2026-09-10T09:45:02Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:27790 2026-09-09T12:19:39Z ddc:004 open_access

Testing the Independent Set Property in Hypergraphs Grigorescu, Elena Nasa, Shreya Seth, Cameron independent set property testing hypergraph container method hypergraph The optimal sample complexity of testing if an n-vertex graph has an independent set of size ρ n, or is ε-far from having an independent set of size ρ n, was established to be Õ(ρ³/ε²), in a notable result by Blais and Seth (SICOMP 2025). In contrast, for q-uniform hypergraphs, there is a significant gap between the best known upper and lower bounds, and there has been no progress on the problem for the last two decades. In this work, we prove a new upper bound of Õ(qρ^{2q-3}/{ε²(q-2)!²}) on the sample complexity of testing the ρ-independent set property. The previous best known upper bound was Õ(2^q q! ρ^{2q}/ε³), due to Langberg (RANDOM 2004). This establishes the optimal dependence on ε and gives an exponential improvement in the dependence on q. We prove our result via a new application of the hypergraph container method. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Elena Grigorescu and Shreya Nasa and Cameron Seth 2026 Is Part Of LIPIcs, Volume 392, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.APPROX/RANDOM.2026.73 urn:nbn:de:0030-drops-277907 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.73 eng https://creativecommons.org/licenses/by/4.0/legalcode