[Paper Review] Fast Approximate Quadratic Programming for Large (Brain) Graph Matching

Joshua T Vogelstein, John M. Conroy+7 more|arXiv (Cornell University)|Dec 23, 2011

Graph Theory and Algorithms40 references19 citations

TL;DR

This paper proposes the Fast Approximate Quadratic assignment (FAQ) algorithm for large-scale graph matching, leveraging an indefinite relaxation of the quadratic assignment problem to achieve superior accuracy and efficiency. FAQ outperforms state-of-the-art methods on over 80% of QAPLIB benchmarks and enables effective matching of large brain connectomes, such as those of C. elegans, with strong scalability and theoretical justification.

ABSTRACT

Quadratic assignment problems (QAPs) arise in a wide variety of domains, ranging from operations research to graph theory to computer vision to neuroscience. In the age of big data, graph valued data is becoming more prominent, and with it, a desire to run algorithms on ever larger graphs. Because QAP is NP-hard, exact algorithms are intractable. Approximate algorithms necessarily employ an accuracy/efficiency trade-off. We developed a fast approximate quadratic assignment algorithm (FAQ). FAQ finds a local optima in (worst case) time cubic in the number of vertices, similar to other approximate QAP algorithms. We demonstrate empirically that our algorithm is faster and achieves a lower objective value on over 80% of the suite of QAP benchmarks, compared with the previous state-of-the-art. Applying the algorithms to our motivating example, matching C. elegans connectomes (brain-graphs), we find that FAQ achieves the optimal performance in record time, whereas none of the others even find the optimum.

Motivation & Objective

• To address the scalability and accuracy trade-off in matching large graphs, particularly brain connectomes.
• To develop an efficient algorithm for solving the graph matching problem (GMP) as a special case of the quadratic assignment problem (QAP).
• To improve upon existing approximate QAP solvers by using an indefinite relaxation that better preserves solution fidelity.
• To enable direct structural comparison of brain graphs, moving beyond graph invariants, for connectomics and neurological disorder analysis.
• To provide a scalable, open-source solution for large-scale graph matching applicable to neuroscience and big data.

Proposed method

• The FAQ algorithm relaxes the graph matching problem into an indefinite quadratic optimization problem, avoiding the convex relaxation used in prior methods.
• It uses an alternating optimization strategy to iteratively solve linear assignment problems (LAPs) to update the permutation matrix.
• The core algorithm operates on the relaxed objective function (6), which is indefinite and provably correct under random graph models.
• It incorporates a power method-like update to iteratively refine the permutation estimate, improving alignment between graph adjacency matrices.
• The method is designed to be computationally efficient, with O(n³) complexity, and is amenable to parallelization of the LAP subroutines.
• The algorithm is generalizable and can be extended to include seeded vertices, vertex attributes, and sparse matrix optimizations.

Experimental results

Research questions

• RQ1 Can an indefinite relaxation of the QAP provide better approximate solutions than convex relaxations for graph matching?
• RQ2 How does the FAQ algorithm compare in accuracy and speed to existing state-of-the-art graph matching algorithms on standard QAP benchmarks?
• RQ3 Can FAQ effectively match large brain connectomes, such as those from C. elegans, with high fidelity and computational efficiency?
• RQ4 What are the theoretical guarantees of the indefinite relaxation approach under random graph models?
• RQ5 How can FAQ be extended to handle real-world challenges like missing vertices, edge errors, and heterogeneous graph sizes?

Key findings

• FAQ achieves a lower objective value than the previous state-of-the-art on over 80% of the QAPLIB benchmark library.
• The algorithm is faster and more accurate than existing methods, including PATH and other convex relaxation-based solvers.
• On C. elegans connectome matching, FAQ efficiently achieves high-quality matches, demonstrating its utility in connectomics.
• The indefinite relaxation used in FAQ is theoretically justified as the correct approach under general random graph models, unlike convex relaxations.
• Extrapolation shows that FAQ would take ~20 years to process n=100,000 on a standard laptop, highlighting the need for further scalability improvements.
• The FAQ algorithm has been integrated into the open-source R package igraph and MATLAB code is publicly available for broader adoption.