Rank-sparsity decomposition for planted quasi clique recovery

In this paper, we apply the Rank-Sparsity Matrix Decomposition to the planted Maximum Quasi-Clique Problem (MQCP). This problem has the planted Maximum Clique Problem (MCP) as a special case. The maximum clique problem is NP-hard. A Quasi-clique or $\gamma$-clique is a dense graph with the edge density of at least $\gamma$, where $\gamma \in (0, 1]$. The maximum quasi-clique problem seeks to find such a subgraph with the largest cardinality in a given graph. Our method of choice is the low-rank plus sparse matrix splitting technique. We present a theoretical basis for when our convex relaxation problem recovers the planted maximum quasi-clique. We derived a new bound on the norm of the dual matrix that certifies the recovery using $l_{\infty,2} norm. We showed that when certain conditions are met, our convex formulation recovers the planted quasi-clique exactly. The numerical experiments we performed corroborated our theory.