Complexity of Chordal Conversion for Sparse Semidefinite Programs with Small Treewidth

TL;DR

This paper demonstrates that for sparse SDPs with small treewidth in an extended graph, chordal conversion enables efficient interior-point solutions, aligning theoretical guarantees with practical success in various applications.

Contribution

The paper introduces an extended graph construction that guarantees efficient SDP solving via chordal conversion when the extended graph has small treewidth, addressing limitations of previous approaches.

Findings

Small treewidth in extended graph guarantees $O(m+n)$ per-iteration time.

Chordal conversion achieves $ ilde{O}( oot{m+n})$ iteration complexity.

Applicable to many practical SDP relaxations like MAX-$k$-CUT and power flow.

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.