exact and approximation algorithms for network flow and disjoint-path problems

146 Pages·2015·0.74 MB·English

EXACT AND APPROXIMATION ALGORITHMS
FOR NETWORK FLOW AND DISJOINT-PATH
PROBLEMS
A Thesis
Submitted to the Faculty
in partial fulfillment of the requirements for the
degree of
Doctor of Philosophy
in
Computer Science
by
Stavros G. Kolliopoulos
DARTMOUTH COLLEGE
Hanover, New Hampshire
August 31, 1998
Examining Committee:
(chair) Clifford Stein
Prasad Jayanti
Fillia Makedon
Aravind Srinivasan
Neal E. Young
Roger D. Sloboda
Dean of Graduate Studies
Copyright by
Stavros G. Kolliopoulos
1998
ii
Abstract
Network flow problems form a core area of Combinatorial Optimization. Their signif-
icance arises both from their very large number of applications and their theoretical
importance. This thesis focuses on efficient exact algorithms for network flow prob-
lems in P and on approximation algorithms for NP-hard variants such as disjoint
paths and unsplittable flow.
Given an n-vertex, m-edge directed network G with real costs on the edges we
give new algorithms to compute single-source shortest paths and the minimum mean
cycle. Our algorithm is deterministic with O(n2logn) expected running time over
a large class of input distributions. This is the first strongly polynomial algorithm
in over 35 years to improve upon some aspect of the O(nm) running time of the
Bellman-Ford shortest-path algorithm.
In the single-source unsplittable flow problem, we are given a network G, a source
vertex s and k commodities with sinks t and real-valued demands ρ , 1 i k.
i i
≤ ≤
We seek to route the demand ρ of each commodity i along a single s-t flow path,
i i
so that the total flow routed across any edge e is bounded by the edge capacity u .
e
This NP-hard problem combines the difficulty of bin-packing with routing through
an arbitrary graph and has many interesting and important variations. We give
a generic framework, which yields approximation algorithms that are simpler than
iii
the previously known and achieve significant improvements upon the approximation
ratios.
Inapacking integer program,weseekavectorxofintegers,whichmaximizescT x,
·
subject to Ax b, A,b,c 0. The edge and vertex-disjoint path problems together
≤ ≥
with their multiple-source unsplittable flow generalization are NP-hard problems with
a multitude of applications in areas such as routing, scheduling and bin packing.
We explore the topic of approximating disjoint-path problems using polynomial-size
packing integer programs. Motivated by the disjoint paths applications, we initiate
the study of a class of packing integer programs, called column-restricted. We derive
approximation algorithms for column-restricted packing integer programs that we
believe are of independent interest.
iv
Acknowledgements
Many thanks to my advisor Cliff Stein, for his wit, support and advice on technical
and non-technical matters. I learned a lot from Cliff. I have greatly benefited from
his perspectiveon Combinatorial Optimization,whichI havecome to share, including
my long-denied affinity for scheduling problems. Cliff has been more than generous
with his time during these last four years and the intellectual climate of dozens of
meetings will be fondly remembered.
Thanks to Prasad Jayanti for his out-of-this-world teaching and for being a calm,
resolute friend. I am grateful to Fillia Makedon for her support and for accepting to
serve on my Thesis Committee in addition to all her other responsibilities. Aravind
Srinivasan provided ample technical advice and shared ideas with characteristic gen-
erosity. I am grateful also to Aravind for serving on my Committee while based in
Singapore. Thanks to NealYoung for reading mythesis and for lengthy, enlightening,
discussions on various algorithmictopics. Neal taught me more things on randomized
rounding than I ever thought I needed to know. Thanks also to Chandra Chekuri for
his late-night remarks on packing integer programs.
I rememberDonald Johnson, myfirst mentor at Dartmouth. He started me work-
ing on shortest paths and guided patiently the motivated but not very knowledgable
first-year student that I was. I want to believe he would have approved of my later
career.
Thanks to Wayne Cripps, our system manager, for his timely and frequent assis-
tance. In my opinion though, he could have given me another Terabyte of disk space.
Thanks to Charles Owen for technical assistance with document preparation.
Dimitris Sofotassios and Thanassis Tsakalidis first got me interested in research
when I was an undergraduate in the University of Patras. I thank them both.
v
I have had the good fortune to have a number of excellent teachers over the
years, teaching, in addition to subject matter, ethos, in the original sense of the
word. I thank them all and I would like to mention at least one more name, Dimitri
Paraskevopoulos. It is definitelythe wrong place to make such a statement, but I feel
I have not learned much more Pure Math since 1988.
Myfriends,fromallaround theglobe, havealwaysbeen closewhen Ineededthem.
Their support has been indispensable. I think they know who they are.
Myparents,Georgios and IoustiniKolliopoulos, instilledto metheloveoflearning
and, among other things, the amount of determinationneededto makeit through ups
and downs. To pronounce the word sea clearly that all the dolphins within it might
shine And the desolation so great it might contain all of God And every waterdrop
ascending steadfastly toward the sun. My elder brother, Alexandros, has never failed
to provide advice, an example, and help. And the small cool domes like blue medusae
Reachedeach time higher to the silverwork The wind so delicately worked As apainting
For other times more distant For other times more distant.
I would like to acknowledge partial support by NSF Award CCR-9308701 and NSF
Career Award CCR-9624828. Uncredited in-text verse by Odysseas Elytis.
vi
Contents
Abstract iii
List of Tables xi
List of Illustrations xiv
1 Introduction 1
2 An Algorithm for Real-Valued Shortest Paths with Fast Average
Case 7
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 The Shortest-Path Algorithm . . . . . . . . . . . . . . . . . . . . . . 13
2.2.1 Skeleton of the new algorithm . . . . . . . . . . . . . . . . . . 13
2.2.2 An implementation with fast average case . . . . . . . . . . . 18
2.3 A Lower Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4 A Fast Algorithm for the Minimum Mean Cycle Problem . . . . . . . 25
2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3 Approximation Algorithms for Single-Source Unsplittable Flow 29
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
vii
3.3 The approximation algorithm for minimum congestion . . . . . . . . 39
3.4 A 3-approximation algorithm for congestion . . . . . . . . . . . . . . 44
3.5 Minimizing congestion with arbitrary capacities . . . . . . . . . . . . 52
3.6 Minimum-cost unsplittable flow . . . . . . . . . . . . . . . . . . . . . 53
3.7 Maximizing the routable demand . . . . . . . . . . . . . . . . . . . . 57
3.8 Minimizing the number of rounds . . . . . . . . . . . . . . . . . . . . 63
3.9 A hardness result for unsplittable flow . . . . . . . . . . . . . . . . . 67
3.10 Restricted sets of demands and applications to scheduling . . . . . . . 70
4 Approximating Disjoint-Path Problems Using Packing Integer Pro-
grams 75
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.1.1 Packing Integer Programs . . . . . . . . . . . . . . . . . . . . 81
4.1.2 Applications of packing to approximation . . . . . . . . . . . . 84
4.1.3 A Greedy Algorithm . . . . . . . . . . . . . . . . . . . . . . . 87
4.2 Approximating a column-restricted PIP . . . . . . . . . . . . . . . . . 89
4.2.1 Notation and basic facts . . . . . . . . . . . . . . . . . . . . . 89
4.2.2 The approximation algorithm . . . . . . . . . . . . . . . . . . 92
4.2.3 Improved approximations for small column values . . . . . . . 98
4.3 Applications of PIP’s to approximation . . . . . . . . . . . . . . . . . 101
4.3.1 Weighted multiple-source unsplittable flow . . . . . . . . . . . 101
4.3.2 Weighted vertex-disjoint paths . . . . . . . . . . . . . . . . . . 106
4.3.3 An Application to Independent Set . . . . . . . . . . . . . . . 108
4.4 Greedy Algorithms for Disjoint Paths . . . . . . . . . . . . . . . . . . 111
5 Conclusions 117
viii
Bibliography 121
ix
x