{"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/parameterized-complexity-of-chordal","title":"Complexity of Chordal Conversion for Sparse Semidefinite Programs with Small Treewidth","arxiv_id":"2306.15288","date":"2023-06-27","proceeding":null,"authors":["Richard Y. Zhang"],"abstract":"If a sparse semidefinite program (SDP), specified over $n\\times n$ matrices and subject to $m$ linear constraints, has an aggregate sparsity graph $G$ with small treewidth, then chordal conversion will sometimes allow an interior-point method to solve the SDP in just $O(m+n)$ time per-iteration, which is a significant speedup over the $\\Omega(n^{3})$ time per-iteration for a direct application of the interior-point method. Unfortunately, the speedup is not guaranteed by an $O(1)$ treewidth in $G$ that is independent of $m$ and $n$, as a diagonal SDP would have treewidth zero but can still necessitate up to $\\Omega(n^{3})$ time per-iteration. Instead, we construct an extended aggregate sparsity graph $\\bar{G}\\supseteq G$ by forcing each constraint matrix $A_{i}$ to be its own clique in $G$. We prove that a small treewidth in $\\bar{G}$ does indeed guarantee that chordal conversion will solve the SDP in $O(m+n)$ time per-iteration, to $\\epsilon$-accuracy in at most $O(\\sqrt{m+n}\\log(1/\\epsilon))$ iterations. This sufficient condition covers many successful applications of chordal conversion, including the MAX-$k$-CUT relaxation, the Lov\\'asz theta problem, sensor network localization, polynomial optimization, and the AC optimal power flow relaxation, thus allowing theory to match practical experience.","url_abs":"https://arxiv.org/abs/2306.15288v2","url_pdf":"https://arxiv.org/pdf/2306.15288v2.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":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"parameterized-complexity-of-chordal","repo_url":"https://github.com/ryz-codes/chordalconv","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":{"status":"ok","spdx":"BSD-2-Clause"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=2306.15288","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}