20-approximations

20-approximations - Approximations to NP Solutions If we...

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Unformatted text preview: Approximations to NP Solutions If we are faced with an NP or NP-complete problem the best we can do on a deterministic computer is a solution in exponential time. Since exponential time solutions are impractical for obvious reasons we could consider approximations to the actual solutions. Here we look at approximation algorithms for the classic Traveling Salesman Problem ( T SP ). p. 1/1 Traveling Salesman The idea is, given a set of cities (nodes) that are connected via roads (weighted edges), find the cheapest route through all the cities (find a Hamiltonian path that minimizes the sum of the weights in the path). Formally, T SP = {h G, s, t, w i| G is directed weighted graph with a minimal Hamiltonian path of weight w from s to t } . p. 2/1 Traveling Salesman Theorem: T SP NP-hard . Proof: Note, a problem is NP-hard if every L NP can be reduced to it in polynomial time but the problem itself is not in NP . No known NP solution exists for T SP ( NP problems have polynomial time verifiers; in T SP it is not possible to verify a certificate in polynomial time). It remains to show that all L NP reduce to it in polynomial time. We will show this by a polynomial time reduction f from HAMP AT H to T SP , h G, s, t i HAMP AT H iff f ( h G, s, t i ) T SP, where f ( h G, s, t i ) = h G , s, t, m i with G the graph G with a weight of 1 on all of its edges and m the number of nodes in G . Clearly, the reduction runs in polynomial time. We verify the reduction condition by first observing that a Hamiltonian path gives rise to a minimal traveling salesman circuit by the virtue that all Hamiltonian paths in G have the same cost. The converse also holds, if we have a traveling salesman circuit this implies that we have a Hamiltonian path. 2 p. 3/1 Approximation Algorithms Approximation algorithms are an approach to attacking difficult optimization problems. Approximation algorithms are often associated with NP-hard problems. Since it is unlikely that there can ever be efficient (Polynomial Time) exact algorithms solving NP-complete/hard problems, one settles for non-optimal solutions, but requires them to...
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This note was uploaded on 10/03/2011 for the course CSC 544 taught by Professor Staff during the Spring '11 term at Rhode Island.

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20-approximations - Approximations to NP Solutions If we...

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