Approximation Algorithms, Corrected Second Printing 2003 - download pdf or read online

By Vijay V. Vazirani

Uploader's Note: Ripped from SpringerLink.

Covering the fundamental recommendations utilized in the most recent learn paintings, the writer consolidates growth made up to now, together with a few very fresh and promising effects, and conveys the sweetness and pleasure of labor within the box. He offers transparent, lucid factors of key effects and concepts, with intuitive proofs, and gives severe examples and various illustrations to assist elucidate the algorithms. a few of the effects offered were simplified and new insights supplied. Of curiosity to theoretical computing device scientists, operations researchers, and discrete mathematicians.

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4. The weight of any edge (a, vA) in G' is defined to be the sum of the weights of edges (a, b) where bE A. Clearly, any cut in G' defines a cut in G. Show that a minimum x-y cut in G' defines a minimum x-y cut in G. 6 Now we are ready to state the Gomory-Hu algorithm. The algorithm maintains a partition of V, (S1, S2, ... St), and a spanning tree T on the vertex set { S 1 , ... , St}. Let w' be the function assigning weights to the edges ofT. Tree T satisfies the following invariant. Invariant: For any edge (Si, SJ) in T there are vertices a and bin Si and SJ respectively, such that w'(Si,SJ) = f(a,b), and the cut defined by edge (Si, Sj) is a minimum a-b cut in G.

Tk- 1 be the degree-weighted functions defined on graphs Go, ... , Gk_ 1 . The vertex cover chosen is C = WoU ... UWk-1· Clearly, V -C = DoU .. UDk. Theorem 2. 7 The layer algorithm achieves an approximation guarantee of factor 2 for the vertex cover problem, assuming arbitrary vertex weights. Proof: We need to show that set C is a vertex cover for G and w(C) < 2 ·OPT. Assume, for contradiction, that Cis not a vertex cover for G. 3 Application to shortest superstring 19 there must be an edge (u, v) with u E Di and v E D1 , for some i, j.

Now, is a clique and, if a peripheral vertex is chosen as the maximal independent set, then the cost of the solution found is 2. 0 a; Next, we will show that 2 is essentially the best approximation factor achievable for the metric k-center problem. Theorem 5. 7 Assuming P =f. NP, there is no polynomial time algorithm achieving a factor of 2- e:, e > 0, for the metric k-center problem. Proof: We will show that such an algorithm can solve the dominating set problem in polynomial time. 6 and involves giving a reduction from the dominating set problem to metric kcenter.

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