Complexity of Chordal Conversion for Sparse Semidefinite Programs with Small Treewidth

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.