Titlebook: Combinatorial Optimization; Theory and Algorithm Bernhard Korte,Jens Vygen Textbook 20084th edition Springer-Verlag Berlin Heidelberg 2008

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Approximation Algorithms,In this chapter we introduce the important concept of approximation algorithms. So far we have dealt mostly with polynomially solvable problems. In the remaining chapters we shall indicate some strategies to cope with .-hard combinatorial optimization problems. Here approximation algorithms must be mentioned in the first place.

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The Knapsack Problem,The . and the . discussed in earlier chapters are among the “hardest” problems for which a polynomial-time algorithm is known. In this chapter we deal with the following problem which turns out to be, in a sense, the “easiest” .-hard problem:

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Springer Series in Materials Scienceextend . to the weighted case and shall again obtain an .(. .)-implementation. This algorithm has many applications, some of which are mentioned in the exercises and in Section 12.2. There are two basic formulations of the weighted matching problem:

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Small Organic Molecules on Surfacesare also many important problems for which no polynomial-time algorithm is known. Although we cannot prove that none exists we can show that a polynomial-time algorithm for one “hard” (more precisely: .-hard) problem would imply a polynomialtime algorithm for almost all problems discussed in this book (more precisely: all .-easy problems).