The document outlines approximation algorithms for NP-hard problems including vertex cover, set cover, and traveling salesman problem (TSP). It discusses why approximation algorithms are useful for intractable but important problems. For vertex cover, it presents a 2-approximation algorithm and proves its performance ratio. For set cover, it presents the greedy algorithm and proves its bound of O(ln|X|). For TSP, it presents an algorithm that returns a tour within a factor of 2 the optimal using minimum spanning trees.

1. Chapter 8 Approximation Algorithms

7. Approximate vertex-cover algorithm The running time of this algorithm is O(E).

11. Figure 6.2 An instance {X, F} of the set covering problem, where X consists of the 12 black points and F = { S 1 , S 2 , S 3 , S 4 , S 5 , S 6 }. A minimum size set cover is C = { S 3 , S 4 , S 5 }. The greedy algorithm produces the set C’ = {S 1 , S 4 , S 5 , S 3 } in order.

17. Thí dụ minh họa giải thuật APPROX-TSP-TOUR

18. The preorder tree walk is not simple tour, since a node be visited many times, but it can be fixed, the tree walk visits the vertices in the order a, b, c, b, h, b, a, d, e, f, e, g, e, d, a. From this order, we can arrive to the hamiltonian cycle H: a, b, c, h, d, e ,f, g, a.

19. The total cost of H is approximately 19.074. An optimal tour H* has the total cost of approximately 14.715. The running time of APPROX-TSP-TOUR is O(E) = O(V 2 ), since the input graph is a complete graph. The optimal tour