AC$\oplus$DC search: behind the winning solution to the FlyWire graph-matching challenge

Daniel Lee, Arie Matsliah, Lawrence K. Saul

Abstract: This paper describes the Alternating Continuous and Discrete Combinatorial (AC$\oplus$DC) optimizations behind the winning solution to the FlyWire Ventral Nerve Cord Matching Challenge. The challenge was organized by the Princeton Neuroscience Institute and held over three months, ending on January 31, 2025. During this period, the challenge attracted teams of researchers with expertise in machine learning, high-performance computing, graph data mining, biological network analysis, and quadratic assignment problems. The goal of the challenge was to align the connectomes of a male and female fruit fly, and more specifically, to determine a one-to-one correspondence between the neurons in their ventral nerve cords. The connectomes were represented as sparse weighted graphs with thousands of nodes and millions of edges, and the challenge was to find the permutation that best maps the nodes and edges of one graph onto those of the other. The winning solution to the challenge alternated between two complementary approaches to graph matching---the first, a combinatorial optimization over the symmetric group of permutations, and the second, a continuous relaxation of this problem to the space of doubly stochastic matrices. For the latter, the doubly stochastic matrices were optimized by combining Frank-Wolfe methods with a fast preconditioner to solve the linear assignment problem at each iteration. We provide a complete implementation of these methods with a few hundred lines of code in MATLAB. Notably, this implementation obtains a winning score to the challenge in less than 10 minutes on a laptop computer.

Submission Length: Long submission (more than 12 pages of main content)

Previous TMLR Submission Url: https://openreview.net/forum?id=cNOjnx8XrR

Changes Since Last Submission:

(12/27/2025) The revised manuscript incorporates the many constructive suggestions from reviewers. It includes not only a fuller discussion of related work (in section 4.3), but also a new appendix and ablation study comparing different preconditioners for perfect matching problems. Though it spills slightly over 12 pages in length, we have observed this to be the norm for accepted papers in TMLR, especially after they are de-anonymized and incorporate reviewer comments. For the revised manuscript, we also regenerated all the results for this paper using the 2025b release of MATLAB. In doing so, we were pleasantly surprised (on exactly the same hardware) to obtain a winning solution in under 10 minutes. We have updated all of the figures and text with these latest numbers. Once again we wish to the editors and reviewers of TMLR for their time and consideration of our work.

(01/13/2026)The re-revised manuscript incorporates the most recent comments of the action editor: (1) In the introduction, the first and penultimate paragraphs now highlight several applications of graph-matching that may be of interest to researchers in machine learning; these include keypoint-matching in computer vision, biological and social network alignment, vulnerability detection in software systems, and learning over the symmetric group of permutations. (2) A new section (4.4) describes five other methods for graph-matching and discusses their suitability for the VNC graph-matching challenge. (3) The appendices have been moved to appear after the references.
(4) The publication date has been updated to 01/2026.

Assigned Action Editor: ~Giannis_Nikolentzos1

Submission Number: 5649