Guruswami and Sinop give a O(1/delta) approximation guarantee for the non-uniform Sparsest Cut problem by solving O(r)-level Lasserre semidefinite constraints, provided that the generalized eigenvalues of the Laplacians of the cost and demand graphs satisfy a certain spectral condition, namely, the (r+1)-th generalized eigenvalue is at least OPT/(1-delta). Their key idea is a rounding technique that first maps a vector-valued solution to [0,1] using appropriately scaled projections onto Lasserre vectors. In this paper, we show that similar projections and analysis can be obtained using only l_2^2 triangle inequality constraints. This results in a O(r/delta^2) approximation guarantee for the non-uniform Sparsest Cut problem by adding only l_2^2 triangle inequality constraints to the usual semidefinite program, provided that the same spectral condition, the (r+1)-th generalized eigenvalue is at least OPT/(1-delta), holds.
@InProceedings{deshpande_et_al:LIPIcs.APPROX-RANDOM.2014.105, author = {Deshpande, Amit and Venkat, Rakesh}, title = {{Guruswami-Sinop Rounding without Higher Level Lasserre}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)}, pages = {105--114}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-74-3}, ISSN = {1868-8969}, year = {2014}, volume = {28}, editor = {Jansen, Klaus and Rolim, Jos\'{e} and Devanur, Nikhil R. and Moore, Cristopher}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2014.105}, URN = {urn:nbn:de:0030-drops-46911}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2014.105}, annote = {Keywords: Sparsest Cut, Lasserre Hierarchy, Metric embeddings} }
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