Design and Analysis of Approximation Algorithms by Ding-Zhu Du, Ker-I Ko, Xiaodong Hu

By Ding-Zhu Du, Ker-I Ko, Xiaodong Hu

This ebook is meant for use as a textbook for graduate scholars learning theoretical computing device technology. it could actually even be used as a reference booklet for researchers within the sector of layout and research of approximation algorithms. layout and research of Approximation Algorithms is a graduate direction in theoretical computing device technology taught broadly within the universities, either within the usa and in another country. There are, besides the fact that, only a few textbooks on hand for this path. between these out there, such a lot books stick with a problem-oriented layout; that's, they accumulated many vital combinatorial optimization difficulties and their approximation algorithms, and arranged them in keeping with the kinds, or purposes, of difficulties, resembling geometric-type difficulties, algebraic-type difficulties, and so on. Such association of fabrics could be handy for a researcher to seem for the issues and algorithms concerning his/her paintings, yet is hard for a scholar to seize the guidelines underlying a few of the algorithms. within the new publication proposed the following, we keep on with a extra dependent, technique-oriented presentation. We set up approximation algorithms into diversified chapters, in line with the layout concepts for the algorithms, in order that the reader can learn approximation algorithms of a similar nature jointly. It is helping the reader to raised comprehend the layout and research recommendations for approximation algorithms, and in addition is helping the instructor to offer the information and strategies of approximation algorithms in a extra unified way.

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Let φ be a 3-CNF formula that is of the form C1 C2 · · · Cm , where each Cj is a clause with three literals. Assume that φ contains Boolean variables v1 , v2 , . . , vn . We are going to define a list of 2n + 2m integers c1 , c2 , . . , c2n+2m, plus an integer K. All integers ci and the integer K are of value between 0 and 10n+m . These integers will satisfy the following property: 2n+2m φ is satisfiable ⇐⇒ ∃ x1 , x2 , . . , x2n+2m ∈ {0, 1} ci xi = K. 4) i=1 Now, let S = K, si = ci for i = 1, 2, .

Suppose 1 ≤ k ≤ n; then it means that τ (vk ) = 0, and Cj contains the literal v¯k . Thus, τ satisfies Cj . On the other hand, if n + 1 ≤ k ≤ 2n, then we know that τ (vk−n ) = 1, and Cj contains the literal vk−n ; and so τ also satisfies Cj . 4). Finally, we remark that the above construction of these integers from the formula φ is apparently polynomial-time computable. Thus, this reduction is a polynomialtime reduction. In addition to the above two problems, thousands of problems from many seemingly unrelated areas have been proven to be NP-complete in the past four decades.

4). First, we note that each integer is between 0 and 10n+m , and so it has a unique decimal representation of exactly n + m digits (with possible leading zeroes). We will define each integer digit by digit, with the kth digit indicating the kth most significant digit. First, we define the first n digits of K to be 1 and the last m digits to be 3. That is, K = 11 · · · 11 33 · · · 33 . n m Next, for each i = 1, 2, . . , n, we define the integer ci as follows: The ith digit and the (n + j)th digits, for all 1 ≤ j ≤ m such that Cj contains the literal v¯i, of ci are 1 and all other digits are 0.

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