Abstract
We give an \(O(g^{1/2} n^{3/2} + g^{3/2} n^{1/2})\)-size extended formulation for the spanning tree polytope of an n-vertex graph embedded in a surface of genus g, improving on the known \(O(n^2 + g n)\)-size extended formulations following from Wong (Proceedings of 1980 IEEE International Conference on Circuits and Computers, pp 149–152, 1980) and Martin (Oper Res Lett 10:119–128, 1991).
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Acknowledgements
We thank William Cook for asking about the extension complexity of the spanning tree polytope of graphs on the torus, which prompted this work. We thank also Jean Cardinal and Hans Raj Tiwary for interesting discussions on the topic, and the two referees for rightfully asking us to shorten the paper. S. Fiorini and T. Huynh are supported by ERC grant FOREFRONT (Grant Agreement No. 615640) funded by the European Research Council under the EU’s 7th Framework Programme (FP7/2007-2013). S. Fiorini also acknowledges support from ARC Grant AUWB-2012-12/17-ULB2 COPHYMA funded by the French community of Belgium. S. Fiorini and K. Pashkovich are grateful for the support of the Hausdorff Institute for Mathematics in Bonn during the trimester program Combinatorial Optimization. G. Joret acknowledges support from an ARC grant from the Wallonia-Brussels Federation of Belgium.
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Fiorini, S., Huynh, T., Joret, G. et al. Smaller Extended Formulations for the Spanning Tree Polytope of Bounded-Genus Graphs. Discrete Comput Geom 57, 757–761 (2017). https://doi.org/10.1007/s00454-016-9852-9
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DOI: https://doi.org/10.1007/s00454-016-9852-9