On Minimum Circular Arrangement

Motivated by a scheduling problem encountered in multicast environments, we study a vertex labelling problem, called Directed Circular Arrangement (DCA), that requires one to find an embedding of a given weighted directed graph into a discrete circle which would minimize the total weighted arc length. Its decision version is already known to be NP-complete when restricted to sparse weighted directed graphs. We prove that the decision version of even un-weighted DCA is NP-complete in case of arbitrary given densities. We also consider complementary version of DCA, called MaxCA. We prove that it is MAX-SNP[π] complete and, therefore, has no PTAS unless P=NP. A similar proof technique shows that DCA is MAX-SNP[π]-hard and hence there is no PTAS for DCA as well.