$k$-Universality of Regular Languages Revisited
Duncan Adamson, Pamela Fleischmann, Annika Huch, Tore Koß, Florin Manea
TL;DR
The paper investigates the k-Existence Subsequence Universality problem ($k$-ESU) for regular languages, where given a language $L$ via an automaton or regex, one asks whether $L$ contains a word with universality index at least $k$. It provides a detailed parameterized complexity analysis with respect to three parameters: the number of states $n$, the alphabet size $ obreak[0]$,$ obreak[0]$ obreak[0]$ $ obreak[0]$Sigma$, and the length of the regular expression, employing a graph-theoretic toolkit (notions like SCCs and arch factorisation) to identify max-universality words along constrained walks. The main results include a faster FPT algorithm in $ obreak[0]$ for constant alphabets, the first FPT algorithm in $n$ (linear for constant $n$), and an $O(n 2^ obreak[0])$-time FPT algorithm for regex-length $n$, along with NP-hardness for $k=1$ in regex representations and ETH-based lower bounds showing tightness. These findings extend and sharpen prior work on $k$-ESU and contribute a comprehensive fine-grained complexity picture for regular languages, while also reinforcing the polynomial-time tractability of the related $k$-USU problem.
Abstract
A subsequence of a word $w$ is a word $u$ such that $u = w[i_1] w[i_2] \cdots w[i_k]$, for some set of indices $1 \leq i_1 < i_2 < \dots < i_k \leq \vert w \vert$. A word $w$ is \emph{$k$-subsequence universal} over an alphabet $Σ$ if every word over $Σ$ up to length $k$ appears in $w$ as a subsequence. In this paper, we revisit the problem $k$-ESU of deciding, for a given integer $k$, whether a regular language, given either as nondeterministic finite automaton or as a regular expression, contains a $k$-universal word. [Adamson et al., ISAAC 2023] showed that this problem is NP-hard, even in the case when $k=1$, and an FPT algorithm w.r.t. the size of the input alphabet was given. In this paper, we improve the aforementioned algorithmic result and complete the analysis of this problem w.r.t. other parameters. That is, we propose a more efficient FPT algorithm for $k$-ESU, with respect to the size of the input alphabet, and propose new FPT algorithms for this problem w.r.t.~the number of states of the input automaton and the length of the input regular expression. We also discuss corresponding lower bounds. Our results significantly improve the understanding of this problem.