{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/convex-optimization-for-the-densest-subgraph","title":"Convex optimization for the densest subgraph and densest submatrix problems","arxiv_id":"1904.03272","date":"2019-04-05","proceeding":null,"authors":["Polina Bombina","Brendan Ames"],"abstract":"We consider the densest $k$-subgraph problem, which seeks to identify the\n$k$-node subgraph of a given input graph with maximum number of edges. This\nproblem is well-known to be NP-hard, by reduction to the maximum clique\nproblem. We propose a new convex relaxation for the densest $k$-subgraph\nproblem, based on a nuclear norm relaxation of a low-rank plus sparse\ndecomposition of the adjacency matrices of $k$-node subgraphs to partially\naddress this intractability. We establish that the densest $k$-subgraph can be\nrecovered with high probability from the optimal solution of this convex\nrelaxation if the input graph is randomly sampled from a distribution of random\ngraphs constructed to contain an especially dense $k$-node subgraph with high\nprobability. Specifically, the relaxation is exact when the edges of the input\ngraph are added independently at random, with edges within a particular\n$k$-node subgraph added with higher probability than other edges in the graph.\nWe provide a sufficient condition on the size of this subgraph $k$ and the\nexpected density under which the optimal solution of the proposed relaxation\nrecovers this $k$-node subgraph with high probability. Further, we propose a\nfirst-order method for solving this relaxation based on the alternating\ndirection method of multipliers, and empirically confirm our predicted recovery\nthresholds using simulations involving randomly generated graphs, as well as\ngraphs drawn from social and collaborative networks.","url_abs":"http://arxiv.org/abs/1904.03272v1","url_pdf":"http://arxiv.org/pdf/1904.03272v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"convex-optimization-for-the-densest-subgraph","repo_url":"https://github.com/pbombina/admmdsm","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1904.03272","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}