Learning Minimally Rigid Graphs with High Realization Counts
Oleksandr Slyvka, Jan Rubeš, Rodrigo Alves, Jan Legerský
TLDR
This paper uses reinforcement learning to discover minimally rigid graphs with record-breaking numbers of realizations, improving bounds for spherical graphs.
Key contributions
- Proposes a reinforcement learning method to construct minimally rigid graphs.
- Uses 0- and 1-extensions (Henneberg moves) to build graphs iteratively.
- Optimizes realization counts with Deep Cross-Entropy Method and GIN-based policy.
- Achieves new record graphs and improved bounds for spherical realization counts.
Why it matters
This work addresses a challenging extremal problem in rigidity theory by leveraging machine learning. It provides a novel approach to discover complex graph structures, leading to new records and advancing our understanding of graph rigidity.
Original Abstract
For minimally rigid graphs, the same edge-length data can admit multiple realizations (up to translations and rotations). Finding graphs with exceptionally many realizations is an extremal problem in rigidity theory, but exhaustive search quickly becomes infeasible due to the super-exponential growth of the number of candidate graphs and the high cost of realization-count evaluation. We propose a reinforcement-learning approach that constructs minimally rigid graphs via 0- and 1-extensions, also known as Henneberg moves. We optimize realization-count invariants using the Deep Cross-Entropy Method with a policy parameterized by a Graph Isomorphism Network encoder and a permutation-equivariant extension-level action head. Empirically, our method matches the known optima for planar realization counts and improves the best known bounds for spherical realization counts, yielding new record graphs.
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