned to any one of the rst (a j ) machines, i.e. through b (aj). Also note that we can assume that (a n )=m, since, otherwise, we can delete all b i s with g(b i ) g(a n ), without changing the structure of the problem. 3. Algorithm LG-LPT The algorithm LG-LPT fetches the rst one from the list of unassigned jobs sorted in LG-LPT order and assigns it to the least-loaded eligible machine, as formally described in Fig. 2. In the next theorem, we analyze the worst-case performance of LG-LPT. Fig. 2. Algorithm LG-LPT.
5 H.-C. Hwang et al. / Computers & Operations Research 31 (200) Theorem 1. For identical parallel machine scheduling problem with GoS eligibility, LG-LPT always yields a schedule with its makespan. 6 5 times the optimum makespan for m =2, 6 2 1=(m 1) times the optimum makespan for m 3. Proof. We prove the theorem in two separate cases. Case 1: m 3. Suppose that the theorem is false for m 3. Then, there must exist a counterexample (J; M) with m 3 for which LG-LPT cannot yield a schedule with makespan no more than 2 1=(m 1) times the optimum makespan. Let us further suppose that the problem instance (J; M) isminimal, meaning that the set J contains the least number of jobs among all counterexamples. Then, due to this minimality, we claim that the makespan of the minimal counterexample is determined by the last job a n. This claim is true due to the following. If (J; M) is a minimal counterexample, LG-LPT should successfully assign jobs a 1 ;:::;a n 1 within time 2 1=(m 1) and fail to do so only for the last job a n. This is because if LG-LPT failed to assign a job a j for j n within time 2 1=(m 1), we can construct another counterexample (J ;M) with J = {a 1 ;:::;a j } which has strictly less number of jobs than (J; M). Hence, a clear contradiction to the minimality of (J; M) results. Let S be the partial schedule that we have at the time right before the job a n is assigned. Then, noting that job a n was not assigned within time 2 1=(m 1), we can easily see that t(s i )+p(a n ) 2 1 for 1 6 i 6 m: (1) m 1 We note, then, S m must contain at least one job, since, otherwise, we have t(s m )+p(a n )= p(a n ) 2 1=(m 1) 1 for m 3 which is a clear contradiction. Hence, let a k be the job contained in S m. Then, a k S m implies (a k )=m and, therefore, p(a k ) should be as large as p(a n ), due to LG-LPT order. Hence, t(s m ) p(a k ) p(a n ): (2) By summing up inequalities (1) for i = 1 through m 1, we get m 1 t(s i )+(m 1)p(a n ) 2m 3: i=1 Then by (2) m t(s i )+(m 2)p(a n ) 2m 3: i=1 And, since m i=1 t(s i)+p(a n ) 6 m, we have (m 3)p(a n ) m 3; (3) which can never hold since m 3 and 0 p(a n ) 6 1. Case 2: m=2. Suppose that the theorem is false. Then, again, there must exist a minimal problem instance (J; M) with m = 2, for which LG-LPT yields a schedule with makespan greater than 5. Then, due to minimality, the makespan is determined by the last job a n and certainly (a n )=2.
6 2060 H.-C. Hwang et al. / Computers & Operations Research 31 (200) Suppose that S is the schedule which we have at the time right before the job a n is assigned. Then we have t(s 1 )+p(a n ) 5 and t(s 2 )+p(a n ) 5 : () Hence, t(s 1 )+t(s 2 )+2p(a n ) 5 2. But, since t(s 1)+t(S 2 )+p(a n ) 6 2, p(a n ) 1 2 : Hence, we can have at most two jobs with (a j )=2 and S 1 contains all but only the jobs with (a j ) = 1. On the other hand, S 2 must contain at least one job, since, otherwise, we have t(s 2 )+ p(a n )=p(a n ) 5. Hence, if we let a k be the job contained in S 2, p(a k ) p(a n ) (6) due to LG-LPT order. Now, suppose that S is the optimum schedule. Then, certainly S 1 is a subset of S 1 and there could be only two possibilities. The set S 1 contains either one of a k or a n. However, neither case is possible, due to () and (6). In fact, when m = 2, our problem is reduced to a case of the non-simultaneous parallel machine scheduling problem where the rst machine must process all the jobs with (a j )=1 before processing any other job with (a j ) = 2. In this case, LG-LPT reduced to the algorithm LPT presented in [8] which is proved to have the worst-case performance ratio of m. This equals to 5 when m =2, which indeed conrms our result. We conclude our paper with the following example which shows that the above bound is tight indeed. Example 2. Case 1: m 3. Consider a problem with J and M, where m 3, g(b i )=i for all 1 6 i 6 m. InJ, there are m 2 small jobs a 1 ;:::;a m 2 and m big jobs a+ 1 ;:::;a+ m. Each small job a j has processing time of 1=(m 1) and g(a j )=m 1. Next, we consider the rst m 2 big jobs, a + 1 ;:::;a+ m 2. These jobs have processing time 1 1=(m 1) with g(a+ j )=m 2. Then for the remaining two big jobs, g(a + m 1 )=g(a+ m)=m and p(a + m 1 )=p(a+ m) = 1. For the given jobs we can list them in the LG-LPT order as follows: a + 1 a+ m 2 a 1 a m 2 a + m 1 a+ m: The LG-LPT will assign the rst m 2 big jobs to machines 1 to m 2, respectively. Then all the m 2 small jobs are assigned to machine m 1. Next, a + m 1 is assigned to machine m. At this time, all the machine loads are the same as 1 1=(m 1) except with the load of 1. Finally, the last big job a + m is assigned to machine 1 yielding a schedule with the makespan of 2 1=(m 1) as in Fig. 3. However, we can construct an optimal schedule S as S i = {a + i ;a i } for 1 6 i 6 m 2; S i = {a + i } for m 1 6 i 6 m: Case 2: m = 2. Let M = { ; }, J = {a 1 ;a 2 ;a 3 ;a }, g( )=1, g( )=2, g(a 1 )=1, g(a 2 )= g(a 3 )=g(a )=2, p(a 1 )= 1, p(a 2)= 3 and p(a 3)=p(a )= 1. Then, LG-LPT yields a schedule 2 (5)
7 H.-C. Hwang et al. / Computers & Operations Research 31 (200) a + 1 a + m a + 1 a + 2 a + 2 {a - 1, a - 2,, a m-2} {a - 1, a - 2,, a m-2} -2 a + m-2-2 a + m a + m-1 (a) a + m-1 (b) a + m Fig. 3. (a) LG-LPT and (b) optimal schedule. S such that S 1 = {a 1 ;a 3 ;a } and S 2 = {a 2 }. The makespan of S is 5, but an optimum schedule S with makespan of 1 can be constructed by setting S1 = {a 1;a 2 } and S2 = {a 3;a }. References [1] Pinedo M. Scheduling: theory, algorithms, and systems. Englewood Clis, NJ: Prentice-Hall, [2] Garey MR, Johnson DS. Computers and intractability: a guide to the theory of NP-completeness. New York: Freeman, [3] Coman Jr. EG, Garey MR, Johnson DS. An application of bin-packing to multiprocessor scheduling. SIAM Journal on Computing 1978;7:1 17. [] Graham RL. Bounds on multiprocessor timing anomalies. SIAM Journal on Computing 1969;17: [5] Yue M. On the exact upper bound for the MULTIFIT processor scheduling algorithm. In: Yue M, editor. Operations research in China. Annals of Operations Research 1990; 2: [6] Azar Y, Naor J, Rom R. The competitiveness of on-line assignments. Journal of Algorithms 1995;18: [7] Lenstra JK, Shmoys DB, Tardos E. Approximation algorithms for scheduling unrelated parallel machines. Mathematical Programming 1990;6: [8] Lee CY. Parallel machine scheduling with nonsimultaneous machine available time. Discrete Applied Mathematics 1991;30:53 61.
Duplicating 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 informationAn improved on-line algorithm for scheduling on two unrestrictive parallel batch processing machines
This is the Pre-Published Version. An improved on-line 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 information! Solve problem to optimality. ! Solve problem in poly-time. ! Solve arbitrary instances of the problem. #-approximation algorithm.
Approximation Algorithms 11 Approximation Algorithms Q Suppose I need to solve an NP-hard problem What should I do? A Theory says you're unlikely to find a poly-time algorithm Must sacrifice one of three
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 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 information! Solve problem to optimality. ! Solve problem in poly-time. ! Solve arbitrary instances of the problem. !-approximation algorithm.
Approximation Algorithms Chapter Approximation Algorithms Q Suppose I need to solve an NP-hard problem What should I do? A Theory says you're unlikely to find a poly-time algorithm Must sacrifice one of
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 NP-hard problem. What should I do? A. Theory says you're unlikely to find a poly-time algorithm. Must sacrifice one
More informationJUST-IN-TIME SCHEDULING WITH PERIODIC TIME SLOTS. Received December May 12, 2003; revised February 5, 2004
Scientiae Mathematicae Japonicae Online, Vol. 10, (2004), 431 437 431 JUST-IN-TIME SCHEDULING WITH PERIODIC TIME SLOTS Ondřej Čepeka and Shao Chin Sung b Received December May 12, 2003; revised February
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 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 informationLoad balancing of temporary tasks in the l p norm
Load balancing of temporary tasks in the l p norm Yossi Azar a,1, Amir Epstein a,2, Leah Epstein b,3 a School of Computer Science, Tel Aviv University, Tel Aviv, Israel. b School of Computer Science, The
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 informationARTICLE IN PRESS. European Journal of Operational Research xxx (2004) xxx xxx. Discrete Optimization. Nan Kong, Andrew J.
A factor 1 European Journal of Operational Research xxx (00) xxx xxx Discrete Optimization approximation algorithm for two-stage stochastic matching problems Nan Kong, Andrew J. Schaefer * Department of
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 informationA binary search algorithm for a special case of minimizing the lateness on a single machine
Issue 3, Volume 3, 2009 45 A binary search algorithm for a special case of minimizing the lateness on a single machine Nodari Vakhania Abstract We study the problem of scheduling jobs with release times
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 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 Hing-Fung Ting 2 1 College of Mathematics and Computer Science, Hebei University,
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 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 informationApproximation Algorithms
Approximation Algorithms or: How I Learned to Stop Worrying and Deal with NP-Completeness Ong Jit Sheng, Jonathan (A0073924B) March, 2012 Overview Key Results (I) General techniques: Greedy algorithms
More informationHow To Find An Optimal Search Protocol For An Oblivious Cell
The Conference Call Search Problem in Wireless Networks Leah Epstein 1, and Asaf Levin 2 1 Department of Mathematics, University of Haifa, 31905 Haifa, Israel. lea@math.haifa.ac.il 2 Department of Statistics,
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 informationNan Kong, Andrew J. Schaefer. Department of Industrial Engineering, Univeristy of Pittsburgh, PA 15261, USA
A Factor 1 2 Approximation Algorithm for Two-Stage Stochastic Matching Problems Nan Kong, Andrew J. Schaefer Department of Industrial Engineering, Univeristy of Pittsburgh, PA 15261, USA Abstract We introduce
More informationDetermination of the normalization level of database schemas through equivalence classes of attributes
Computer Science Journal of Moldova, vol.17, no.2(50), 2009 Determination of the normalization level of database schemas through equivalence classes of attributes Cotelea Vitalie Abstract In this paper,
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 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 informationTU e. Advanced Algorithms: experimentation project. The problem: load balancing with bounded look-ahead. Input: integer m 2: number of machines
The problem: load balancing with bounded look-ahead Input: integer m 2: number of machines integer k 0: the look-ahead numbers t 1,..., t n : the job sizes Problem: assign jobs to machines machine to which
More informationOnline Scheduling with Bounded Migration
Online Scheduling with Bounded Migration Peter Sanders, Naveen Sivadasan, and Martin Skutella Max-Planck-Institut für Informatik, Saarbrücken, Germany, {sanders,ns,skutella}@mpi-sb.mpg.de Abstract. Consider
More informationCompletion 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 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@ens-rennes.fr ENS Rennes Academic Year 2013-2014
More informationOn the k-path cover problem for cacti
On the k-path cover problem for cacti Zemin Jin and Xueliang Li Center for Combinatorics and LPMC Nankai University Tianjin 300071, P.R. China zeminjin@eyou.com, x.li@eyou.com Abstract In this paper we
More informationBi-objective approximation scheme for makespan and reliability optimization on uniform parallel machines
Bi-objective approximation scheme for makespan and reliability optimization on uniform parallel machines Emmanuel Jeannot 1, Erik Saule 2, and Denis Trystram 2 1 INRIA-Lorraine : emmanuel.jeannot@loria.fr
More informationFull and Complete Binary Trees
Full and Complete Binary Trees Binary Tree Theorems 1 Here are two important types of binary trees. Note that the definitions, while similar, are logically independent. Definition: a binary tree T is full
More informationApproximability of Two-Machine No-Wait Flowshop Scheduling with Availability Constraints
Approximability of Two-Machine No-Wait Flowshop Scheduling with Availability Constraints T.C. Edwin Cheng 1, and Zhaohui Liu 1,2 1 Department of Management, The Hong Kong Polytechnic University Kowloon,
More informationClustering and scheduling maintenance tasks over time
Clustering and scheduling maintenance tasks over time Per Kreuger 2008-04-29 SICS Technical Report T2008:09 Abstract We report results on a maintenance scheduling problem. The problem consists of allocating
More informationEvery tree contains a large induced subgraph with all degrees odd
Every tree contains a large induced subgraph with all degrees odd A.J. Radcliffe Carnegie Mellon University, Pittsburgh, PA A.D. Scott Department of Pure Mathematics and Mathematical Statistics University
More informationAnalysis of Approximation Algorithms for k-set Cover using Factor-Revealing Linear Programs
Analysis of Approximation Algorithms for k-set Cover using Factor-Revealing Linear Programs Stavros Athanassopoulos, Ioannis Caragiannis, and Christos Kaklamanis Research Academic Computer Technology Institute
More informationChapter 3. Cartesian Products and Relations. 3.1 Cartesian Products
Chapter 3 Cartesian Products and Relations The material in this chapter is the first real encounter with abstraction. Relations are very general thing they are a special type of subset. After introducing
More informationLocal search for the minimum label spanning tree problem with bounded color classes
Available online at www.sciencedirect.com Operations Research Letters 31 (003) 195 01 Operations Research Letters www.elsevier.com/locate/dsw Local search for the minimum label spanning tree problem with
More informationStiffie's On Line Scheduling Algorithm
A class of on-line scheduling algorithms to minimize total completion time X. Lu R.A. Sitters L. Stougie Abstract We consider the problem of scheduling jobs on-line on a single machine and on identical
More informationNP-Completeness and Cook s Theorem
NP-Completeness and Cook s Theorem Lecture notes for COM3412 Logic and Computation 15th January 2002 1 NP decision problems The decision problem D L for a formal language L Σ is the computational task:
More information3. Mathematical Induction
3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)
More informationSo let us begin our quest to find the holy grail of real analysis.
1 Section 5.2 The Complete Ordered Field: Purpose of Section We present an axiomatic description of the real numbers as a complete ordered field. The axioms which describe the arithmetic of the real numbers
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 informationCloud Storage and Online Bin Packing
Cloud Storage and Online Bin Packing Doina Bein, Wolfgang Bein, and Swathi Venigella Abstract We study the problem of allocating memory of servers in a data center based on online requests for storage.
More informationPh.D. Thesis. Judit Nagy-György. Supervisor: Péter Hajnal Associate Professor
Online algorithms for combinatorial problems Ph.D. Thesis by Judit Nagy-György Supervisor: Péter Hajnal Associate Professor Doctoral School in Mathematics and Computer Science University of Szeged Bolyai
More informationA Note on Maximum Independent Sets in Rectangle Intersection Graphs
A Note on Maximum Independent Sets in Rectangle Intersection Graphs Timothy M. Chan School of Computer Science University of Waterloo Waterloo, Ontario N2L 3G1, Canada tmchan@uwaterloo.ca September 12,
More informationWhy? A central concept in Computer Science. Algorithms are ubiquitous.
Analysis of Algorithms: A Brief Introduction Why? A central concept in Computer Science. Algorithms are ubiquitous. Using the Internet (sending email, transferring files, use of search engines, online
More informationComplexity Theory. IE 661: Scheduling Theory Fall 2003 Satyaki Ghosh Dastidar
Complexity Theory IE 661: Scheduling Theory Fall 2003 Satyaki Ghosh Dastidar Outline Goals Computation of Problems Concepts and Definitions Complexity Classes and Problems Polynomial Time Reductions Examples
More informationClass constrained bin covering
Class constrained bin covering Leah Epstein Csanád Imreh Asaf Levin Abstract We study the following variant of the bin covering problem. We are given a set of unit sized items, where each item has a color
More informationPartitioned real-time scheduling on heterogeneous shared-memory multiprocessors
Partitioned real-time scheduling on heterogeneous shared-memory multiprocessors Martin Niemeier École Polytechnique Fédérale de Lausanne Discrete Optimization Group Lausanne, Switzerland martin.niemeier@epfl.ch
More informationImproved Algorithms for Data Migration
Improved Algorithms for Data Migration Samir Khuller 1, Yoo-Ah Kim, and Azarakhsh Malekian 1 Department of Computer Science, University of Maryland, College Park, MD 20742. Research supported by NSF Award
More information8 Divisibility and prime numbers
8 Divisibility and prime numbers 8.1 Divisibility In this short section we extend the concept of a multiple from the natural numbers to the integers. We also summarize several other terms that express
More informationVirtual Machine Allocation in Cloud Computing for Minimizing Total Execution Time on Each Machine
Virtual Machine Allocation in Cloud Computing for Minimizing Total Execution Time on Each Machine Quyet Thang NGUYEN Nguyen QUANG-HUNG Nguyen HUYNH TUONG Van Hoai TRAN Nam THOAI Faculty of Computer Science
More information20 Selfish Load Balancing
20 Selfish Load Balancing Berthold Vöcking Abstract Suppose that a set of weighted tasks shall be assigned to a set of machines with possibly different speeds such that the load is distributed evenly among
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 information5 INTEGER LINEAR PROGRAMMING (ILP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1
5 INTEGER LINEAR PROGRAMMING (ILP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1 General Integer Linear Program: (ILP) min c T x Ax b x 0 integer Assumption: A, b integer The integrality condition
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 informationA Study of Crossover Operators for Genetic Algorithm and Proposal of a New Crossover Operator to Solve Open Shop Scheduling Problem
American Journal of Industrial and Business Management, 2016, 6, 774-789 Published Online June 2016 in SciRes. http://www.scirp.org/journal/ajibm http://dx.doi.org/10.4236/ajibm.2016.66071 A Study of Crossover
More informationOn closed-form solutions of a resource allocation problem in parallel funding of R&D projects
Operations Research Letters 27 (2000) 229 234 www.elsevier.com/locate/dsw On closed-form solutions of a resource allocation problem in parallel funding of R&D proects Ulku Gurler, Mustafa. C. Pnar, Mohamed
More information6.2 Permutations continued
6.2 Permutations continued Theorem A permutation on a finite set A is either a cycle or can be expressed as a product (composition of disjoint cycles. Proof is by (strong induction on the number, r, of
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 informationInteger factorization is in P
Integer factorization is in P Yuly Shipilevsky Toronto, Ontario, Canada E-mail address: yulysh2000@yahoo.ca Abstract A polynomial-time algorithm for integer factorization, wherein integer factorization
More informationON THE COMPLEXITY OF THE GAME OF SET. {kamalika,pbg,dratajcz,hoeteck}@cs.berkeley.edu
ON THE COMPLEXITY OF THE GAME OF SET KAMALIKA CHAUDHURI, BRIGHTEN GODFREY, DAVID RATAJCZAK, AND HOETECK WEE {kamalika,pbg,dratajcz,hoeteck}@cs.berkeley.edu ABSTRACT. Set R is a card game played with a
More information1 if 1 x 0 1 if 0 x 1
Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or
More informationHYBRID GENETIC ALGORITHMS FOR SCHEDULING ADVERTISEMENTS ON A WEB PAGE
HYBRID GENETIC ALGORITHMS FOR SCHEDULING ADVERTISEMENTS ON A WEB PAGE Subodha Kumar University of Washington subodha@u.washington.edu Varghese S. Jacob University of Texas at Dallas vjacob@utdallas.edu
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES Contents 1. Random variables and measurable functions 2. Cumulative distribution functions 3. Discrete
More informationBounded Cost Algorithms for Multivalued Consensus Using Binary Consensus Instances
Bounded Cost Algorithms for Multivalued Consensus Using Binary Consensus Instances Jialin Zhang Tsinghua University zhanggl02@mails.tsinghua.edu.cn Wei Chen Microsoft Research Asia weic@microsoft.com Abstract
More information8 Primes and Modular Arithmetic
8 Primes and Modular Arithmetic 8.1 Primes and Factors Over two millennia ago already, people all over the world were considering the properties of numbers. One of the simplest concepts is prime numbers.
More informationResearch Article Batch Scheduling on Two-Machine Flowshop with Machine-Dependent 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 Two-Machine Flowshop with Machine-Dependent
More informationScheduling Single Machine Scheduling. Tim Nieberg
Scheduling Single Machine Scheduling Tim Nieberg Single machine models Observation: for non-preemptive problems and regular objectives, a sequence in which the jobs are processed is sufficient to describe
More information1 Approximating Set Cover
CS 05: Algorithms (Grad) Feb 2-24, 2005 Approximating Set Cover. Definition An Instance (X, F ) of the set-covering problem consists of a finite set X and a family F of subset of X, such that every elemennt
More informationAlgorithm Design for Performance Aware VM Consolidation
Algorithm Design for Performance Aware VM Consolidation Alan Roytman University of California, Los Angeles Sriram Govindan Microsoft Corporation Jie Liu Microsoft Research Aman Kansal Microsoft Research
More informationDimensioning an inbound call center using constraint programming
Dimensioning an inbound call center using constraint programming Cyril Canon 1,2, Jean-Charles Billaut 2, and Jean-Louis Bouquard 2 1 Vitalicom, 643 avenue du grain d or, 41350 Vineuil, France ccanon@fr.snt.com
More informationOptimal Online-list Batch Scheduling
Optimal Online-list Batch Scheduling Jacob Jan Paulus a,, Deshi Ye b, Guochuan Zhang b a University of Twente, P.O. box 217, 7500AE Enschede, The Netherlands b Zhejiang University, Hangzhou 310027, China
More informationMinimizing the Number of Machines in a Unit-Time Scheduling Problem
Minimizing the Number of Machines in a Unit-Time Scheduling Problem Svetlana A. Kravchenko 1 United Institute of Informatics Problems, Surganova St. 6, 220012 Minsk, Belarus kravch@newman.bas-net.by Frank
More informationNotes on Complexity Theory Last updated: August, 2011. Lecture 1
Notes on Complexity Theory Last updated: August, 2011 Jonathan Katz Lecture 1 1 Turing Machines I assume that most students have encountered Turing machines before. (Students who have not may want to look
More informationQuotient Rings and Field Extensions
Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F.
More informationRobust Geometric Programming is co-np hard
Robust Geometric Programming is co-np hard André Chassein and Marc Goerigk Fachbereich Mathematik, Technische Universität Kaiserslautern, Germany Abstract Geometric Programming is a useful tool with a
More informationNear Optimal Solutions
Near Optimal Solutions Many important optimization problems are lacking efficient solutions. NP-Complete problems unlikely to have polynomial time solutions. Good heuristics important for such problems.
More informationOffline sorting buffers on Line
Offline sorting buffers on Line Rohit Khandekar 1 and Vinayaka Pandit 2 1 University of Waterloo, ON, Canada. email: rkhandekar@gmail.com 2 IBM India Research Lab, New Delhi. email: pvinayak@in.ibm.com
More informationModeling and Performance Evaluation of Computer Systems Security Operation 1
Modeling and Performance Evaluation of Computer Systems Security Operation 1 D. Guster 2 St.Cloud State University 3 N.K. Krivulin 4 St.Petersburg State University 5 Abstract A model of computer system
More informationOn the Relationship between Classes P and NP
Journal of Computer Science 8 (7): 1036-1040, 2012 ISSN 1549-3636 2012 Science Publications On the Relationship between Classes P and NP Anatoly D. Plotnikov Department of Computer Systems and Networks,
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 informationThe Student-Project Allocation Problem
The Student-Project Allocation Problem David J. Abraham, Robert W. Irving, and David F. Manlove Department of Computing Science, University of Glasgow, Glasgow G12 8QQ, UK Email: {dabraham,rwi,davidm}@dcs.gla.ac.uk.
More information1 The Line vs Point Test
6.875 PCP and Hardness of Approximation MIT, Fall 2010 Lecture 5: Low Degree Testing Lecturer: Dana Moshkovitz Scribe: Gregory Minton and Dana Moshkovitz Having seen a probabilistic verifier for linearity
More informationJoint Optimization of Overlapping Phases in MapReduce
Joint Optimization of Overlapping Phases in MapReduce Minghong Lin, Li Zhang, Adam Wierman, Jian Tan Abstract MapReduce is a scalable parallel computing framework for big data processing. It exhibits multiple
More informationModern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh
Modern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh Peter Richtárik Week 3 Randomized Coordinate Descent With Arbitrary Sampling January 27, 2016 1 / 30 The Problem
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 informationDynamic TCP Acknowledgement: Penalizing Long Delays
Dynamic TCP Acknowledgement: Penalizing Long Delays Karousatou Christina Network Algorithms June 8, 2010 Karousatou Christina (Network Algorithms) Dynamic TCP Acknowledgement June 8, 2010 1 / 63 Layout
More informationProperties of Stabilizing Computations
Theory and Applications of Mathematics & Computer Science 5 (1) (2015) 71 93 Properties of Stabilizing Computations Mark Burgin a a University of California, Los Angeles 405 Hilgard Ave. Los Angeles, CA
More information14.1 Rent-or-buy problem
CS787: Advanced Algorithms Lecture 14: Online algorithms We now shift focus to a different kind of algorithmic problem where we need to perform some optimization without knowing the input in advance. Algorithms
More informationThe Binomial Distribution
The Binomial Distribution James H. Steiger November 10, 00 1 Topics for this Module 1. The Binomial Process. The Binomial Random Variable. The Binomial Distribution (a) Computing the Binomial pdf (b) Computing
More informationWHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT?
WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? introduction Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certainly
More informationHigh degree graphs contain large-star factors
High degree graphs contain large-star factors Dedicated to László Lovász, for his 60th birthday Noga Alon Nicholas Wormald Abstract We show that any finite simple graph with minimum degree d contains a
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 informationLarge induced subgraphs with all degrees odd
Large induced subgraphs with all degrees odd A.D. Scott Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, England Abstract: We prove that every connected graph of order
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 informationLoad Balancing and Switch Scheduling
EE384Y Project Final Report Load Balancing and Switch Scheduling Xiangheng Liu Department of Electrical Engineering Stanford University, Stanford CA 94305 Email: liuxh@systems.stanford.edu Abstract Load
More informationPRIME FACTORS OF CONSECUTIVE INTEGERS
PRIME FACTORS OF CONSECUTIVE INTEGERS MARK BAUER AND MICHAEL A. BENNETT Abstract. This note contains a new algorithm for computing a function f(k) introduced by Erdős to measure the minimal gap size in
More information