The zoo of tree spanner problems

Tree spanner problems have important applications in network design, e.g. in the telecommunications industry. Mathematically, there have been considered quite a number of max-stretch tree spanner problems and of average stretch tree spanner problems.We propose a unified notation for 20 tree spanner problems, which we investigate for graphs with general positive weights, with metric weights, and with unit weights. This covers several prominent problems of combinatorial optimization. Having this notation at hand, we can clearly identify which problems coincide. In the case of unweighted graphs, the formally 20 problems collapse to only five different problems.Moreover, our systematic notation for tree spanner problems enables us to identify a tree spanner problem whose complexity status has not been solved so far. We are able to provide an NP-hardness proof. Furthermore, due to our new notation of tree spanner problems, we are able to detect that an inapproximability result that is due to Galbiati [On min–max cycle bases, in: P. Eades, T. Takaoka (Eds.), ISAAC, Lecture Notes in Computer Science, vol. 2223, Springer, Berlin, 2001, pp. 116–123; On finding cycle bases and fundamental cycle bases with a shortest maximal cycle, Inform. Process. Lett. 88(4) (2003) 155–159] in fact applies to the classical max-stretch tree spanner problem. We conclude that the inapproximability factor for this problem thus is 2-ɛ2-ɛ, instead of only 1+52≈1.618 according to Peleg and Reshef [A variant of the arrow distributed directory with low average complexity, in: J. Wiedermann, P. van Emde Boas, M. Nielsen (Eds.), ICALP, Lecture Notes in Computer Science, vol. 1644, Springer, Berlin, 1999, pp. 615–624].