Optimized parametrization of systems of incidences between rigid bodies

Graphs of pairwise incidences between collections of rigid bodies occur in many practical applications and give rise to large algebraic systems for which all solutions have to be found. Such pairwise incidences have explicit, simple and rational parametrizations that, in principle, allow us to partially resolve these systems and arrive at a reduced, parametrized system in terms of the rational parameters. However, the choice of incidences and the partial order of incidence resolution strongly determine the algebraic complexity of the reduced, parametrized system – measured primarily in the number of variables and secondarily in the degree of the equations. Using a pairwise overlap graph, we introduce a combinatorial class of incidence tree parametrizations. Each parametrized system in the class represents the same collection of incident rigid bodies but they typically widely differ in their algebraic complexity. Minimizing the algebraic complexity over this class reduces to a purely combinatorial optimization problem that is a special case of the set cover problem. We quantify the exact improvement of algebraic complexity obtained by optimization and illustrate the improvement by examples that can not be solved without optimization. Since incidence trees represent only a subclass of possible parametrizations, we characterize when optimizing over this class useful. That is, we show what properties of standard collections of rigid bodies are necessary for an optimal incidence tree to have minimal algebraic complexity. For a standard collections of rigid bodies, the optimal incidence tree parameterization offers lower algebraic complexity than any other known parameterization.