2026-08-17T18:52:10Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:11472 2024-03-06T10:48:07Z ddc:004 open_access

Hardness of FO Model-Checking on Random Graphs Dreier, Jan Rossmanith, Peter random graphs FO model-checking It is known that FO model-checking is fixed-parameter tractable on Erdős - Rényi graphs G(n,p(n)) if the edge-probability p(n) is sufficiently small [Grohe, 2001] (p(n)=O(n^epsilon/n) for every epsilon>0). A natural question to ask is whether this result can be extended to bigger probabilities. We show that for Erdős - Rényi graphs with vertex colors the above stated upper bound by Grohe is the best possible. More specifically, we show that there is no FO model-checking algorithm with average FPT run time on vertex-colored Erdős - Rényi graphs G(n,n^delta/n) (0 < delta < 1) unless AW[*]subseteq FPT/poly. This might be the first result where parameterized average-case intractability of a natural problem with a natural probability distribution is linked to worst-case complexity assumptions. We further provide hardness results for FO model-checking on other random graph models, including G(n,1/2) and Chung-Lu graphs, where our intractability results tightly match known tractability results [E. D. Demaine et al., 2014]. We also provide lower bounds on the size of shallow clique minors in certain Erdős - Rényi and Chung - Lu graphs. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Jan Dreier and Peter Rossmanith 2019 Is Part Of LIPIcs, Volume 148, 14th International Symposium on Parameterized and Exact Computation (IPEC 2019) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.IPEC.2019.11 urn:nbn:de:0030-drops-114721 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2019.11 eng https://creativecommons.org/licenses/by/3.0/legalcode