We give the first constant-factor approximation algorithm for quasi-bipartite instances of Directed Steiner Tree on graphs that exclude fixed minors. In particular, for K_r-minor-free graphs our approximation guarantee is O(r⋅√(log r)) and, further, for planar graphs our approximation guarantee is 20. Our algorithm uses the primal-dual scheme. We employ a more involved method of determining when to buy an edge while raising dual variables since, as we show, the natural primal-dual scheme fails to raise enough dual value to pay for the purchased solution. As a consequence, we also demonstrate integrality gap upper bounds on the standard cut-based linear programming relaxation for the Directed Steiner Tree instances we consider.
@InProceedings{friggstad_et_al:LIPIcs.APPROX/RANDOM.2023.13, author = {Friggstad, Zachary and Mousavi, Ramin}, title = {{A Constant-Factor Approximation for Quasi-Bipartite Directed Steiner Tree on Minor-Free Graphs}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)}, pages = {13:1--13:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-296-9}, ISSN = {1868-8969}, year = {2023}, volume = {275}, editor = {Megow, Nicole and Smith, Adam}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.13}, URN = {urn:nbn:de:0030-drops-188389}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2023.13}, annote = {Keywords: Directed Steiner tree, Combinatorial optimization, approximation algorithms} }
Feedback for Dagstuhl Publishing