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

Abstract

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.

Problems on Finite Automata and the Exponential Time Hypothesis

“…In [29], algorithms with running times of O˚p2 q q and of O˚p|Σ| q were given, complemented by proofs that show that there is neither an O˚p2 opqq q-time (with unbounded input alphabet size) nor an O˚pp|Σ|´ q q-time algorithm (for any ą 0) under ETH or SETH, respectively.…”

“…From the reduction presented in ( [29] Theorem 12), we cannot get any lower bounds for bounded input alphabets; the dependency on Σ for the mentioned O˚p2 q q-time algorithm is only linear.…”

“…It was already observed in [29] that the algorithm given there for SW transfers to Q sync -SW, as this is only a breadth-first search algorithm on an auxiliary graph (of exponential size, with vertex set 2 Q ). The PSPACE-hardness proof contained in ([30] Theorem 1.22), based on [31], reduces from DFA-INTERSECTION.…”

“…Notice that in [29], another SETH-based result relating to synchronizing words was derived. Namely, it was shown that (under SETH) there is no algorithm that determines, given a deterministic finite semi-automaton A " pQ, Σ, tµ a | a P Σuq and an integer , whether or not there is a synchronizing word for A, and that runs in time O˚´p|Σ|´εq ¯f or any ε ą 0.…”

“…From these reductions, we can borrow quite a lot of other results from [29], dealing with inapproximability and parameterized intractability.…”