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Extracting structured subgraphs inside large graphs - often known as theplanted subgraph problem - is a fundamental question that arises in a range ofapplication domains. This problem is NP-hard in general, and as a result,significant efforts have been directed towards the development of tractableprocedures that succeed on specific families of problem instances. We propose anew computationally efficient convex relaxation for solving the plantedsubgraph problem; our approach is based on tractable semidefinite descriptionsof majorization inequalities on the spectrum of a symmetric matrix. Thisprocedure is effective at finding planted subgraphs that consist of fewdistinct eigenvalues, and it generalizes previous convex relaxation techniquesfor finding planted cliques. Our analysis relies prominently on the notion ofspectrally comonotone matrices, which are pairs of symmetric matrices that canbe transformed to diagonal matrices with sorted diagonal entries uponconjugation by the same orthogonal matrix.

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