By Vijay V. Vazirani

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Covering the elemental concepts utilized in the most recent study paintings, the writer consolidates growth made to date, together with a few very contemporary and promising effects, and conveys the wonder and pleasure of labor within the box. He supplies transparent, lucid causes of key effects and ideas, with intuitive proofs, and offers severe examples and various illustrations to assist elucidate the algorithms. the various effects awarded were simplified and new insights supplied. Of curiosity to theoretical machine scientists, operations researchers, and discrete mathematicians.

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This problem is NP-hard. Perhaps the first algorithm that comes to mind for finding a short superstring is the following greedy algorithm. Define the overlap of two strings s, t E E* as the maximum length of a suffix of s that is also a prefix oft. The algorithm maintains a set of strings T; initially T = S. At each step, the algorithm selects from T two strings that have maximum overlap and replaces them with the string obtained by overlapping them as much as possible. After n - 1 steps, T will contain a single string.

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 , ... , Vk_ 1 : each edge corresponds to the set it comes out of in the rooted tree. 2 The minimum k-cut problem 43 Suppose edge (u, v) E B' corresponds to set Vi in this manner. The weight of a minimum u-v cut in G is w' (u, v). Since Ai is a u-v cut in G, w(Ai) ~ w'(u,v).

Today, this problem occupies a central place in the field of approximation algorithms. The problem has a wide range of applications, all the way from finding minimum length interconnection of terminals in VLSI design to constructing phylogeny trees in computational biology. This problem and its generalizations will be studied extensively in this book, see Chapters 22 and 23. 1 (Steiner tree) Given an undirected graph G = (V, E) with nonnegative edge costs and whose vertices are partitioned into two sets, required and Steiner, find a minimum cost tree in G that contains all the required vertices and any subset of the Steiner vertices.

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