Lower bounds for approximation algorithms for the Steiner tree problem

Abstract

The Steiner tree problem asks for a shortest subgraph connecting a given set of terminals in a graph. It is known to be APX-complete, which means that no polynomial time approximation scheme can exist for this problem, unless P=NP. Currently, the best approximation algorithm for the Steiner tree problem has a performance ratio of <span class='mathrm'>1.55</span>, whereas the corresponding lower bound is smaller than <span class='mathrm'>1.01</span>. In this paper, we provide for several Steiner tree approximation algorithms lower bounds on their performance ratio that are much larger. For two algorithms that solve the Steiner tree problem on quasi-bipartite instances, we even prove lower bounds that match the upper bounds. Quasi-bipartite instances are of special interest, as currently all known lower bound reductions for the Steiner tree problem in graphs produce such instances.