Transcript NP Completeness

Linear Programming
Piyush Kumar
Graphing 2-Dimensional LPs
Optimal
Solution
y
Example 1:
4
Maximize
x+y
Subject to: x + 2 y  2
x3
3
Feasible
Region
2
y4
x0 y0
1
0
These LP animations were
created by Keely Crowston.
x
0
1
2
3
Graphing 2-Dimensional LPs
y
Example 2:
4
Minimize **
Multiple
Optimal
Solutions!
x-y
Subject to: 1/3 x + y  4
-2 x + 2 y  4
3
2
Feasible
Region
x3
x0 y0
1
0
0
1
2
3
x
Graphing 2-Dimensional LPs
y
Example 3:
40
Minimize
x + 1/3 y
Subject to: x + y  20
-2 x + 5 y  150
30
Feasible
Region
20
x5
x 0 y 0
10
x
Optimal
Solution
0
0
10
20
30
40
Do We Notice Anything From
These 3 Examples?
Extreme point
y
y
y
4
4
40
3
3
30
2
2
20
1
1
10
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3
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x
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x
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40
A Fundamental Point
y
y
y
4
4
40
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3
30
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2
20
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1
10
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x
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If an optimal solution exists, there is
always a corner point optimal solution!
40
Graphing 2-Dimensional LPs
Second
Corner pt.
Example 1:
Optimal
Solution
y
4
Maximize
x+y
Subject to: x + 2 y  2
x3
3
Feasible
Region
2
y4
x 0 y 0
1
Initial
0
Corner pt.
x
0
1
2
3
And We Can Extend this to Higher
Dimensions
Then How Might We Solve an LP?
 The constraints of an LP give rise to a
geometrical shape - we call it a polyhedron.
 If we can determine all the corner points of the
polyhedron, then we can calculate the objective
value at these points and take the best one as our
optimal solution.
 The Simplex Method intelligently moves from corner
to corner until it can prove that it has found the
optimal solution.
Linear Programs in higher
dimensions
maximize
subject to
z=
-4x1 + x2 - x3
-7x1 + 5x2 + x3 <= 8
-2x1 + 4x2 + 2x3 <= 10
x1,
x2,
x3

0
In Matrix terms
T
Max c x
subject to Ax  b
Anxd , cdx1, xdx1
LP Geometry
Forms a d dimensional polyhedron
Is convex : If z1 and z2 are two feasible
solutions then λz1+ (1- λ)z2 is also feasible.
Extreme points can not be written as a
convex combination of two feasible points.
LP Geometry
Extreme point theorem: If there exists
an optimal solution to an LP Problem,
then there exists one extreme point
where the optimum is achieved.
Local optimum = Global Optimum
LP: Algorithms
Simplex. (Dantzig 1947)
 Developed shortly after WWII in response to logistical problems:
used for 1948 Berlin airlift.
 Practical solution method that moves from one extreme point to a
neighboring extreme point.
 Finite (exponential) complexity, but no polynomial
implementation known.
Courtesy Kevin Wayne
LP: Polynomial Algorithms
Ellipsoid. (Khachian 1979, 1980)
 Solvable in polynomial time: O(n4 L) bit operations.
o n = # variables
o L = # bits in input
 Theoretical tour de force.
 Not remotely practical.
Karmarkar's algorithm. (Karmarkar 1984)
 O(n3.5 L).
 Polynomial and reasonably efficient
implementations possible.
Interior point algorithms.
 O(n3 L).
 Competitive with simplex!
o Dominates on simplex for large problems.
 Extends to even more general problems.
LP: The 2D case
Let's suppose we are given n linear inequalities h1, h2 ,..., h n
hi : ai , x x  ai , y y  bi
Wlog, we can assume that c=(0,-1).
So now we want to find the
Extreme point with the smallest y coordinate.
Lets also assume, no degeneracies, the solution is given by two
Halfplanes intersecting at a point.
Incremental Algorithm?
How would it work to solve a 2D LP
Problem?
How much time would it take in the
worst case?
Can we do better?