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, \delta, \gamma) and DkSReg(n, k, d, \delta, d', \lambda) models introduced in [KL20], when the planted subgraph \mathcal{G is a clique instead of an arbitrary d-regular graph.
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.
A
FK00,
A
for a variety of graph problems
A
(for a "large" range of input parameters)
R
DkSReg(n, k, d, \delta, \gamma) and DkSReg(n, k, d, \delta, d', \lambda) models introduced
models introduced for the Densest k-subgraph problem
D
\mathcal{G