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

arXiv: https://arxiv.org/abs/1605.04008 · HTML: https://mlsanity.com/paper/1605.04008/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 May 2016.

## Built on by (most cited first, 1 of 1)

- [Random Subgraph Detection Using Queries](https://mlsanity.com/paper/2110.00744/) (2021)

## Abstract (arXiv)

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.