1. ADVANCED COMPUTATIONAL MODELS AND ALGORITHMS
Instructor: Dr. Gautam Das
March 10, 2009
Class notes by Alexandra Stefan

2. Topics covered Linear Programming (LP) Integer Programming (IP)
Problem instance:
Set of n real variables
Set of restrictions in the form of linear inequalities
Goal/optimizing function
Integer Programming (IP)
Set of n integer variables

3. Linear programming Example: Find real values x,y, s.t. Constraints:
Goal:
(G) 4x + y is maximized
Solution for this problem: (x,y) = (5, 0)

4. Geometric interpretation of the linear problemy
(3) = 0 G
(solution plane) 5 4
(1) = 0 3
Feasible region 2
Solution point 1
(4) = 0
0.2 x 1 2 3 4 5 6
(2) = 0
(G) = 0
(G) = 4

5. Feasible region is convex
Convex region: given any two points from the region, the line segment between them is part of the region
Proof idea: the feasible region is the intersection of half-planes which are convex regions.
Each constraint generates at most one edge in the feasible region
Intuition on finding the solution:
as you move the goal line, G, to increase it’s value, the point that will maximize it is one of the ‘corners’.

6. Naive algorithm – special case (n =2)
Problem definition:
2 variables and m constraints
Solution:
For all pairs of constraints ci,cj
Find the intersection point: pij (pij satisfies both ci and cj)
Check whether pij is part of the feasible region
If yes, apply goal function. Keep track of which point gave maximum value for the goal function.
Complexity: O(m3)
O(m2) pairs (ci, cj). For each pair perform O(m) evaluations to see if the point pij satisfies all the m constraints

7. Naive algorithm – general case
General problem definition:
n variables and m constraints,
Solution
A corner is now defined by intersecting n hyper-planes (that is satisfying n constraints)
This gives a total of m choose n corners => complexity O(mn) (exponential in n)

8. Methods that solve LP Simplex (by George B. Dantzig, 1947)
Finds optimum: (because feasible region is convex)
Runtime: average time is linear; worst case time is exponential (covers all feasible corners)
Ellipsoidal method (by Leonid Khachiyan, 1839)
Runtime: polynomial
Theoretical value; not useful in practice
Interior point method (by Narendra Karmarkar, 1984)
Runtime : polynomial
Practical value

9. Simplex Start from a corner STOPPING condition
Note that note all corners are part of the feasible region. The algorithm will find a feasible one.
Look at the neighboring corners
Intuition of how you find the neighboring corners: remove one constraint and add another
Move to the one that gives the highest value to the goal function
STOPPING condition
: none of the neighboring points gives a higher value than the current one.

10. Integer Programming (IP)
Set of n integer variables
Set of restrictions in the form of linear inequalities
Goal/optimizing function
IP is NP-Complete
Decision problem:
Input: IP and a target value C
Output: is there a point in the region that evaluates the goal function to a value greater than C?
Easy to verify it is NP
NP-Complete
reduce Vertex Cover to IP