NP-Completeness of Searches for Smallest Possible Feature Sets

In many learning problems, the learning system is presented with values for features that are actually irrelevant to the concept it is trying to learn. The FOCUS algorithm, due to Almuallim and Dietterich, performs an explicit search for the smallest possible input feature set S that permits a consistent mapping from the features in S to the output feature. The FOCUS algorithm can also be seen as an algorithm for learning determinations or functional dependencies, as suggested in [6]. Another algorithm for learning determinations appears in [7]. The FOCUS algorithm has superpolynomial runtime, but Almuallim and Dietterich leave open the question of tractability of the underlying problem. In this paper, the problem is shown to be NP-complete. We also describe briefly some experiments that demonstrate the benefits of determination learning, and show that finding lowest-cardinality determinations is easier in practice than finding minimal determinations.

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