This document summarizes the lecture 0 introduction to a course on generalized network flows. It discusses why network flows are an important topic that is applicable to many real-world problems. It also outlines some of the limitations of existing algorithms and how this course will approach the topic using discrete calculus and handling non-linear problems. Finally, it provides an outline of topics to be covered in subsequent lectures.

1. Lecture 0: Generalized Network Flows:
Theory, Algorithms, and Applications
Wai-Shing Luk (陆伟成)
Fudan University
2012 年 8 月 11 日
W.-S. Luk (Fudan Univ.) Lecture 0: Generalized Network Flows 2012 年 8 月 11 日 1 / 10

2. Motivation
Q. Why this topic is important?
A. Well, “network” is everywhere:
Transportation network, logistics network
Power network (smart grid)
Electronics circuits
Wireless network
Social network
Neural network, Bayesian network
and ...more
W.-S. Luk (Fudan Univ.) Lecture 0: Generalized Network Flows 2012 年 8 月 11 日 2 / 10

3. Motivation (cont’d)
Q. Why should I learn it as it is already a “mature” topic?
A. At least we still need to know
How to choose the existing algorithms wisely
How to transform a problem into a standard network flow formulation
How to handle new problems: e.g. non-linear problems.
W.-S. Luk (Fudan Univ.) Lecture 0: Generalized Network Flows 2012 年 8 月 11 日 3 / 10

4. Motivation (cont’d)
Q. What are the limitations of the existing algorithms?
A. The existing algorithms
mostly handle linear problems, whereas most engineering problems are
non-linear.
can handle only single parameter (for parametric problems), whereas
most realistic problems are multi-parameter.
mostly rely on finding “cycles” rather than “cuts”. Dual problems are
first transformed into their primal counterparts via Lagrange duality
theory, which make the problem more complicated.
W.-S. Luk (Fudan Univ.) Lecture 0: Generalized Network Flows 2012 年 8 月 11 日 4 / 10

5. Motivation (cont’d)
Q. Why should I learn this course instead of many others?
A In this course, we will
explain the concept using “Discrete Calculus”
describe how to transform a problem into a standard network flow
formulation.
describe the fundamental mechanism of algorithms so that we can
tackle new problems.
W.-S. Luk (Fudan Univ.) Lecture 0: Generalized Network Flows 2012 年 8 月 11 日 5 / 10

6. Why Generalization?
1 Unify network flows and physical flows. In fact, same terminology in
both sides is not coincident!
2 Develop co-domain algorithms for nonlinear scheduling problems.
W.-S. Luk (Fudan Univ.) Lecture 0: Generalized Network Flows 2012 年 8 月 11 日 6 / 10

7. Applications in Electronic Design Automation
Primal domain: Co-domain:
Escape routing (flip-chip) Clock skew scheduling
Assignment problem Re-timing
Resource allocation Delay padding
Circuit partitioning Buffer insertion
Bipartite matching Transportation
Perfect matching Clock concurrent optimization
W.-S. Luk (Fudan Univ.) Lecture 0: Generalized Network Flows 2012 年 8 月 11 日 7 / 10

8. Theory
Discrete Calculus (1-complex = Network)
Concept of Pairing: Generalized Stokes’ theorem
Scheduling problem in co-domain
Important Note
Not direction, but orientation
Not duality, but pairing
W.-S. Luk (Fudan Univ.) Lecture 0: Generalized Network Flows 2012 年 8 月 11 日 8 / 10

9. Course Outline
Lecture 1: Network and flows
Lecture 2: feasibility problems
Lecture 3: Parametric problems (single parameter)
Lecture 4: Min-cost flow/potential problems (linear)
Lecture 5: Min-cost flow/potential problems (convex)
Lecture 6: Parametric problems (multi-parameter)
W.-S. Luk (Fudan Univ.) Lecture 0: Generalized Network Flows 2012 年 8 月 11 日 9 / 10

10. References
1 R. T. Rockafellar, Network flows and monotropic optimization, John Wiley
and & Sons, 1984.
2 Network optimization
3 Network flows: theory, algorithms and applications
4 S. M. Burns, Performance Analysis and Optimization of Asynchronous
Circuits. PhD thesis, CalTech, Pasadena, CA, December 1991.
5 N. E. Young, R. E. Tarjan, and J. B. Orlin, “Faster parametric shortest path
and minimum balance algorithms,” Networks, 1991.
6 Yi Wang, Wai-Shing Luk et al., Yield-driven clock skew scheduling
7 Yan-Ling Zhi, Wai-Shing Luk et al., Multi-domain clock skew scheduling
W.-S. Luk (Fudan Univ.) Lecture 0: Generalized Network Flows 2012 年 8 月 11 日 10 / 10