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An NP-optimization problem NPO is a combinatorial optimization problem with the following additional conditions.

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This implies that the corresponding decision problem is in NP. In computer science, interesting optimization problems usually have the above properties and are therefore NPO problems. A problem is additionally called a P-optimization PO problem, if there exists an algorithm which finds optimal solutions in polynomial time. Often, when dealing with the class NPO, one is interested in optimization problems for which the decision versions are NP-complete.

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Note that hardness relations are always with respect to some reduction. Due to the connection between approximation algorithms and computational optimization problems, reductions which preserve approximation in some respect are for this subject preferred than the usual Turing and Karp reductions.

An example of such a reduction would be the L-reduction. For this reason, optimization problems with NP-complete decision versions are not necessarily called NPO-complete.

Combinatorial optimization

NPO is divided into the following subclasses according to their approximability: [7]. From Wikipedia, the free encyclopedia.

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Retrieved Then divide by two because it does not matter in which direction in time they come after each other: 14! Categories : Combinatorial optimization Computational complexity theory Theoretical computer science.


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It is indeed an excellent introduction, it seems to me, taking the reader from an all but tabula rasa state to a very decent grasp of some serious combinatorial mathematics. I cannot pass up the chance to end this review with a recollection of two professors of mine, both deceased, God rest their souls, whom I idolized in my undergraduate years at UCLA, namely Ernst Straus and Basil Gordon. I was also part of their seminar on transcendental number theory: what a fond memory. Both Straus and Gordon were famous for their acumen in combinatorics, as well as being major players in a number of other areas.

Additionally, they were very kind men, to whom I think back very fondly. The book under review reminds me of them.


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