Some aspects of studying an optimization or decision problem in different computational models

  • In this paper we want to discuss some of the features coming up when analyzing a problem in different complexity theoretic frameworks. The focus will be on two problems. The first is related to mathematical optimization. We consider the quadratic programming problem of minimizing a quadratic polynomial on a polyhedron. We discuss how the complexity of this problem might change if we consider real data together with an algebraic model of computation (the Blum–Shub–Smale model) instead of rational inputs together with the Turing machine model. The results obtained will lead us to the second problem; it deals with the intrinsic structure of complexity classes in models over real- or algebraically closed fields. A classical theorem by Ladner for the Turing model is examined in these different frameworks. Both examples serve well for working out in how far different approaches to the same problem might shed light upon each other. In some cases this will lead to quite diverse results with respect to the different models. On the other hand,In this paper we want to discuss some of the features coming up when analyzing a problem in different complexity theoretic frameworks. The focus will be on two problems. The first is related to mathematical optimization. We consider the quadratic programming problem of minimizing a quadratic polynomial on a polyhedron. We discuss how the complexity of this problem might change if we consider real data together with an algebraic model of computation (the Blum–Shub–Smale model) instead of rational inputs together with the Turing machine model. The results obtained will lead us to the second problem; it deals with the intrinsic structure of complexity classes in models over real- or algebraically closed fields. A classical theorem by Ladner for the Turing model is examined in these different frameworks. Both examples serve well for working out in how far different approaches to the same problem might shed light upon each other. In some cases this will lead to quite diverse results with respect to the different models. On the other hand, for some problems the more general approach can also give a unifying idea why results hold true in several frameworks. The paper is of tutorial character in that it collects some results into the above direction obtained previously.show moreshow less

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Metadaten
Author: Klaus MeerGND, Gerhard Wilhelm Weber
URL:http://www.sciencedirect.com/science/article/pii/S0377221702002692
DOI:https://doi.org/10.1016/S0377-2217(02)00269-2
ISSN:0377-2217
Title of the source (English):European Journal of Operational Research
Document Type:Scientific journal article peer-reviewed
Language:English
Year of publication:2002
Tag:Algebraic models of computation; Complexity theory Saturation; Linear and quadratic programming; Structure of complexity classes
Volume/Year:143
Issue number:2
First Page:406
Last Page:418
Faculty/Chair:Fakultät 1 MINT - Mathematik, Informatik, Physik, Elektro- und Informationstechnik / FG Theoretische Informatik
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