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 \(\sqrt{2}\) 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.
Work supported by NSF Grant CCF-0728640.
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Chakrabarty, D., Devanur, N.R., Vazirani, V.V. (2008). New Geometry-Inspired Relaxations and Algorithms for the Metric Steiner Tree Problem. In: Lodi, A., Panconesi, A., Rinaldi, G. (eds) Integer Programming and Combinatorial Optimization. IPCO 2008. Lecture Notes in Computer Science, vol 5035. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-68891-4_24
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