Finite Automata Intersection Non-Emptiness: Parameterized Complexity Revisited
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The problem DFA-Intersection-Nonemptiness asks if a given number of deterministic automata accept a common word. In general, this problem is PSPACE-complete. Here, we investigate this problem for the subclasses of commutative automata and automata recognizing sparse languages. We show that in both cases DFA-Intersection-Nonemptiness is complete for NP and for the parameterized class W[1], where the number of input automata is the parameter, when the alphabet is fixed. Additionally, we establish the same result for Tables Non-Empty Join, a problem that asks if the join of several tables (possibly containing null values) in a database is non-empty. Lastly, we show that Bounded NFA-Intersection-Nonemptiness, parameterized by the length bound, is W[2]-hard with a variable input alphabet and for nondeterministic automata recognizing finite strictly bounded languages, yielding a variant leaving the realm of W[1].
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