Complexity of Road Coloring with Prescribed Reset Words

TL;DR

This paper investigates the computational complexity of the Road Coloring problem with prescribed reset words, providing a classification for binary words and highlighting differences in strongly connected multigraphs.

Contribution

It offers a complete classification of binary words for the complexity of Road Coloring with prescribed reset words, and explores the problem's complexity in strongly connected multigraphs.

Findings

NP-complete for certain words

Polynomial-time solvable for others

Classification differs for strongly connected graphs

Abstract

By the Road Coloring Theorem (Trahtman, 2008), the edges of any aperiodic directed multigraph with a constant out-degree can be colored such that the resulting automaton admits a reset word. There may also be a need for a particular reset word to be admitted. For certain words it is NP-complete to decide whether there is a suitable coloring of a given multigraph. We present a classification of all words over the binary alphabet that separates such words from those that make the problem solvable in polynomial time. We show that the classification becomes different if we consider only strongly connected multigraphs. In this restricted setting the classification remains incomplete.