New geometry-inspired relaxations and algorithms for the metric Steiner tree problem

Abstract

Determining the integrality gap of the bidirected cut relaxation for the metric Steiner tree problem, and exploiting it algorithmically, is a long-standing open problem. We use geometry to define an LP whose dual is equivalent to this relaxation. This opens up the possibility of using the primal-dual schema in a geometric setting for designing an algorithm for this problem. Using this approach, we obtain a 4/3 factor algorithm and integrality gap bound for the case of quasi-bipartite graphs; the previous best integrality gap upper bound being 3/2 (Rajagopalan and Vazirani in On the bidirected cut relaxation for the metric Steiner tree problem, 1999). We also obtain a factor ${\sqrt{2}}$ strongly polynomial algorithm for this class of graphs. A key difficulty experienced by researchers in working with the bidirected cut relaxation was that any reasonable dual growth procedure produces extremely unwieldy dual solutions. A new algorithmic idea helps finesse this difficulty—that of reducing the cost of certain edges and constructing the dual in this altered instance—and this idea can be extracted into a new technique for running the primal-dual schema in the setting of approximation algorithms.