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ADVANCED COMPUTATIONAL MODELS AND ALGORITHMS
Instructor: Dr. Gautam Das Lecture 10 March 03, 2009 Class notes by Rachit Shah
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Overview Linear Programming Integer Programming
Approximation Algorithm based on LP relaxation of IP formulation
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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
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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
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X2 Optimization Function (4x1+x2) x1+x2 <= 5 x1+2x2 <= 6 Convex Region Maximized (4x1+x2) x2 >= .1 X1 x1 >= 0
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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)
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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
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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
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Greedy Algorithm (Hill Climbing)
Optimization function Optimum location Global Hilltop Climb Climb Start Greedy Algorithm (Hill Climbing) Dangzeg Algorithm
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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
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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
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X2 Optimization Function Integer Grid x1+x2 <= 5 x1+2x2 <= 6 Convex Region x2 >= .1 X1 x1 >= 0
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