Complexity of chordal conversion for sparse semidefinite programs with small treewidth

If a sparse semidefinite program (SDP), specified over n×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 Ω(n3) 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 Ω(n3) time per-iteration. Instead, we construct an extended aggregate sparsity graph G¯⊇G by forcing each constraint matrix Ai to be its own clique in G. We prove that a small treewidth in G¯ does indeed guarantee that chordal conversion will solve the SDP in O(m+n) time per-iteration, to ϵ-accuracy in at most O(m+nlog(1/ϵ)) iterations. This sufficient condition covers many successful applications of chordal conversion, including the MAX-k-CUT relaxation, the Lovász theta problem, sensor network localization, polynomial optimization, and the AC optimal power flow relaxation, thus allowing theory to match practical experience.