Hardness of FO Model-Checking on Random Graphs

摘要

It is known that FO model-checking is fixed-parameter tractable on Erd??s - R??nyi graphs G(n,p(n)) if the edge-probability p(n) is sufficiently small [Grohe, 2001] (p(n)=O(n^epsilon) for every epsilon0). A natural question to ask is whether this result can be extended to bigger probabilities. We show that for Erd??s - R??nyi 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 Erd??s - R??nyi graphs G(n,n^delta) (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 Erd??s - R??nyi and Chung - Lu graphs.