Rank-sparsity decomposition for planted quasi clique recovery

Abstract 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 $$\gamma $$ γ -clique is a dense graph with the edge density of at least $$\gamma $$ γ , $$\gamma \in (0, 1]$$ γ ∈ ( 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 have derived a new bound on the norm of the dual matrix that certifies the recovery using $$l_{\infty , 2}$$ l ∞ , 2 norm. We have showed that when certain conditions are met, our convex formulation recovers the planted quasi-clique exactly. The numerical experiments we have performed corroborate our theoretical findings.