Exact recovery of Planted Cliques in Semi-random graphs
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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 [Ste17, MMT20]. 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., 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 DkSReg(n, k, d, δ, γ) and DkSReg(n, k, d, δ, d', λ) models introduced in [KL20], when the planted subgraph 𝒢[𝒮] is a clique instead of an arbitrary d-regular graph.
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