We study the problem of estimating the number of edges in a graph with access to only an independent set oracle. Independent set queries draw motivation from group testing and have applications to the complexity of decision versus counting problems. We give two algorithms to estimate the number of edges in an n-vertex graph: one that uses only polylog(n) bipartite independent set queries, and another one that uses n^{2/3} polylog(n) independent set queries.
@InProceedings{beame_et_al:LIPIcs.ITCS.2018.38, author = {Beame, Paul and Har-Peled, Sariel and Natarajan Ramamoorthy, Sivaramakrishnan and Rashtchian, Cyrus and Sinha, Makrand}, title = {{Edge Estimation with Independent Set Oracles}}, booktitle = {9th Innovations in Theoretical Computer Science Conference (ITCS 2018)}, pages = {38:1--38:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-060-6}, ISSN = {1868-8969}, year = {2018}, volume = {94}, editor = {Karlin, Anna R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2018.38}, URN = {urn:nbn:de:0030-drops-83552}, doi = {10.4230/LIPIcs.ITCS.2018.38}, annote = {Keywords: Approximate Counting, Independent Set Queries, Sparsification, Importance Sampling} }
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