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### Approximation Algorithms

Course: ITSD 4312, Fall 2009
School: Clayton
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Word Count: 668

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4312 ITSD ITSD Advanced Programming II Approximation Algorithms Motivation Motivation Many important problems are NPcomplete We cant abandon them But we can certainly try different approaches If n is small, an exponential solution is OK Special cases might be identified Solve for a near-optimal solution A Well-known Approximation Well-known n 1 1 e = = lim n x = 0 x! x = 0 x! 111 1 1 e= + + + + 1 2 6...

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4312 ITSD ITSD Advanced Programming II Approximation Algorithms Motivation Motivation Many important problems are NPcomplete We cant abandon them But we can certainly try different approaches If n is small, an exponential solution is OK Special cases might be identified Solve for a near-optimal solution A Well-known Approximation Well-known n 1 1 e = = lim n x = 0 x! x = 0 x! 111 1 1 e= + + + + 1 2 6 24 120 Another Approximation Another x 2 k +1 k sin x = lim (1) n (2k + 1)! k =0 n x3 x5 x7 sin x = x + + 3! 5! 7! Two Classes of Problems Two Return a minimum Traveling salesman (minimize hamiltonian cycle) Return a maximum Subset-sum (maximize up to limit) Getting Close Getting Lets allow a approximation ratio, (n) C is cost using our approx. algorithm C* is optimal cost C C * max , ( n) C* C 0 < C C* (for maximizing problem) 0 < C* C (for minimizing problem) Traveling Salesman Traveling Given G(V, E) Cost c(u,v) 1, (u,v) E Cost c( A) = c(u , v) ( u ,v ) A Where A E Find a hamiltonian cycle (tour) that minimizes c(A) on G TSP Approximation (1977) TSP Given G(V,E) Pick root r V Generate MST(G,r) L = pre-order traverse MST(G,r) Return hamiltonian cycle generated by visiting vertices in the order L This is a polynomial time 2-approximation algorithm (n)=2 i.e. its max runtime is 2C* TSP Approx Visualized TSP a d e b c h f g c h b f a d e g abchdefg a d e a d e b c h f g b f g c(A)=19.074 c h C*=14.715 Subset-Sum Subset-Sum Defined by (S, t) A set of positive integers, S={x1, x2, , xn} A positive integer, t Is there a subset of S that adds up exactly to t? Many practical applications Truck packing without exceeding weight limit Waiter holding meals NP-complete Solution Start NP-complete with empty list L For each x in S L += (L + x) Ex S = { 1, 4, 5 } P1 = { 0, 1 } P2 = { 0, 1, 4, 5 } P3 = { 0, 1, 4, 5, 6, 9, 10 } Subset-sum Approximation Subset-sum Same as before Except, trim list after each iteration Select trimming factor: 0 < d < 1 Remove y and keep z such that y z y 1+ d Remove elements larger than t Return largest in list Subset-sum Approx Ex Subset-sum d = 0.1 L = { 10, 11, 12, 15, 20, 21, 22, 23, 24, 29 } L = { 10, 12, 15, 20, 23, 29 } - or L = { 11, 15, 22, 24, 29 } Trimming Example d = 0.1 L = { 10, 11, 12, 15, 20, 21, 22, 23, 24, 29 } L = { 10, 12, 15, 20, 23, 29 } - or L = { 11, 15, 22, 24, 29 } Subset-sum Approx Ex Subset-sum S = { 104, 102, 201, 101 }, t = 308, = 0.40 d = /(2n) = 0.40/8 = 0.05 P1 = { 0, 104 } P2 = { 0, 102, 104, 206 } P3 = { 0, 102, 201, 206, 303, 407 } P4 = { 0, 101, 102, 201, 302, 303, 404 } Optimal: 101 + 102 + 104 = 307 Backtracking Backtracking N-queens problem = correctly place N queens onto an NxN board such that no queen attacks any other queen. Q Q Q ...

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