ADVANCED COMPUTATIONAL MODELS AND ALGORITHMS

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ADVANCED COMPUTATIONAL MODELS AND ALGORITHMS

• Instructor Dr. Gautam Das
• March 10, 2009
• Class notes by Alexandra Stefan

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Topics covered

• Linear Programming (LP)
• 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
• Set of restrictions in the form of linear
  inequalities
• Goal/optimizing function

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Linear programming

• Example
• Find real values x,y, s.t.
• Constraints
• (1) xy lt 5
• (2) x 2y lt 6
• (3) x gt 0
• (4) y gt 0.2
• Goal
• (G) 4x y is maximized
• Solution for this problem (x,y) (5, 0)

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Geometric interpretation of the linear problem
y
(3) 0
G (solution plane)
5
4
(1) 0
3
Feasible region
2
Solution point
1
0.2
(4) 0
x
5
4
3
2
1
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 its
  value, the point that will maximize it is one of
  the corners.

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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

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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
  gt complexity O(mn) (exponential in n)

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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

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Simplex

• Start from a corner
• 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.

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Integer Programming (IP)

• 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