Differential approximation results for the Steiner tree problem

Marc Demange (1, 2) , Jérôme Monnot (3) , Vangelis Th. Paschos (3)

1 CERMSEM - CEntre de Recherche en Mathématiques, Statistique et Économie Mathématique
2 ESSEC Business School
3 LAMSADE - Laboratoire d'analyse et modélisation de systèmes pour l'aide à la décision

We study the approximability of three versions of the Steiner tree problem. For the first one where the input graph is only supposed connected, we show that it is not approximable within better than |V \setminus N|^{-r} for any r in [0,1], where V and N are the vertex-set of the input graph and the set of terminal vertices, respectively. For the second of the Steiner tree versions considered, the one where the input graph is supposed complete and the edge distances are arbitrary, we prove that it can be differentially approximated within 1/2. For the third one defined on complete graphs with edge distances 1 or 2, we show that it is differentially approximable within 0.82. Also, we extend the result of (M. Bern and P. Plassmann, The Steiner problem with edge lengths 1 and 2, Inform. Process. Lett. 32, 1989), we show that the Steiner tree problem with edge lengths 1 and 2 is MaxSNP-complete even in the case where |V|< r|N|, for any r > 0. This allows us to finally show that Steiner tree problem with edge lengths 1 and 2 cannot by approximated by polynomial time differential approximation schemata.

Mots clés: Approximate algorithms; Differential ratio; Steiner tree; Performance ratio; Analysis of Algorithms; Reductions

Domaines: Mathématique discrète [cs.DM]

Marc Demange, Jérôme Monnot, Vangelis Th. Paschos. Differential approximation results for the Steiner tree problem. Applied Mathematics Letters, 2003, 16, pp.733-739. ⟨hal-00004058⟩