We additionally say that a class \(\textbf{C}\) is closed under \(\leq_P\) if \(Y \in \textbf{C}\) and \(X \leq_P Y\) implies \(X \in \textbf{C}\). [16] A problem \(Y\) is said to be hard for a class \(\textbf{C}\) if \(X \leq_P Y\) for all \(X \in \textbf{C}\). Finally \(Y\) is said to be complete for \(\textbf{C}\) if it is both hard for \(\textbf{C}\) and also a member of \(\textbf{C}\). e. so-called NP-complete problems. A canonical example of such a problem is a time-bounded variant of the