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The Nature of Computation$
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Cristopher Moore and Stephan Mertens

Print publication date: 2011

Print ISBN-13: 9780199233212

Published to Oxford Scholarship Online: December 2013

DOI: 10.1093/acprof:oso/9780199233212.001.0001

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Optimization and Approximation

Optimization and Approximation

(p.351) Chapter 9 Optimization and Approximation
The Nature of Computation

Cristopher Moore

Stephan Mertens

Oxford University Press

This chapter focuses on the relationships between decision problems and their optimisation versions. It shows that, for most problems, the optimal solution can be realised in polynomial time if and only if we can tell whether a solution with a given quality exists. It then explores approximation algorithms for the traveling salesman problem and considers some large families of optimisation problems that can be solved in polynomial time, including linear programming and semidefinite programming. It demonstrates that the duality between MAX FLOW and MIN CUT is no accident — that linear programming problems come in pairs. In addition, the chapter looks at integer linear programming, which can be solved in polynomial time before concluding with a discussion of optimisation problems from a practical point of view.

Keywords:   decision problems, optimisation, polynomial time, approximation, algorithms, traveling salesman problem, linear programming, semidefinite programming, duality, integer linear programming

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