Computational Considerations for Exclusive \((M, N)\)-Transitive Augmentations

Digraph \(D\) is defined to be exclusive \((M, N)\)-transitive if, for each pair of vertices \(x\) and \(y\), for each \(xy\)-path \(P_1\) of length \(M\), there is an \(xy\)-path \(P_2\) of length \(N\) such that \(P_1 \cap P_2 = \{x, y\}\). It is proved that computation of a minimal edge augmentation to make \(K\) exclusive \((M, N)\)-transitive is NP-hard for \(M > N \geq 2\), even if \(D\) is acyclic. The corresponding decision problems are NP-complete. For \(N = 1\) and \(D = (V, E)\) with \(|V| = n\), an \(O(n^{M+3})\) algorithm to compute the exclusive \((M, 1)\)-transitive closure of an arbitrary digraph is provided.