Sparse Semidefinite Programs with Near-Linear Time Complexity

Some of the strongest polynomial-time relaxations to NP-hard combinatorial optimization problems are semidefinite programs (SDPs), but their solution complexity of up to O(nL) time and O(n) memory for L accurate digits limits their use in all but the smallest problems. Given that combinatorial SDP relaxations are often sparse, a technique known as chordal conversion can sometimes reduce complexity substantially. In this paper, we describe a modification of chordal conversion that allows any general-purpose interiorpoint method to solve a certain class of sparse SDPs with a guaranteed complexity of O(nL) time and O(n) memory. To illustrate the use of this technique, we solve the MAX kCUT relaxation and the Lovasz Theta problem on power system models with up to n = 13659 nodes in 5 minutes, using SeDuMi v1.32 on a 1.7 GHz CPU with 16 GB of RAM. The empirical time complexity for attaining L decimal digits of accuracy is ≈ 0.001nL seconds.