Designing small keyboards is hard

We study the problem of placing symbols of an alphabet onto the minimum number of keys of a small keyboard so that any word of a given dictionary can be recognized univoquely only by looking at the corresponding sequence of keys. This problem is motivated by the design of small keyboards for mobile devices. We show that the problem is hard in general, and NP-complete even if we only wish to decide whether two keys are sufficient. We also consider two variants of the problem. In the first one, symbols on a key must be contiguous in an ordered alphabet. In the second variant, a well-chosen measure of ambiguity in the recognition of the words is minimized given the number of keys. Hardness and approximability results are given.