Exact recovery of planted cliques in semi-random graphs
Yash Khanna
TL;DR
This work addresses exact recovery of planted cliques in a semi-random graph model that blends a planted clique, random cross edges, sparse outside structures, and monotone adversarial deletions. It introduces an SDP-based relaxation and a rounding scheme that concentrates mass on the planted clique, enabling exact recovery with high probability under a broad but explicit parameter regime. The recovery procedure combines a mass-based SDP rounding step to extract a core clique subset with a subsequent greedy completion, and its robustness to monotone deletions distinguishes it from many spectral or combinatorial approaches. The results extend planted clique and semi-random analysis, offering a pathway to exact recovery in robust graph settings and suggesting future work on even more general semi-random models.
Abstract
In this paper, we study the Planted Clique problem in a semi-random model. Our model is inspired from the Feige-Kilian model [16] which has been studied in many other works [8,11,17,26,35,38] 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 work of Khanna and Louis [25]. As a by-product of our main result, we give an alternate SDP-based rounding algorithm (with similar guarantees) for solving the Planted Clique problem in a random graph.