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

This e-book is meant for use as a textbook for graduate scholars learning theoretical desktop technological know-how. it could possibly even be used as a reference publication for researchers within the sector of layout and research of approximation algorithms. layout and research of Approximation Algorithms is a graduate path in theoretical machine technology taught greatly within the universities, either within the usa and in another country. There are, even though, only a few textbooks to be had for this direction. between these in the market, such a lot books stick with a problem-oriented layout; that's, they gathered many vital combinatorial optimization difficulties and their approximation algorithms, and arranged them in response to the kinds, or functions, of difficulties, comparable to geometric-type difficulties, algebraic-type difficulties, and so forth. Such association of fabrics could be handy for a researcher to appear for the issues and algorithms on the topic of his/her paintings, yet is tough for a scholar to seize the information underlying a number of the algorithms. within the new booklet proposed right here, we stick to a extra based, technique-oriented presentation. We arrange approximation algorithms into assorted chapters, in response to the layout thoughts for the algorithms, in order that the reader can research approximation algorithms of an identical nature jointly. It is helping the reader to raised comprehend the layout and research thoughts for approximation algorithms, and likewise is helping the trainer to give the information and methods of approximation algorithms in a extra unified way.

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Show that the following problems are NP-complete: (a) N OT-A LL -E QUAL 3-S AT: Given a 3-CNF F , determine whether F has an assignment which assigns, for each clause C, value 1 to a literal in C and value 0 to another literal in C. (b) O NE - IN -T HREE 3-S AT: Given a 3-CNF F , determine whether F has an assignment which, for each clause C, assigns value 1 to exactly one literal in C. (c) P LANAR 3-S AT: Given a planar 3-CNF formula F , determine whether F is satisﬁable. 20 A subset D of vertices in a graph G = (V, E) is called a dominating set if every vertex v ∈ V either is in D or is adjacent to a vertex in D.

B) The total distance of P is exactly twice that of T , and so at most twice that of the optimal solution. G: (a) the minimum spanning tree; (b) the Euler tour; and (c) the shortcut. (c) By the triangle inequality, the total distance of the shortcut Q is no greater than that of tour P . Christoﬁdes [1976] introduced a new idea into this approximation algorithm and improved the performance ratio to 3/2. This new idea requires another basic graph algorithm: Minimum Perfect Matching Algorithm: Given a complete graph G of an even number of vertices and a distance function d on edges, this algorithm ﬁnds a perfect matching with the minimum total distance.

5 that if an NP-complete problem is in P, then P = NP. Thus, in view of our inability to solve the P vs. NP question, the next best way to prove a problem intractable is to show that it is NP-complete (and so it is most likely not in P, unless P = NP). Among all problems, S AT was the ﬁrst problem proved NP-complete. It was proved by Cook [1971], who showed that for any polynomial-time nondeterministic Turing machine, its computation on any input x can be encoded by a Boolean formula φx of polynomially bounded length such that the formula φx is satisﬁable if and only if M accepts x.