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15853:Algorithms in the Real World
g
Linear and Integer Programming III
– Integer Programming
• Applications
•A
lgorithms
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g
Integer (linear) Programming
minimize:
c
T
x
subject to:
Ax
≤
b
Related Problems
– Mixed Integer Programming (MIP)
–
ero ne programming
x
≥
0
x
∈
Z
n
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Zero one programming
– Integer quadratic programming
– Integer nonlinear programming
History
• Introduced in 1951 (Dantzig)
•T
S
P
a
s
special case in 1954 (Dantzig)
• First convergent algorithm in 1958 (Gomory)
• General branchandbound technique 1960
(Land and Doig)
• Frequently used to prove bounds on approximation
algorithms (late 90s)
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Current Status
• Has become “dominant” over linear programming in
past decade
• Saves industry Billions of Dollars/year
• Can solve 10,000+ city TSP problems
• 1 million variable LP approximations
• Branchandbound, Cutting Plane, and Separation
all used in practice
eneral purpose packages do not tend to work as
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•
General purpose packages do not tend to work as
well as with linear programming  knowledge of
the domain is critical.
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Subproblems/Applications
•
Facility location
Locating warehouses or franchises (e.g. a Burger King)
•
Set covering and partitioning
Scheduling airline crews
•
Multicomodity distribution
Distributing auto parts
•
Traveling salesman and extensions
Routing deliveries
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g
•
Capital budgeting
•
Other Applications
VLSI layout, clustering
Knapsack Problem
Integer (zeroone) Program:
maximize
c
T
x
where:
b = maximum weight
= utility of item i
subject to:
ax
≤
b
x
binary
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c
i
ut ty of t m
a
i
= weight of item i
x
i
= 1 if item i is selected, or 0 otherwise
The problem is NPhard.
Traveling Salesman Problem
Find shortest tours that visit all of n cities.
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courtesy:
Applegate
,
Bixby
,
Chvátal
, and
Cook
Traveling Salesman Problem
∑∑
==
n
i
n
j
ij
ij
x
c
11
minimize:
c
ij
= c
ji
= distance from city i to city j
ssuming
ymmetric version
n
i
x
n
j
ij
≤
≤
=
∑
=
1
2
0
(path enters and leaves)
subject to:
binary
,
ij
ji
x
x
=
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(assuming
symmetric version
)
x
ij
if tour goes from i to j or j to i, and 0 otherwise
Anything missing?
3
Traveling Salesman Problem
∑∑
==
n
i
n
j
ij
ij
x
c
11
minimize:
n
j
i
n
nx
t
t
n
j
x
n
i
x
ij
j
i
n
i
ij
n
j
ij
≤
≤
−
≤
+
−
≤
≤
=
≤
≤
=
∑
∑
=
=
,
2
1
1
1
1
1
0
0
(out degrees = 1)
(in degrees = 1)
(??)
subject to:
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c
ij
= distance from city i to city j
x
ij
= 1 if tour visits i then j, and 0 otherwise (binary)
t
i
= arbitrary real numbers we need to solve for
Traveling Salesman Problem
n
i
n
j
ij
ij
x
c
minimize
:
n
j
i
n
nx
t
t
n
j
x
n
i
x
ij
j
i
n
i
ij
n
j
ij
≤
≤
−
≤
+
−
≤
≤
=
≤
≤
=
∑
∑
=
=
,
2
1
1
1
1
1
0
0
subject to
:
(out degrees = 1)
(in degrees = 1)
(??)
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c
ij
= distance from city i to city j
x
ij
= 1 if tour visits i then j, and 0 otherwise (binary)
t
i
= arbitrary real numbers we need to solve for
Traveling Salesman Problem
The last set of constraints:
prevents “subtours”:
n
j
i
n
nx
t
t
ij
j
i
≤
≤
−
≤
+
−
,
2
1
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This note was uploaded on 01/26/2010 for the course COMPUTER S 15853 taught by Professor Guyblelloch during the Fall '09 term at Carnegie Mellon.
 Fall '09
 GuyBlelloch
 Algorithms

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