Integer Programming
Sample PhD Qualifier Questions
1.
cfw_
Let X = (x, y)
2
+
Z 2 : 3x1 + x2 + 2 y1 + 4 y2
13 . Given the feasible point (x ,y ) =
(0, 0.6, 3.1, 1.55), find a valid inequality for X that cuts off this point.
2.
(Branch and bound can take

2-2, 2-5, 2-6
Stable Set IP Solution
Node Covering IP solution
Since the objective functions are the same in all problems,
it is only necessary to consider the feasible regions.
\barcfw_w > 0 because
the LHS must be > 0,
otherwise \barcfw_x
would not be i

vf$vfAkwhhk$H
W3TAW333WATV
j1
`WVj7fw
d x i
n n ~ d d yx cfw_ d r
k7f`7jkbHh7x`h`|SSzfowjvy
yfxwv3uk`7f`fdWsq7 hfdjkfFmfkjhfd e
d t r p o n l dig
x y
d`WS)PP`1`x
wv`ts3pWShfdb`X
uc r q i acgeca Y
BWVU0TSSRQP4FFHFCA976531)(&
E 46 D ' I I & G E D B @ 8

Shortest Path/Tree Problem (Jensen & Bard)
Given a directed network of n nodes and m arcs from a specified node s, called the
source, to a second specified node t, called the destination or sink. Figure 11 shows a typical
network where s = 1 and t = 10. T

Vehicle Routing Problem with Time Windows (VRPTW)
Problem: Find the minimum cost to visit a set of customers subject to time window constraints
and vehicle capacity limitations. It is assumed that the vehicles are located at a central depot and
are homoge

Maximum Flow Problem
(from Chapter 6, Jensen and Bard 2003)
1
Excel Add-in Model (note, back arc (6,1) added)
Network Model
15
9
TRUE
TRUE
TRUE
100
100
Change
Name: Max_Flow
Type: Net
Goal: Max
Cost: 15
Solver:
Type:
Sens.:
Side:
Excel Solver
Linear
Yes
N

Running CPLEX in ME High Performance Computing Lab
The ME High Performance Computing (HPC) Lab in ETC 3.140 consists of 11 rack
mounted Dell Poweredge 2950 Workstations running Ubuntu Linux. Each computer has 2
dualcore, hyperthreading 3.73 GHz Xeon proce

INTEGER PROGRAMMING COMPUTER ASSIGNMENTS
During the semester, you will be asked to write a number of computer programs implementing
some of the methods discussed in class. These programs can be written in any language and run on any
machine you choose, ci

ORI 391Q.4 (IP)
HW# 8
Benders Decomposition Homework
1. The Fixed-Charge Network Flow (FCNF) problem: We are given a directed network with a
set of nodes V (facilities) and a set of arcs A. An arc e = (i,j) that points from node i to node j
means that the

ORI 391Q.4 (IP)
HW# 8
Benders Decomposition Homework
1. The Fixed-Charge Network Flow (FCNF) problem: We are given a directed network with a
set of nodes V (facilities) and a set of arcs A. An arc e = (i,j) that points from node i to node j
means that the

8.1 Greedy Algorithms (From Jensen & Bard, OR Models & Methods, Wiley 2003)
A greedy algorithm for solving an optimization problem is one that iteratively constructs a
solution so that at each step the current partial solution is augmented by seeking maxi

Cycles in an Undirected Graph
1. Determine if an undirected graph G = (V, E) is connected, where |V| = n and |E| = m.
2. Find all the cycles in an undirected graph G.
3. (Optional) Find all articulation points in G.
Terminology: Given an undirected graph,

A new way to enumerate cycles in graph
Hongbo Liu, Jiaxin Wang
State Key Lab of Intelligent Technology and System
Department of Computer Science and Technology, Tsinghua University
Beijing 100084, China
[email protected]
Abstract
In many cases,

& !
1 "
" Rr scfw_wG
$" Rr H
&
1 "
y
y
s y wt
u x
y
y
y
y
#r
s
dy
y
y
y
t 5r
s
t 5r
s
t x
)det 5r
s
5r
s s
5r
s s
y
y
#r
s
y
y t x s
eD#Gzw 5r
udG)wRu 5r
s
t x y s
y | y
8
y | y
b
5r
s s
t x s
D#Gzw 5r
dy
&
&
1
" Rr k51 % zr 5H

Proving a Decision Problem is NP-Complete
1. Show that is in NP
2. Select a known NP-complete problem
3. Construct a transformation f from to
4. Prove that f is a polynomial transformation
5. Show that an instance of has a solution if and only if the co

Lecture 6 Integer Programming
Models
Topics
General model
Defining decision variables
Continuous vs. integral solution
Logic constraints
Applications: staff scheduling, fixed charge, TSP
Piecewise linear approximations to nonlinear
functions
Linear

Tutorial on Computational Complexity
Craig A. Tovey
School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332
[email protected]
This paper was refereed.
Computational complexity measures how much work is re

Knapsack and Related Problems and Their Complexity
Some notation:
p
objective function weight
w weight
x
decision variable
Basic 0-1 Knapsack Problem
n
Maximize
p x
j =1
j
j
n
subject to
w x
j =1
j
j
c
x j = 0 or 1,
j = 1,n
Bounded Knapsack Problem
n
Maxi

Classification of Integer Programming Problems
In integer programming and the related field of combinatorics, it is common to
differentiate between easy and difficult problems. To develop a method of classification,
just four concepts are needed:
1. a cla

um: humuumqa am 22. 2. t in . 5:25;. 2 ab. .3 23. mini. an:
:5 waumtu. 2. BE Emmi n h .22.; .H Eo 85. :5. H Spa 3:0 _.
-253 2. Ta .3 awn. :23 2.2.5. 52. .3 SE 3.5%. 2 nag
.3. $3. 2.: am .35 23: 2.: E 5an E :32? 3.3. gumm
. m. 2. :25: 25-5325 .mm. .3.
.3

Derivation of Gomory Cut
(from Jensen and Bard OR Models and Methods, Wiley 2003)
Consider an IP in equality form with all-integer data, and note that any set of values of
the decision variables satisfying the constraints must also satisfy any relations d

J.F. Bard
Summary of Branch and Bound Methodology Using LP Relaxation
The problem under consideration is
maxcfw_cx : x S
where S = cfw_x : Ax = b, x 0 and integer. The corresponding continuous relaxation will be
denoted by T = cfw_x : Ax = b, x 0, where S

Data Structures for Integer Branch and Bound Search Tree
(Depth-First Search)
Separation: Assume that in an LP solution to a mixed integer program x1 = 4.6. The separation
or partitioning step leading to the development of a search tree requires adding tw

Rudimentary Branch and Bound Algorithm Using LP Relaxation
The problem under consideration is
maxcfw_cx : x S
where S = cfw_x : Ax = b, x 0 and integer. The corresponding continuous relaxation will be
denoted by T = cfw_x : Ax = b, x 0, where S T.
To simp