approximation algorithms for scheduling problems - The Stanford

145 Pages·1998·0.82 MB·English

APPROXIMATION ALGORITHMS FOR SCHEDULING
PROBLEMS
a dissertation
submitted to the department of computer science
and the committee on graduate studies
of stanford university
in partial fulfillment of the requirements
for the degree of
doctor of philosophy
By
Chandra Chekuri
August 1998
(cid:13)c Copyright 1998 by Chandra Chekuri
All Rights Reserved
ii
I certify that I have read this dissertation and that in my
opinion it is fully adequate, in scope and quality, as a disser-
tation for the degree of Doctor of Philosophy.
Rajeev Motwani
(Principal Adviser)
I certify that I have read this dissertation and that in my
opinion it is fully adequate, in scope and quality, as a disser-
tation for the degree of Doctor of Philosophy.
Serge Plotkin
I certify that I have read this dissertation and that in my
opinion it is fully adequate, in scope and quality, as a disser-
tation for the degree of Doctor of Philosophy.
Cli(cid:11)ord Stein
ApprovedfortheUniversityCommitteeonGraduateStudies:
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Abstract
This thesis describes e(cid:14)cient approximation algorithms for some NP-Hard deterministic
machine scheduling and related problems. An approximation algorithm for an NP-Hard
optimization problem is a polynomial time algorithm which, given any instance of the
problem,returnsasolutionwhosevalueiswithinsomeguaranteedmultiplicativefactor(cid:11)of
theoptimalsolutionvalueforthatinstance. Thequantity(cid:11)iscalledtheapproximationratio
ofthealgorithm. Atypicalprobleminmachineschedulingconsistsofasetofjobsthatareto
be executed in either preemptive or non-preemptive fashion on a set of available machines
subject to a variety of constraints. Two common objectives are minimizing makespan
(the time to complete all jobs) and minimizing average completion time. Constraints that
we study include precedence constraints between jobs and release dates on jobs. Brief
descriptions of the problems we study and highlights of our results follow.
We study single machine and parallel machine scheduling problems with the objective
of minimizing average completion time and its weighted generalization. We introduce new
techniquesthateitherimproveearlierresultsand/orresultinsimpleande(cid:14)cientalgorithms.
e
For thesinglemachineproblemwithjobshavingrelease dates onlywe obtainan e(cid:0)1 1:58
’
approximation. For the parallel machine case we obtain a 2:85 approximation. We then
focus on the case when jobs have precedence constraints. For the single machine problem
we obtain an 2-approximation algorithmthat in contrast to earlieralgorithms does not rely
on linear programming relaxations. We also give a general algorithm that converts an (cid:11)-
approximatesinglemachinescheduleintoa(2(cid:11)+2)-approximate parallelmachineschedule.
The conversion algorithm is simple and yields e(cid:14)cient and combinatorial constant factor
algorithms for several variants.
Wethenconsidertheproblemofminimizingmakespanonmachineswithdi(cid:11)erentspeeds
when jobs have precedence constraints. We obtain an O(logm) approximation (m is the
3
numberofmachines) inO(n )time. Ourapproximationratio matches thebestknownratio
v
up to constant factors. However, our algorithm is e(cid:14)cient and easy to implement and is
based on a natural heuristic.
We introduce a new class of scheduling problems that arise from query optimization in
paralleldatabases. The novel aspect isinmodelingcommunicationcosts between operators
in a task system that represents a query execution plan. We address one of the problems
that we introduce, namely, the pipelined operator tree problem. An instance of the prob-
lem consists of a tree with node weights that represent processing times of the associated
operators, and edge weights that represent communication costs. Scheduling two nodes
connected by an edge on di(cid:11)erent processors adds communication cost equal to the edge
weight to both the nodes. The objective is to schedule the nodes on parallel processors to
minimize makespan. This is a generalization of the well known multi-processor scheduling
problem. We obtain a polynomial time approximation scheme for this problem. We also
obtain e(cid:14)cient O(nlogn) time algorithms that have ratios of 3:56 and 2:58 respectively.
Finally we study multi-dimensional generalizations of three well known problems in
combinatorial optimization: multi-processor scheduling, bin packing, and knapsack. We
study versions of these problems when items are multi-dimensional vectors instead of real
numbers. We obtain several approximability and inapproximability results. For the vector
scheduling problem we obtain a PTAS when d, the dimensionof the vectors, is (cid:12)xed. For d
2
arbitraryweobtainaO(min logdm;log d ) approximationandalsoshowthatno constant
f g
factor approximation is possible unless NP=ZPP. For the vector bin packing problem we
obtain a (1+(cid:15)d+O(ln(1=(cid:15)))) approximation for all d and show APX-Hardness even for
d =2. The vector version of knapsack captures a broad class of hard integer programming
problemscallingpackingintegerprograms. Theonlygeneraltechniqueknowntosolvethese
problems is randomized rounding. We show that results obtained by randomized rounding
are the best possible for a large class of these problems.
Practical applications motivating some of our problems include instruction scheduling
in compilers and query optimization in parallel databases.
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Acknowledgements
This thesis and the growth in my knowledge and maturity over the last few years owe a
greatdealtomanyteachers, colleagues, andfriends. FirstamongthemismyadviserRajeev
Motwani. He accepted me as his student at a time when I was unsure of my interests and
capabilities, and gave me direction by suggesting concrete interesting problems. The many
hours he spent with me in my (cid:12)rst few years here, when I needed his time most, have
contributed much to my scholarship. I express my sincere thanks to him for his support,
guidance, and wisdom. His persistenceintackling problems, con(cid:12)dence, and great teaching
will always be an inspiration.
Cli(cid:11)Stein’s sabbaticalat Stanfordhas beeninstrumentalinshapingthisthesis. Collab-
oration with him on parts of Chapter 2 gave me impetus to (cid:12)nish my thesis on scheduling.
His calm and diligent approach to research and its responsibilities are worth emulating. I
thank him for carefully reading my thesis and for coming to my defense all the way from
Dartmouth. He has been like a second adviser to me and I am grateful for his kindness.
Serge Plotkin’s classes have taught me many new ideas and approaches for which I am
most thankful. In several ways he has made the the theory community at Stanford a nicer
place. Thanks to him also for being on my reading committee.
My co-authors, colleagues, and many others from the community have given me their
valuabletimeandinsights,andIamgratefultoallofthem. Inadditionto RajeevandCli(cid:11),
I would especially like to thank Sanjeev Khanna and Moses Charikar for their in(cid:13)uence on
my work. Sanjeev has been a mentor to me since my (cid:12)rst year. His encouragement and
constant advice about working on hard and meaningful problems have guided me through
out. I am extremely happy to have a chapter in this thesis that is joint work with him.
I could always count on sound and intelligent advice from Moses, and his clear thinking
helpeduntangle manyof myideas. Thoughno papers writtenjointlywithhim(andothers)
arepartofthisthesis,theyhavehadasubstantialin(cid:13)uenceinopeningnewareasofresearch
vii
to me.
I had (cid:12)ve wonderful years at Stanford during which time I have not only matured
academically but also as an individual. I have been lucky to have many beautiful and
cherished friendships, and it is with more than a tinge of sadness that I leave this place. I
would like to thank Harish in particular for adding a great deal to my life. He has lifted
my spiritsmany a time and memories of our trip to South America willalways be with me.
Julien and Sanjeev with their worldly advice have been more than generous friends. The
longhoursspentinmyo(cid:14)cewouldnothavebeenpossiblebutforthecompanyofwonderful
studentsaroundme. Iwillmissthemanyenjoyablehoursspenttalkingaboutsundrytopics
to Ashish, Sudipto, Piotr, Shiva, and Suresh, Craig’s sense of humour, Donald’s generosity,
Je(cid:11)rey’s personality, lunch with Scott, co(cid:11)ee with Julien, Harish’s cookies, tennis with
Ashish, Shiva, and Suresh (all credit for my improved game goes to them), theory lunch,
and many such seemingly small but important things that will be remembered for a long
time. My sincere thanks also to Eric, Luca, Michael, Nikolai, Oliver, Ram, and Tomas
for their company. Many others are not mentioned here, but know my a(cid:11)ection for them.
Stanford University and the Computer Science Department have provided an incredible
academic and cultural environment. I am thankful to both institutions and not least the
fabulous weather of the bay area.
Finally, I thank my parents and my brother for their unconditional love and support
through ups and downs. Their faith and con(cid:12)dence in my abilities, though sometimes
irrational, have kept me going. I dedicate this thesis to my brother for the many happy
times we shared.
Chandra Chekuri
Stanford, California
August 1998
I gratefully acknowledge funding support from a Stanford University School of Engineering
Fellowship, an IBM Cooperative Fellowship, and from NSF underthe Award CCR-9357849
to Rajeev Motwani with matching support from IBM, Mitsubishi, Schlumberger Founda-
tion, Shell Foundation, and Xerox Corporation.
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Contents
Abstract v
Acknowledgements vii
1 Introduction 1
1.1 Background and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.1 Scheduling Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.2 Approximation Algorithms and Inapproximability . . . . . . . . . . 6
1.2 Organization and Overview of Contributions . . . . . . . . . . . . . . . . . 10
1.2.1 Scheduling to Minimize Average Completion Time . . . . . . . . . . 10
1.2.2 Minimizing Makespan on Machines with Di(cid:11)erent Speeds . . . . . . 12
1.2.3 Scheduling Problems in Parallel Query Optimization . . . . . . . . . 12
1.2.4 Approximability of Vector Packing Problems . . . . . . . . . . . . . 13
2 Minimizing Average (Weighted) Completion Time 15
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Scheduling on a Single Machine . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.1 One-Machine Scheduling with Release Dates . . . . . . . . . . . . . 21
2.2.2 One-Machine Scheduling with Precedence Constraints . . . . . . . . 28
2.3 Scheduling on Parallel Machines . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3.1 Parallel Machine Scheduling with Release Dates . . . . . . . . . . . 36
2.3.2 Precedence Constraints and a Generic Conversion Algorithm . . . . 38
2.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3 Makespan on Machines with Di(cid:11)erent Speeds 49
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
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3.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.3 A New Lower Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.4 The Approximation Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.4.1 Release Dates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.4.2 Scheduling Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4 Scheduling Problems in Parallel Query Optimization 63
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2 A Model for Scheduling Problems . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2.1 Forms of Parallelism . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.2.2 Operator Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.2.3 Model of Communication . . . . . . . . . . . . . . . . . . . . . . . . 68
4.2.4 Scheduling Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.3 POT Scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.3.1 Problem De(cid:12)nition and Prior Results . . . . . . . . . . . . . . . . . 70
4.3.2 A Two-stage Approach . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.4 The LocalCuts Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.5 The BoundedCuts Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.6 An Approximation Scheme for POT Scheduling . . . . . . . . . . . . . . . . 84
4.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5 Approximability of Vector Packing Problems 91
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.1.1 Problem De(cid:12)nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.1.2 Related Work and Our Results . . . . . . . . . . . . . . . . . . . . . 94
5.1.3 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.2 Approximation Algorithms for Vector Scheduling . . . . . . . . . . . . . . . 96
5.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.2.2 A PTAS for (cid:12)xed d . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.2.3 The General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.3 Vector Bin Packing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.4 Inapproximability Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.4.1 Packing Integer Programs . . . . . . . . . . . . . . . . . . . . . . . . 108
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