Learning Minimally Rigid Graphs with High Realization Counts

TL;DR

This paper introduces a reinforcement learning method to construct minimally rigid graphs with high realization counts, achieving new record graphs and matching known optima.

Contribution

It presents a novel approach combining deep reinforcement learning and graph neural networks to optimize realization counts in rigidity theory.

Findings

Matches known optima for planar realization counts.

Improves bounds for spherical realization counts.

Yields new record graphs in extremal rigidity problems.

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.