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TSP_PTAS - CS468 Wed Feb 15 th 2006 Journal of the ACM...

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Unformatted text preview: CS468, Wed Feb 15 th 2006 Journal of the ACM, 45(5):753–782, 1998 PTAS for Euclidean Traveling Salesman and Other Geometric Problems Sanjeev Arora PTAS 1 S. Arora — Euclidean TSP and other related problems → same as LTAS, with ”Linear” replaced by ”Polynomial” Def Given a problem P and a cost function | . | , a PTAS of P is a one- parameter family of PT algorithms, { A ε } ε> , such that, for all ε > and all instance I of P , | A ε ( I ) | ≤ (1 + ε ) | OPT( I ) | . PTAS 1 S. Arora — Euclidean TSP and other related problems → same as LTAS, with ”Linear” replaced by ”Polynomial” Def Given a problem P and a cost function | . | , a PTAS of P is a one- parameter family of PT algorithms, { A ε } ε> , such that, for all ε > and all instance I of P , | A ε ( I ) | ≤ (1 + O ( ε )) | OPT( I ) | . • PT means time complexity n O (1) , where the constant may depend on 1 / ε and on the dimension d (when pb in R d ) • As far as we get n O (1) , we do not care about the constant • the constant in (1 + O ( ε )) must not depend on I nor on ε TSP K 7 2 3 7 Given a complete graph G = ( V, E ) with non- negative weights, find the Hamiltonian tour of minimum total cost. S. Arora — Euclidean TSP and other related problems 5 1 . 2 8 17 2 TSP K 7 2 3 7 Given a complete graph G = ( V, E ) with non- negative weights, find the Hamiltonian tour of minimum total cost. S. Arora — Euclidean TSP and other related problems OPT 5 1 . 2 8 17 | OPT | = 36 . 2 2 TSP K 7 2 3 7 Given a complete graph G = ( V, E ) with non- negative weights, find the Hamiltonian tour of minimum total cost. S. Arora — Euclidean TSP and other related problems OPT 5 1 . 2 8 17 | OPT | = 36 . 2 TSP is NP-hard ⇒ no PT algorithm, unless P = NP . 2 TSP K 7 2 3 7 Given a complete graph G = ( V, E ) with non- negative weights, find the Hamiltonian tour of minimum total cost. S. Arora — Euclidean TSP and other related problems OPT 5 1 . 2 8 17 | OPT | = 36 . 2 TSP is NP-hard ⇒ no PT algorithm, unless P = NP . Thm For all PT computable function α ( n ) , TSP cannot be approxi- mated in PT within a factor of (1 + α ( n )) , unless P = NP . 2 TSP K 7 Given a complete graph G = ( V, E ) with non- negative weights, find the Hamiltonian tour of minimum total cost. S. Arora — Euclidean TSP and other related problems TSP is NP-hard ⇒ no PT algorithm, unless P = NP . Thm For all PT computable function α ( n ) , TSP cannot be approxi- mated in PT within a factor of (1 + α ( n )) , unless P = NP . Proof Reduction of Hamiltonian Cycle: Let G = ( V, E ) unweighted, incomplete → G = ( V , E ) where: • V = V • ∀ e ∈ E , add ( e, 1) to E • ∀ e / ∈ E , add ( e, (1 + α ( n )) n ) to E 2 TSP K 7 Given a complete graph G = ( V, E ) with non- negative weights, find the Hamiltonian tour of minimum total cost....
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TSP_PTAS - CS468 Wed Feb 15 th 2006 Journal of the ACM...

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