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approx-2x2 - Approximation algorithms As weve seen some...

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Approximation algorithms As we’ve seen, some optimisation problems are “hard” (by hardness of related decision problem), little chance of finding poly-time algorithm that computes optimal solution largest clique smallest vertex cover largest independent set . . . But: sometimes sub-optimal solutions are kind of OK pretty large clique pretty small vertex cover pretty large independent set . . . if algorithms run in poly time (preferrably small exponents). Approximation algorithms compute near-optimal solutions. Known for thousands of years. For instance, approximations of value of π ; some engineers still use 4 these days :-) Approximation algorithms 1 Consider optimisation problem . Each potential solution has positive cost (or quality ). We want near-optimal solution. Depending on problem, optimal solution may be one with maximum possible cost (maximisation problem), like maximum clique, or one with minimum possible cost (minimisation prob- lem), like minimum vertex cover. Algorithm has approximation ratio of ρ ( n ) , if for any input of size n , the cost C of its solution is within factor ρ ( n ) of cost of optimal solution C , i.e. max C C , C C ρ ( n ) For maximisation problems, 0 < C C , thus C /C gives factor by which optimal solution is better than approximate solution (note: C /C 1 and C/C 1 ). For minimisation problems, 0 < C C , thus C/C gives factor by which optimal solution is better than approximate solution (note C/C 1 and C /C 1 ). Approximation algorithms 2 Approximation ratio is never less than one: C C < 1 C C > 1 An algorithm with guaranteed approximation ration of ρ ( n ) is called a ρ ( n ) -approximation algorithm . A 1 -approximation algorithm is optimal, and the larger the ratio, the worse the solution. For many N P -complete problems, constant-factor ap- proximations (i.e. computed clique is always at least half the size of maximum-size clique), sometimes in best known approx ratio grows with n , and sometimes even proven lower bounds on ratio ( for every approximation alg, the ratio is at least this and that, unless P = N P ). Sometimes better ratio when spending more computation time. An approximation scheme for an optimisation problem is an approximation algorithm that takes as input an instance plus a parameter > 0 s.t. for any fixed , the scheme is a (1 + ) -approximation ( trade-off ). Approximation algorithms 3 A scheme is a poly-time approximation scheme (PAS) if for any fixed > 0 , it runs in time polynomial in input size. Running time can increase dramatically with decreasing , consider e.g. T ( n ) = n 2 / . 2 1 1 / 2 1 / 4 1 / 100 n T ( n ) n n 2 n 4 n 8 n 200 10 1 10 1 10 2 10 4 10 8 10 200 10 2 10 2 10 4 10 8 10 16 10 400 10 3 10 3 10 6 10 12 10 24 10 600 10 4 10 4 10 8 10 16 10 32 10 800 We want: if decreases by constant factor, then running time increases by at most some other constant factor, i.e., running time is polynomial in n and 1 / .
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