The discrete basis problem

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The discrete basis problem (2005)

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by Pauli Miettinen

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41 - 13 self

BibTeX

@MISC{Miettinen05thediscrete,
    author = {Pauli Miettinen},
    title = {The discrete basis problem},
    year = {2005}
}

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Abstract

We consider the Discrete Basis Problem, which can be described as follows: given a collection of Boolean vectors find a collection of k Boolean basis vectors such that the original vectors can be represented using disjunctions of these basis vectors. We show that the decision version of this problem is NP-complete and that the optimization version cannot be approximated within any finite ratio. We also study two variations of this problem, where the Boolean basis vectors must be mutually otrhogonal. We show that the other variation is closely related with the well-known Metric k-median Problem in Boolean space. To solve these problems, two algorithms will be presented. One is designed for the variations mentioned above, and it is solely based on solving the k-median problem, while another is a heuristic intended to solve the general Discrete Basis Problem. We will also study the results of extensive experiments made with these two algorithms with both synthetic and real-world data. The results are twofold: with the synthetic data, the algorithms did rather well, but with the real-world data the results were not as good.

Keyphrases

discrete basis problem boolean basis vector real-world data finite ratio general discrete basis problem synthetic data decision version optimization version cannot basis vector original vector well-known metric k-median problem boolean space k-median problem extensive experiment boolean vector

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