On the absolute and relative oriented clique problems’ time complexity

Abstract

Given a graph $G=(V\left(G\right),E\left(G\right))$, the size of the largest clique $\omega \left(G\right)$ is always less than or equal to the chromatic number $\chi \left(G\right)$ of $G$. The oriented coloring of an oriented graph $\vec{G}$ assigns colors to the vertices of $\vec{G}$, such that the arcs connecting vertices in different color classes always have the same direction and the smallest number ${\chi }_o\left(\vec{G}\right)$ of colors in an oriented coloring is the oriented chromatic number of $\vec{G}$. Oriented colorings have fundamental implications for homomorphisms of oriented graphs and significant applications in distributed processing and task scheduling. In 2004, Klostermeyer and MacGillivray defined the concept of an “analogue of clique” for oriented coloring in which a subgraph $\overrightarrow{C_{ao}}$ of $\vec{G}$ is an absolute oriented clique if the oriented distance between a pair of vertices of $\overrightarrow{C_{ao}}$ in $\overrightarrow{C_{ao}}$ is at most 2. The authors defined the absolute oriented clique number of $\vec{G}$ as the number of vertices $\left|V\left(\overrightarrow{C_{ao}}\right)\right|={\chi }_o\left(\overrightarrow{C_{ao}}\right)={\omega }_{ao}\left(\vec{G}\right)$ in a maximum absolute oriented clique $\overrightarrow{C_{ao}}$ of $\vec{G}$ satisfying that ${\omega }_{ao}\left(\vec{G}\right)\le {\chi }_o\left(\vec{G}\right)$. Ever since, for 20 years, the time complexity status of the problem of finding this parameter remained open. Recently in 2016, Sopena et al. defined the relative oriented clique number ${\omega }_{ro}\left(\vec{G}\right)$ of an oriented graph $\vec{G}$ as the size $\left|R_{ro}\right|$ of a maximum set of vertices $R_{ro}$, such that the oriented distance between a pair of vertices of $R_{ro}$ in $\vec{G}$ is at most 2. For every oriented graph $\vec{G}$, ${\omega }_{ao}\left(\vec{G}\right)\le {\omega }_{ro}\left(\vec{G}\right)\le {\chi }_o\left(\vec{G}\right)$. In this paper we classify the Klostermeyer and MacGillivray’s decision open problem proving that given an oriented graph $\vec{G}$ and a positive integer $k$ it is $NP$-complete to decide whether ${\omega }_{ao}\left(\vec{G}\right)\ge k$, even if the underlying graph $G$ is bipartite. We analyze the approximation and parameterization behaviors of the both associated optimization problems: absolute oriented clique and relative oriented clique. We prove that if $P\ne NP$, then the polynomial approximation ratio of both problems is greater than $n^{1-ϵ}$ for all $ϵ>0$. If the maximum degree $\Delta$ of the underlying graph $G$ is bounded, then there is a polynomial-time $\lfloor \frac{3}{1+\Delta +\lfloor \frac{{\Delta }^2}{2}\rfloor }\rfloor$-approximation algorithm and an $FPT$ parameterized algorithm in the parameter $\Delta$ for both absolute oriented clique and relative oriented clique is devised. We prove that absolute oriented clique is $W\left[2\right]$ and is $W\left[1\right]$-hard and that relative oriented clique is $W\left[1\right]$-complete in the parameter $k$ of the size of the solution. We proved a general result that if the absolute oriented clique or the relative oriented clique sizes are bounded for a given class of graphs, then there is a polynomial time algorithm for the corresponding problem in this class. This result implies that both problems are polynomial in the class of oriented planar graphs.