🤖 AI Summary
This work addresses the combinatorial explosion inherent in searching for minimally rigid graphs with a maximal number of realizations within rigidity theory, a challenge that renders traditional exhaustive methods infeasible at scale. The authors propose a novel progressive construction framework that integrates deep reinforcement learning with graph isomorphism networks, leveraging Henneberg operations to iteratively build candidate graphs. Their policy network combines a graph isomorphism encoder with a permutation-equivariant action head and is optimized via the deep cross-entropy method to maximize an invariant proxy for the number of realizations. The approach successfully reproduces known optimal results in the planar case and establishes new records on the sphere by discovering previously unknown minimally rigid graphs with exceptionally high realization counts, thereby substantially overcoming the scalability limitations of conventional search strategies.
📝 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.