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Network Flows
by
Ravindra K. Ahuja, Thomas L. Magnanti
and
James B. Orlin
OR 185-88 August 1988
ma
NETWORK FLOWS
Ravindra K. Ahuja* , Thomas L. Magnanti, and James B. Orlin
Sloan School of Management
Massachusetts Institute of Technology
Cambridge, MA. 02139
* On leave from Indian Institute of Technology, Kanpur - 208016, INDIA
I
il
NETWORK FLOWS
OVERVIEW
1. Introduction
1.1 Applications
1.2 Complexity Analysis
1.3 Notation and Definitions
1.4 Network Representations
1.5 Search Techniques
1.6 Developing Polynomial Time Algorithms
2. Basic Properties of Network Flows
21 Flow Decomposition Properties and Optimality Conditions
22 Cycle Free and Spanning Tree Solutions
2.3 Networks, Linear and Integer Programming
2.4 Network Transformations
3. Shortest Path Problems
3.1 Dijkstra's Algorithm
3.2 Dial's Algorithm
3.3 R-Heap Implementation
3.4 Label Correcting Algorithms
3.5 All Pair Shortest Path Algorithm
4. Maximum Flow Problem
4.1 Labeling Algorithm and the Max-Flow Min-Cut Theorem
4.2 Decreasing the Number of Augmentations
4.3 Shortest Augmenting Path Algorithm
4.4 Preflow-Push Algorithms
4.5 Excess-Scaling Algorithm
4.6 Networks with Positive Lower Bounds
S. Minimum Cost Flow Problem
5.1 Optimality Conditions
52 Relationship to Shortest Path and Maximum Flow Problems
5.3 Negative Cycle Algorithm
5.4 Successive Shortest Path Algorithm
5.5 Primal-Dual and Out-of-Kilter Algorithms
5.6 Network Simplex Algorithm
5.7 Right-Hand-Side Scaling Algorithm
5.8 Cost Scaling Algorithm
5.9 Double Scaling Algorithm
5.10 Sensitivity Analysis
5.11 Assignment Problem
Reference Notes
References
so.
1
Network Flows
Perhaps no subfield of mathematical programming is more alluring than
network optimization. Highway, rail, electrical, communication and may other physical
networks pervade in our everyday lives. As a consequence, even non-specialists
recognize the practical importance and the wide ranging applicability of networks.
Moreover, because the physical operating characteristics of networks (e.g., flows on arcs
and mass balance at nodes) have natural mathematical representations, practitioners
and non-specialists can readily understand the mathematical descriptions of network
optimization problems and the basic nature of techniques used to solve these problems.
This combination of widespread applicability and ease of assimilation has undoubtedly
been instrumental in the evolution of network planning models as one of the most
widely used modeling techniques in all of operations research and applied mathematics.
Network optimization is also alluring to methodologists. Networks provide a
concrete setting for testing and devising new theories. Indeed, network optimization has
inspired many of the most fundamental results in all of optimization. For example,
price directive decomposition algorithms for both linear programming and
combinatorial optimization had their origins in network optimization. So did cutting
plane methods and branch and bound procedures of integer programming, primal-dual
methods of linear and nonlinear programming, and polyhedral methods of
combinatorial optimization. In addition, networks have served as the major prototype
for several theoretical domains (for example, the field of matroids) and as the core model
for a wide variety of min/max duality results in discrete mathematics.
Moreover, network optimization has served as a fertile meeting ground for ideas
from optimization and computer science. Many results in network optimization are
routinely used to design and evaluate computer systems, and ideas from computer
science concerning data structures and efficient data manipulation have had a major
impact on the design and implementation of many network optimization algorithms.
The aim of this paper is to summarize many of the fundamental ideas of network
optimization. In particular, we concentrate on network flow problems and highlight a
number of recent theoretical and algorithmic advances. We have divided the discussion
into the following broad major topics:
2
* Applications
* Basic Properties of Network Flows
* Shortest Path Problems
Maximum Flow Problem
* Minimum Cost Flow Problem
* Assignment Problem
Much of our discussion focuses on the design of provably good (e.g., polynomial)
algorithms. Among good algorithms, we have presented those algorithms which are
simple and are likely to be efficient in practice. We have attempted to structure our
discussion so that it not only provides a survey of the field for the specialists, but also
serves as an introduction and summary to the non-specialists who have a basic working
knowledge of the rudiments of optimization, particularly linear programming.
As a prelude to our discussion in the next section, we present in this section
several important preliminaries . We discuss (i) different ways to measure the
performance of algorithms; (ii) graph notation and various ways to represent networks
quantitively; (iii) a few basic ideas from computer science that underlie the design of
many algorithms; and (iv) a couple of generic proof techniques that have been useful in
designing polynomial algorithms.
1.1 Applications
Networks arise in numerous application settings and in a variety of guises. In
this section, we briefly describe a few prototypical applications. Our discussion is
intended to illustrate a range of applications and to be suggestive of how network flow
problems arise in practice; a more extensive survey would take us far beyond the scope
of our discussion. To illustrate the breadth of network applications, we consider some
models requiring solution techniques that we will not describe in this chapter.
For the purposes of this discussion we will consider four different types of
networks arising in practice:
· Physical networks (Streets, railbeds, pipelines, wires)
· Route networks
· Space-time networks (Scheduling networks)
· Derived networks (Through problem transformations)