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Published March 15, 2018 | Published + Submitted
Journal Article Open

Finding Planted Subgraphs with Few Eigenvalues using the Schur-Horn Relaxation


Extracting structured subgraphs inside large graphs---often known as the planted subgraph problem---is a fundamental question that arises in a range of application domains. This problem is NP-hard in general and, as a result, significant efforts have been directed towards the development of tractable procedures that succeed on specific families of problem instances. We propose a new computationally efficient convex relaxation for solving the planted subgraph problem; our approach is based on tractable semidefinite descriptions of majorization inequalities on the spectrum of a symmetric matrix. This procedure is effective at finding planted subgraphs that consist of few distinct eigenvalues, and it generalizes previous convex relaxation techniques for finding planted cliques. Our analysis relies prominently on the notion of spectrally comonotone matrices, which are pairs of symmetric matrices that can be transformed to diagonal matrices with sorted diagonal entries upon conjugation by the same orthogonal matrix.

Additional Information

© 2018 Society for Industrial and Applied Mathematics. Received by the editors May 13, 2016; accepted for publication (in revised form) October 24, 2017; published electronically March 15, 2018. Funding: The authors were supported in part by National Science Foundation grants CCF-1350590 and CCF-1637598, by Air Force Office of Scientific Research grants FA9550-14-1-0098 and FA9550-16-1-0210, and by a Sloan research fellowship.

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Published - 16m1075144.pdf

Submitted - 1605.04008.pdf


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