Unavoidable Sets of Partial Words of Uniform Length

TL;DR

This paper investigates the classification of unavoidable sets of partial words of uniform length over finite alphabets, focusing on the maximum number of holes that can be filled without losing unavoidability, addressing a complex NP-hard problem.

Contribution

It explores the maximum fill-in of holes in unavoidable sets of partial words of fixed length, advancing understanding of their structural properties and classification.

Findings

NP-hardness of avoidability decision problem

Maximum holes fillable in unavoidable sets analyzed

Insights into structural limits of partial words

Abstract

A set X of partial words over a finite alphabet A is called unavoidable if every two-sided infinite word over A has a factor compatible with an element of X. Unlike the case of a set of words without holes, the problem of deciding whether or not a given finite set of n partial words over a k-letter alphabet is avoidable is NP-hard, even when we restrict to a set of partial words of uniform length. So classifying such sets, with parameters k and n, as avoidable or unavoidable becomes an interesting problem. In this paper, we work towards this classification problem by investigating the maximum number of holes we can fill in unavoidable sets of partial words of uniform length over an alphabet of any fixed size, while maintaining the unavoidability property.