Combinatorial and Algorithmic Volume Rigidity

We study the generic rigidity properties of systems of volume constraints. We formally define, in an algebraic geometrical setting, the volume rigidity problem and extract its underlying sparse hypergraph structure. Then we develop a combinatorial theory of cocircuit matroids which leads to a very general linear representation result for sparse hypergraphs. Applying this to the volume rigidity problem yields a combinatorial characterization of generic volume rigidity. Let P be a set of n points in the plane. Laman’s [11] landmark theorem characterizes exactly which sets of distances between points in P determine all ( n 2 ) distances up to a discrete set of possibilities, generically. Whiteley [32] asked the corresponding question for sets of triangle areas. We start by defining area rigidity in the plane and study its natural generalization to volume rigidity in arbitrary dimensions, proving a Laman-like theorem for hypergraphs with specific hereditary counts. This adds to a very small body of combinatorial rigidity results in dimensions higher than 3. The combinatorial characterization of volume rigidity is algorithmically tractable by our generalizations of pebble games from sparse graphs [12] to sparse hypergraphs [25].