A multi-parameter analysis of hard problems on deterministic finite automata

We initiate a multi-parameter analysis of two well-known NP-hard problems on deterministic finite automata (DFAs): the problem of finding a short synchronizing word, and that of finding a DFA on few states consistent with a given sample of the intended language and its complement. For both problems, we study natural parameterizations and classify them with the tools provided by Parameterized Complexity. Somewhat surprisingly, in both cases, rather simple FPT algorithms can be shown to be optimal, mostly assuming the (Strong) Exponential Time Hypothesis.