🤖 AI Summary
This work addresses the exact recovery of planted cliques in semi-random graphs. We consider the Feige–Kilian-type semi-random model under adversarial edge perturbations and propose a novel semidefinite programming (SDP) algorithm. Our method introduces a tailored rectangular constraint rounding scheme and develops refined structural analysis techniques for the SDP solution. We prove that the algorithm achieves exact recovery of the planted clique with high probability (w.h.p.), attaining the same detection threshold as the optimal result in purely random graphs. Moreover, it resolves a key special case of the DkSReg model studied in KL20. To our knowledge, this is the first work to establish tight theoretical guarantees for exact recovery in a non-uniform semi-random setting, significantly extending both the applicability and robustness of existing SDP-based approaches.
📝 Abstract
In this short 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 a special case of the D$k$SReg$(n, k, d, delta, gamma)$ model introduced in [KL20], when the planted subgraph $G[S]$ is a clique.