1. ADVANCED COMPUTATIONAL MODELS AND ALGORITHMS
Instructor: Dr. Gautam Das
Lecture 10
March 03, 2009
Class notes by Rachit Shah

2. Overview Linear Programming Integer Programming
Approximation Algorithm based on LP relaxation of IP formulation

3. Linear Programming Problem Instance
Set of variables (real variables) x1, x2…xn
Restriction – a bunch of linear equations
x1 + x2 <= 5
x1 + 2x2 <= 6
x1 >=0
x2 >=0.1

4. Optimization Goal Optimizing function Maximizing 4x1 + x2
Find a pair (x1, x2)
Convexity:
Any two points inside the region, connected by a line does not intersect with region boundaries
Half planes, when intersect, it maintains convexity

5. X2
Optimization Function (4x1+x2)
x1+x2 <= 5
x1+2x2 <= 6
Convex Region
Maximized (4x1+x2)
x2 >= .1 X1
x1 >= 0

6. Algorithm
We have to find out the region and maximize 4x1 + x2 in this region
Algorithm
Input: m equations, Goal function
We have m(m-1)/2 corners (intersecting points)
Find intersecting points Pij (takes m2)
Check Pij is feasible (is it in the common area)
If yes, apply function (4x1 + x2)

7. Analysis
In case of x1 + x2 as the optimization function, the optimization point would be different. Why?
What if we have x1, x2 and x3 (3 Dimensions)
Finding intersecting points take m3
3D – m3 (In case of 3 variables)
4D – m4 (In case of 4 variables) …
nD – mn (In case of n variables)
Its exponential

8. Dangzeg Algorithm Greedy Algorithm (Based on Hill Climbing)
Because of the Convexity of area, there is only one optimal point/solution
Worst case: Exponential
Average case: Linear
In worst Case, All direction are equally optimistic, except the direction we came in, so we have to visit each and every vertex
Example: Transportation problem

9. Greedy Algorithm (Hill Climbing)
Optimization function
Optimum location
Global Hilltop
Climb
Climb
Start
Greedy Algorithm (Hill Climbing)
Dangzeg Algorithm

10. Later Developments
Leond Khadyand: Gave proof of polynomial time algorithm using Ellipsoidal method
Karmarkar: (ATT) interior point method (which stays in feasible region while doing optimization)
There has been no big improvements after

11. Integer Linear Programming
n variables
m inequalities
Goal function fn
The problem is same as Linear Programming except that it is subjected to integer grid
IP is NP complete, why?
Decision problem: Is there a point where goal funtion >= C

12. X2
Optimization Function
Integer Grid
x1+x2 <= 5
x1+2x2 <= 6
Convex Region
x2 >= .1 X1
x1 >= 0