$k$-Universality of Regular Languages Revisited

2025-03-24

AI Summary

This paper investigates the *k-Existence of a Universal Word* (k-ESU) problem for regular languages: given a regular language $L$ (specified by an NFA or a regular expression) and an integer $k$, decide whether $L$ contains a *k-universal word*—a word containing every string of length at most $k$ over its alphabet as a subsequence. While k-ESU is NP-hard, this work establishes the first comprehensive parameterized complexity landscape. It introduces improved fixed-parameter tractable (FPT) algorithms: one parameterized by alphabet size $|Sigma|$, running in $2^{O(k|Sigma|)} cdot mathrm{poly}(n)$ time (where $n$ is the number of NFA states), and the first FPT algorithm parameterized by regular expression length $m$, running in $2^{O(km)} cdot mathrm{poly}(|Sigma|)$ time. Matching conditional lower bounds are provided, confirming the optimality of these algorithms under standard complexity assumptions. The results advance the structural understanding of subsequence universality and substantially extend the practical scalability of exact solving.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization
• Machine Learning: Kernel Methods

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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 $Sigma$ if every word over $Sigma$ 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.

Problem

• Decide if a regular language contains a k-universal word.
• Improve FPT algorithms for k-ESU problem.
• Analyze problem complexity with different parameters.

Innovation

• FPT algorithm for alphabet size
• FPT algorithm for state count
• FPT algorithm for expression length