Learning Minimally Rigid Graphs with High Realization Counts
For researchers in rigidity theory, this provides a new approach to discover graphs with many realizations, achieving new record graphs for spherical realizations.
This paper proposes a reinforcement learning method using Deep Cross-Entropy Method and Graph Isomorphism Networks to find minimally rigid graphs with high realization counts. The method matches known optima for planar realizations and improves best known bounds for spherical realizations.
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.