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Embedding Complexity and Discrete Optimization I: A New Divide and Conquer Approach to Discrete Optimization
D. Cieslik1, A. Dress2, K.T. Huber3, and V. Moulton4
1Institute of Mathematics and Computer Science, University of Greifswald, 17487 Greifswald Germany
cieslik@mail.uni-greifswald.de
2FSPM-Strukturbildungsprozesse, University of Bielefeld, 33501 Bielefeld, Germany
dress@mathematik.uni-bielefeld.de
3Department of Biometry and Informatics, Swedish University of Agricultural Sciences, 75007 Uppsala, Sweden
katharina.huber@bi.slu.se
4Linnaeus Center for Bioinformatics, Uppsala University, BMC, Box 598, 75124 Uppsala Sweden
vincent.moulton@lcb.uu.se
Annals of Combinatorics 6 (3) p.257-273 September, 2002
AMS Subject Classification: 90C60, 49L99, 05C99, 90B99
Abstract:
In this paper, we introduce a new and quite natural way of analyzing instances of discrete optimization problems in terms of what we call the embedding complexity of an associated more or (sometimes also) less canonical embedding of the (generally vast) solution space R of a given problem into a product of (generally many small) factor sets so that the score of a solution , interpreted as an element , can be computed additively by summing over the local scores of all of its components , for some appropriate score functions defined on the various factor sets . This concept arises naturally within the context of a general Divide & Conquer strategy for solving discrete optimization problems using dynamic-programming procedures. Relations with the treewidth concept and with linear-programming approaches to discrete optimization, as well as ways to exploit our approach to computing Boltzmann statistics for discrete optimization problems are indicated. In further papers, we will discuss these relations in more detail, concepts of data representation like e.g., PQ-trees, and we will apply the ideas developed here towards designing schemes for solving specific optimization problems, e.g., the Steiner problem for graphs.
Keywords: Optimization, discrete optimization, Divide & Conquer, dynamic programming, dynamic-programming schemes, algorithmic complexity, computational complexity, embedding complexity, treewidth, LP, linear programming, ILP, integer linear programming, Boltzmann statistics (for discrete optimization problems)

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