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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2208.03251 |
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Table of Contents:
- 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 $γ$-clique is a dense graph with the edge density of at least $γ$, where $γ\in (0, 1]$. 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.