CO 351 Network Flows, Final Exam, Winter 2007
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UNIVERSITY OF WATERLOO FINAL EXAMINATION WINTER TERM 2007
Surname: First Name: Id.#:
Course Number Course Title Instructor
CO 351 Network Flows J. K nemann o
Date of Exam Time Period Number of Exam Page

CO 351 Spring 2014
Introduction to Clutters
July 22-24, 2014
Observe the following special relationship between st-dipaths and st-cuts in a digraph: every
st-dipath intersects every st-cut. We would like to analyze this relationship from a global
perspect

CO 351: Assignment 5
Due Thursday, July 24, in class
Assignment Policy: While it is acceptable for students to discuss the course material and
the assignments, the work you hand in should be your own and in your own words. Outright
copying is a very serio

CO 351 Network Flow Theory
University of Waterloo
Fall 2007
Instructor: Ashwin Nayak
Homework 7, due November 16, 2007
Question 1. In this question, we analyze an alternative greedy rule for pushing flow, which came up
in the lectures on the Ford-Fulkerso

CO 351 Network Flow Theory
University of Waterloo
Fall 2007
Instructor: Ashwin Nayak
Suggested Problems, October 15, 2007
(not to be submitted for grading)
Question 1. Consider the following project:
Jobs
Duration
(days)
Predecessors
J1 J2 J3 J4 J5 J6 J7

CO 351 Network Flow Theory
University of Waterloo
Fall 2007
Instructor: Ashwin Nayak
Homework 1: due by September 21, 2007, 12 noon
Question 1. A matching M in a graph G = (V, E) is a subset of the edges E such that no two edges in
it are incident on the

CO 351 Network Flow Theory
University of Waterloo
Fall 2007
Instructor: Ashwin Nayak
Suggested problems, November 26, 2007
(not to be submitted for grading)
Question 1. Consider the minimum cost flow instance given below. The arc labels are capacity-weigh

CO 351 Network Flow Theory
University of Waterloo
Fall 2007
Instructor: Ashwin Nayak
Homework 6, due November 9, 2007
Question 1. [4 marks] Give an example of a digraph with nodes s, t, non-negative arc-capacities, and a
feasible (s, t)-flow such that som

CO 351 Network Flow Theory
University of Waterloo
Fall 2007
Instructor: Ashwin Nayak
Homework 4: due by 2:25pm, October 12, 2007
Question 1. [4 marks] Let A Rnm be a totally unimodular (TU) matrix. Let A = [A|e1 ] be the
matrix obtained by adding the colu

CO 351 Network Flow Theory
University of Waterloo
Fall 2007
Instructor: Ashwin Nayak
Homework 8, due November 23, 2007
Question 1. [4 marks] Find feasible flows in the digraphs in Figure 1 if they exist, or explain why there
is none.
a 0
a 1
(1,2)
0b
(0,2

1
UNIVERSITY OF WATERLOO MIDTERM EXAMINATION CO351, Winter Term 2007
Surname: First Name: Id.#:
Course Number Course Title Instructor
CO 351 Network Flows J. K nemann o
Date of Exam Time Period Number of Exam Pages (including this cover sheet) Exam Type A

1
UNIVERSITY OF WATERLOO MIDTERM EXAMINATION CO351, Winter Term 2007
SOLUTIONS
2 1. Modeling with Shortest Paths (20 Points) A factory requires the following number of units of certain chemical component X for each of the next 4 days. Days Requirements (i

ID number:
Page 2 of 11
Problem 1: [10 marks] Use Dantzigs Algorithm to either nd a tree of shortest dipaths rooted at s or nd a negative dicycle in the digraph below. (Start with the tree indicated in bold. Briey describe each step.)
2 a 1 d 5 2 c 2 b 2

ID:
Page 2 of 8
a
-2 -2
c
d
1
s
-2
-1
f
1 -1
b
2 4
e
1
Problem 1: [10 marks] Using Dantzig's Algorithm, either find a tree of shortest dipaths rooted at s, or find a negative dicycle in the digraph above. (Start with the spanning tree indicated by the edg

CO 351 - Final Examination
1
CO 351 Network Flow Theory Final Examination
December 6, 2007 12:30 3:00 p.m.
Instructor: Ashwin Nayak Instructions
Please print your name and student identication number.
NAME STUDENT IDENTIFICATION NUMBER
Make sure you ha

CO 351 - Final Examination
1
CO 351 Network Flow Theory Solutions to the Final Examination
December 6, 2007 Instructor: Ashwin Nayak
CO 351 - Final Examination
2
Question 1. [10 marks] Find an arborescence of shortest dipaths rooted at node 1 in the follo

CO 351 - Midterm Examination
1
CO 351 Network Flow Theory Midterm Examination
October 22, 2007 7:00 8:30 p.m.
Instructor: Ashwin Nayak Instructions
Please print your name and student identication number.
NAME STUDENT IDENTIFICATION NUMBER
Make sure you

CO 351 Network Flow Theory University of Waterloo Fall 2007 Instructor: Ashwin Nayak Solutions to Midterm Examination October 22, 2007 Note: these are one of many correct solutions. Question 1. Let D = (N, A) be a digraph with s, t N , s = t, with arc-len

CO 351 Network Flow Theory
University of Waterloo
Fall 2007
Instructor: Ashwin Nayak
Homework 2: due by September 28, 2007, 12 noon
Question 1. Let D = (N, A) be a digraph with arc-lengths (we : e A) and let s, t N .
For 0 i n 1 and v N , let OPT(i, v) be