Interval number for tournaments in P3-convexity
We study the complexity of determining the interval numbers of tournaments in the $\overrightarrow{P_3}$ and $\overrightarrow{P_3^*}$ convexities, denoted by $\overrightarrow{\mathrm{in}}_{P_3}(T)$ and $\overrightarrow{\mathrm{in}}_{P_3^*}(T)$ on a tournament $T$. For each $\overrightarrow{\mathcal{X}} \in \{\overrightarrow{P_3}, \overrightarrow{P_3^*}\}$, we show that determining whether $\overrightarrow{\mathrm{in}}_{\mathcal{X}}(T) \leq k$ is W[2]-complete when parameterized by $k$. Moreover, under ETH, we show that there is no parameterized algorithm for that problem with running time $f(k)\, n^{o(k)}$ on an $n$-vertex tournament, where $f$ is any computable function. For the $\overrightarrow{P_3}$-convexity, we also show that $\overrightarrow{\mathrm{in}}_{P_3}(T) = \mathcal{O}(\log n)$, which yields a simple quasi-polynomial $n^{\mathcal{O}(\log n)}$ brute-force algorithm. On the other hand, under ETH, we show that the problem is NP-intermediate, that is, it is neither NP-hard nor in P. For the $\overrightarrow{P_3^*}$-convexity, the same brute force algorithm is not quasi-polynomial, since we present a family of instances with $\overrightarrow{\mathrm{in}}_{P_3^*}(T) = Θ(n)$. We conjecture that this problem is NP-complete.