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 being 3/2 [RV99]. We also obtain a factor 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

• Type: Conference
• Year: 2008
• Where: IPCO
• Authors: Deeparnab Chakrabarty, Nikhil R. Devanur, Vijay V. Vazirani
• Keywords: Cut Relaxation, Integrality Gap, IPCO 2008, Optimization, Primal-dual Schema