On exact and approximation algorithms for distinguishing substring selection

The NP-complete DISTINGUISHING SUBSTRING SELECTION problem (DSSS for short) asks, given a set of "good" strings and a, set of "bad" strings, for a solution string which is, with respect to Hamming metric, "away" from the good strings and "close" to the bad strings. Studying the parameterized complexity of DSSS, we show that DSSS is W[1]-hard with respect to its natural parameters. This, in particular, implies that a recently given polynomial-time approximation scheme (PTAS) by Deng et al. cannot be replaced by a so-called efficient polynomial-time approximation scheme (EPTAS) unless an unlikely collapse in parameterized complexity theory occurs. By way of contrast, for a special case of DSSS, we present an exact fixed-parameter algorithm solving the problem efficiently. In this way, we exhibit a sharp border between fixed-parameter tractability and intractability results.