Sparse Semidefinite Programs with Near-Linear Time Complexity

Richard Y. Zhang, Javad Lavaei

Abstract: 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(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">6.5</sup> L) time and O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sup> ) 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 interior-point method to solve a certain class of sparse SDPs with a guaranteed complexity of O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l.5</sup> L) time and O(n) memory. To illustrate the use of this technique, we solve the MAX k- CUT 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 ≈ 0.001n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l.l</sup> L seconds.