Beyond Worst-Case Analysis: Lecture 7

Scribed by Jeff Xu

In which we discussed planted clique distribution, specifically, we talked about how to find a planted clique in a random graph. We heavily relied upon our material back in lecture 2 and lecture 3 in which we covered the upper bound certificate for max clique in {G_{n,\frac{1}{2}}}. At the end of this class, we wrapped up this topic and started the topic of {k}-SAT.

1. Planted Clique

To start with, we describe a distribution of graphs with a planted clique. Suppose that we sample {G} from {G_{n,\frac{1}{2}}} and we want to modify {G} s.t. it has a size {k} clique, i.e., we have a clique {S \subseteq V} with {\left|S\right|=k}. The following code describes a sampler for the distribution.

  • {G \leftarrow G_{n,\frac{1}{2}}}
  • Pick a subset of vertices {S} from {V} s.t. {|S|=k}
  • Independently for each pair {\{u,v\}}, make {\{u,v\}} an edge with probability
    • {1} if {u,v \in S}
    • {\frac 12} otherwise

Note: We are only interested in the case {k \geq 2\log n}, which is the case in which the planted clique is, with high probability, larger than any pre-existing clique

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