Approximation Algorithms for Sequencing Problems

184 Pages·2009·1.12 MB·English

Approximation Algorithms for Sequencing Problems
Viswanath Nagarajan
March 2009
Tepper School of Business
Carnegie Mellon University
Pittsburgh, PA 15213
Thesis Committee:
R. Ravi (Chair)
Gerard Cornuejols
Anupam Gupta
Michael Trick
Submitted in partial fulfillment of the requirements for the degree of
Doctor of Philosophy in Algorithms, Combinatorics and Optimization.

Abstract
This thesis presents approximation algorithms for some sequencing
problems, with an emphasis on vehicle routing. Vehicle Routing Prob-
lems (VRPs) form a rich class of variants of the basic Traveling Salesman
Problem, thatarealsopracticallymotivated. TheVRPsconsideredinthis
thesisincludesingleandmultiplevehicleDialaRide, VRPwithStochastic
Demands, Directed Orienteering and Directed Minimum Latency. Other
sequencing problems studied in this thesis are Permutation Flowshop
Scheduling and Maximum Quadratic Assignment.

Acknowledgements
I begin by thanking my advisor R. Ravi, who has been a great mentor, colleague
and friend. I am particularly grateful to him for the patience he showed early in my
graduate studies, and for lessons in good research style that I have learnt from him.
I also thank Anupam Gupta, Nikhil Bansal and Maxim Sviridenko, with whom
it has been a pleasure collaborating. The downs of failed attempts and the ups of
discovering proofs were exciting indeed.
I feel fortunate to have been at CMU along with many other graduate students
with similar research interests. I thank Dan Golovin, Vineet Goyal, and Mohit Singh
for numerous discussions on research problems (and troubles). I also thank the
excellent faculty members to whom I could turn to for advice: Egon Balas, Gerard
Cornuejols, Alan Frieze, Francois Margot, and Michael Trick.
I would like to thank the Algorithms group at IBM T.J. Watson Research for sum-
mer internships in 2007 and 2008, which was a productive and learning experience
for me. I am also grateful to the financial support I received during my Ph.D. from
the William Larimer Mellon Fellowship, the Aladdin grant, R. Ravi’s NSF grants
CCR-0430751 and CCF-0728841, and an IBM graduate fellowship.
I also thank my other collaborators not mentioned above, Barbara Anthony,
Deeparnab Chakrabarty, Inge Li Gørtz, MohammadTaghi Hajiaghayi, Rohit Khan-
dekar, Jochen Ko¨nemann, Ravishankar Krishnaswamy, Jon Lee, Aranyak Mehta,
Vahab Mirrokni, Britta Peis, Abhiram Ranade, and Vijay Vazirani.
My stay in Pittsburgh has been memorable thanks to my friends Amitabh, John,
both Mohits, Prasad, Varun, and Vineet. These years would have seemed a lot
longer without them. I also thank my relatives and friends in the US whom I visited
several times during holidays.
I thank my parents and sister for their endless support throughout my Ph.D. I am
also thankful to my parents for instigating in me the value of knowledge, without
which my research motivation would have been incomplete.
v

This thesis is dedicated to my parents.
vii

Contents
1 Introduction 1
1.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Thesis Contribution and Results . . . . . . . . . . . . . . . . . . . . . 3
1.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Single vehicle Dial-a-Ride 9
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1 The k-Forest Problem . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.2 The Dial-a-Ride Problem . . . . . . . . . . . . . . . . . . . . . 11
2.1.3 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 Algorithms for the k-forest problem . . . . . . . . . . . . . . . . . . . 14
2.2.1 An O(√k) approximation algorithm . . . . . . . . . . . . . . 14
2.2.2 An O(√n) approximation algorithm . . . . . . . . . . . . . . 16
2.2.3 Approximation algorithm for k-forest . . . . . . . . . . . . . . 17
2.3 Applications to Dial-a-Ride problems . . . . . . . . . . . . . . . . . . 18
2.3.1 Classical Dial-a-Ride . . . . . . . . . . . . . . . . . . . . . . . 20
2.3.2 Non-uniform Dial-a-Ride . . . . . . . . . . . . . . . . . . . . . 21
2.3.3 Weighted Dial-a-Ride . . . . . . . . . . . . . . . . . . . . . . . 23
2.4 The Effect of Preemptions . . . . . . . . . . . . . . . . . . . . . . . . 27
3 Multi vehicle Dial-a-Ride 33
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x CONTENTS
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1.1 Problem Definition and Preliminaries . . . . . . . . . . . . . . 34
3.1.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.1.3 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2 Uncapacitated Preemptive mDaR . . . . . . . . . . . . . . . . . . . . 37
3.2.1 Reduction to depot-demand instances . . . . . . . . . . . . . 38
3.2.2 Algorithm for depot-demand instances . . . . . . . . . . . . . 39
3.2.3 Tight example for uncapacitated mDaR lower bounds. . . . . 41
3.2.4 Improved guarantee for metrics excluding a fixed minor. . . . 42
3.3 Preemptive multi-vehicle Dial-a-Ride . . . . . . . . . . . . . . . . . . 43
3.3.1 Capacitated Vehicle Routing with Bounded Delay . . . . . . . 44
3.3.2 Algorithm for preemptive mDaR . . . . . . . . . . . . . . . . . 46
3.3.3 Weighted preemptive mDaR . . . . . . . . . . . . . . . . . . . 51
4 Stochastic Demands Vehicle Routing 55
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.1.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.1.2 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.2 SVRP under Independent Demands . . . . . . . . . . . . . . . . . . . 59
4.3 SVRP under Explicit Demands . . . . . . . . . . . . . . . . . . . . . . 65
4.3.1 Two Auxiliary Problems . . . . . . . . . . . . . . . . . . . . . 67
4.3.2 Algorithm for the Metric Isolation Problem . . . . . . . . . . . 73
4.3.3 Optimal Split Tree Problem . . . . . . . . . . . . . . . . . . . 79
4.3.4 Issue of Observing Demands . . . . . . . . . . . . . . . . . . . 81
4.3.5 Hardness of Approximation . . . . . . . . . . . . . . . . . . . 82
4.4 SVRP under Black-box Distribution . . . . . . . . . . . . . . . . . . . 87
5 VRPs on Asymmetric Metrics 93
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93