The Hardness of the Functional Orientation 2-Color Problem

TL;DR

This paper investigates the computational complexity of the Functional Orientation 2-Color problem, establishing new hardness results and efficient algorithms for specific graph classes, thereby closing existing gaps in the problem's complexity landscape.

Contribution

It proves NP-completeness for planar graphs of maximum degree 6 and provides a linear time solution for graphs of maximum degree 5, resolving open questions in the problem's complexity.

Findings

NP-complete for planar graphs of maximum degree 6

Linear time algorithm for graphs of maximum degree 5

Problem is always solvable for graphs of maximum degree 5

Abstract

We consider the Functional Orientation 2-Color problem, which was introduced by Valiant in his seminal paper on holographic algorithms [SIAM J. Comput., 37(5), 2008]. For this decision problem, Valiant gave a polynomial time holographic algorithm for planar graphs of maximum degree 3, and showed that the problem is NP-complete for planar graphs of maximum degree 10. A recent result on defective graph coloring by Corr\^ea et al. [Australas. J. Combin., 43, 2009] implies that the problem is already hard for planar graphs of maximum degree 8. Together, these results leave open the hardness question for graphs of maximum degree between 4 and 7. We close this gap by showing that the answer is always yes for arbitrary graphs of maximum degree 5, and that the problem is NP-complete for planar graphs of maximum degree 6. Moreover, for graphs of maximum degree 5, we note that a linear time algorithm for finding a solution exists.