[Paper Review] A Faster Small Treewidth SDP Solver

Yuzhou Gu, Zhao Song|arXiv (Cornell University)|Nov 11, 2022

Advanced Optimization Algorithms Research4 citations

TL;DR

This paper presents a novel algorithm for solving semidefinite programs (SDPs) with small treewidth in time linear in the number of variables, achieving $ O(n \cdot \tau^{2\omega + 0.5} \log(1/\epsilon)) $, where $ \omega $ is the matrix multiplication exponent and $ \tau $ is the treewidth. It also improves the running time for low-treewidth linear programs to $ O(n \cdot \tau^{(\omega+1)/2} \log(1/\epsilon)) $, marking the first nearly linear-time SDP solver under this setting.

ABSTRACT

Semidefinite programming is a fundamental tool in optimization and theoretical computer science. It has been extensively used as a black-box for solving many problems, such as embedding, complexity, learning, and discrepancy. One natural setting of semidefinite programming is the small treewidth setting. The best previous SDP solver under small treewidth setting is due to Zhang-Lavaei '18, which takes $n^{1.5} τ^{6.5}$ time. In this work, we show how to solve a semidefinite programming with $n imes n$ variables, $m$ constraints and $τ$ treewidth in $n τ^{2ω+0.5}$ time, where $ω< 2.373$ denotes the exponent of matrix multiplication. We give the first SDP solver that runs in time in linear in number of variables under this setting. In addition, we improve the running time that solves a linear programming with tau treewidth from $n τ^2$ (Dong-Lee-Ye '21) to $n τ^{(ω+1)/2}$.

Motivation & Objective

• To address the open question of whether semidefinite programs with small treewidth can be solved in nearly linear time in the number of variables.
• To improve upon the prior best-known SDP solver under small treewidth, which ran in $ n^{1.5} \tau^{6.5} $ time.
• To extend the improvement to low-treewidth linear programming, surpassing the previous $ n\tau^2 $ bound.
• To achieve this by leveraging treewidth structure and advanced matrix multiplication techniques in interior-point methods.

Proposed method

• The algorithm uses a novel decomposition of the SDP problem based on tree decomposition of the SDP graph, exploiting low treewidth to reduce computational complexity.
• It applies a customized barrier function and central path method tailored to the treewidth structure, avoiding the $ \Omega(m^\omega) $ bottleneck of prior methods.
• The method introduces a recursive Schur complement computation over tree decomposition nodes, with matrix operations bounded by $ \widetilde{O}(\tau^{\omega}) $ per node.
• It uses a hierarchical clustering of constraints and variables to maintain sparsity and enable efficient matrix inversion via low-rank updates.
• The algorithm integrates a refined analysis of condition numbers and condition number reduction to ensure fast convergence.
• It combines dynamic programming over tree decomposition with fast matrix multiplication to achieve $ \widetilde{O}(n \cdot \tau^{2\omega + 0.5}) $ runtime.

Experimental results

Research questions

• RQ1 Can semidefinite programs with small treewidth be solved in time linear in the number of variables?
• RQ2 Can the dependency on treewidth be reduced below $ \tau^6 $ in the exponent, especially to $ \tau^{2\omega + 0.5} $?
• RQ3 Can the same structural insights be applied to improve the running time of low-treewidth linear programs?
• RQ4 Is it possible to achieve nearly linear time in $ n $ for SDP under small treewidth, even with matrix multiplication exponent $ \omega \approx 2.373 $?

Key findings

• The paper achieves an SDP solver runtime of $ O(n \cdot \tau^{2\omega + 0.5} \log(1/\epsilon)) $, which is linear in $ n $ and improves upon the prior $ n^{1.5} \tau^{6.5} $ bound.
• The algorithm is the first to solve small-treewidth SDPs in time nearly linear in $ n $, under standard linear independence assumptions.
• For low-treewidth linear programs, the runtime is improved to $ O(n \cdot \tau^{(\omega+1)/2} \log(1/\epsilon)) $, surpassing the previous $ n\tau^2 $ bound.
• The improvement relies on a new analysis of matrix operations within tree decomposition, reducing the effective cost per node to $ \widetilde{O}(\tau^{\omega}) $.
• The method avoids the $ \Omega(m^\omega) $ dependency of prior interior-point methods by exploiting sparsity and treewidth structure.
• The results are tight up to logarithmic factors and matrix multiplication exponent, with explicit dependence on $ \omega $, the current best-known exponent.