By Vijay V. Vazirani

**Uploader's Note**: Ripped from SpringerLink.

Covering the elemental recommendations utilized in the newest learn paintings, the writer consolidates growth made to this point, together with a few very fresh and promising effects, and conveys the sweetness and pleasure of labor within the box. He supplies transparent, lucid factors of key effects and ideas, with intuitive proofs, and gives serious examples and diverse illustrations to aid elucidate the algorithms. some of the effects awarded were simplified and new insights supplied. Of curiosity to theoretical desktop scientists, operations researchers, and discrete mathematicians.

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**Extra resources for Approximation Algorithms, Corrected Second Printing 2003**

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The next 4 exercises provide background and an algorithm for finding Gomory-Hu trees. 3 Let G = (V, E) be a graph and w : E ~ R+ be an assignment of nonnegative weights to its edges. For u, v E V let f( u, v) denote the weight of a minimum u-v cut in G. 1. Let u, v, w E V, and suppose f(u, v) ~ f(u, w) ~ f(v, w). , the two smaller numbers are equal. 2. Show that among the (~) values f(u, v), for all pairs u, v E V, there are at most n - 1 distinct values. 3. Show that for u, v, w E V, f(u,v) 2: min{f(u,w),f(w,v)}.

This path is also a spanning tree 0 on R. Hence, the MST on R has cost at most 2 ·OPT. 3 gives a straightforward factor 2 algorithm for the metric Steiner tree problem: simply find an MST on the set of required vertices. As in the case of set cover, the "correct" way of viewing this algorithm is in the setting of LP-duality theory. In Chapters 22 and 23 we will see that LP-duality provides the lower bound on which this algorithm is based and also helps solve generalizations of this problem. 4 For a tight example, consider a graph with n required vertices and one Steiner vertex.

The k- 1 cuts are identified as follows. Let B be the set of edges of T that connect across two of the sets V1 , V2 , ... , Vk. Consider the graph on vertex set V and edge set B, and shrink each ofthe sets V1 , V2 , ... , Vk to a single vertex. This shrunk graph must be connected (since Twas connected). Throw edges away until a tree remains. Let B' ~ B be the left over edges, IB'I = k- 1. The edges of B' define the required k - 1 cuts. Next, root this tree at Vk (recall that Ak was assumed to be the heaviest cut among the cuts Ai)· This helps in defining a correspondence between the edges in B' and the sets V1, V2 , ...