On percolation and -hardness

We consider the robustness of computational hardness of problems whose input is obtained by applying independent random deletions to worst-case instances. For some classical NPhard problems on graphs, such as Coloring, Vertex-Cover, and Hamiltonicity, we examine the complexity of these problems when edges (or vertices) of an arbitrary graph are deleted independently with probability 1−p > 0. We prove that for n-vertex graphs, these problems remain as hard as in the worst-case, as long as p > 1 n1− for arbitrary ∈ (0, 1), unless NP ⊆ BPP. We also prove hardness results for Constraint Satisfaction Problems, where random deletions are applied to clauses or variables, as well as the Subset-Sum problem, where items of a given instance are deleted at random. ∗Department of Computer Science, Cornell University, Ithaca, NY, USA. Email: daniel.reichman@gmail.com. Supported in part by NSF grants IIS-0911036 and CCF-1214844, AFOSR grant FA9550-08-1-0266, and ARO grant W911NF-14-1-0017 †Courant Institute of Mathematical Sciences, New York University. Research supported by NSF grants CCF 1422159, 1061938, 0832795 and Simons Collaboration on Algorithms and Geometry grant.

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