Consider an Erdos---Renyi random graph in which each edge is present independently with probability $$1/2$$1/2, except for a subset $$\mathsf{C}_N$$CN of the vertices that form a clique (a complete...

In this paper, we study the Planted Clique problem in a semi-random model. Our model is inspired from the Feige-Kilian model [FK01] which has been studied in many other works [FK00, Ste17, MMT20] for a variety of graph problems. Our algorithm and analysis is on similar lines to the one studied for the Densest $k$-subgraph problem in the recent work of Khanna and Louis [KL20]. However since our algorithm fully recovers the planted clique w.h.p. (for a "large" range of input parameters), we require some new ideas. 
As a by-product of our main result, we give an alternate SDP based rounding algorithm (with matching guarantees) for solving the Planted Clique problem in a random graph. Also, we are able to solve special cases of the models introduced for the Densest $k$-subgraph problem in [KL20], when the planted subgraph is a clique instead of an arbitrary $d$-regular graph.

Finding Hidden Cliques of Size $$\sqrt{N/e}$$N/e in Nearly Linear Time

Citations

Exact recovery of Planted Cliques in Semi-random graphs.

References

Exact recovery of Planted Cliques in Semi-random graphs.

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