Optimization of mixed volume over valid selections

Determine the computational complexity of minimizing the mixed volume among valid square subsystems when the cost varies and valid selections are known or can be found efficiently, and establish polynomial-time approximation guarantees for this optimization problem.

Background

The paper proves NP-completeness of deciding whether a valid selection exists even for a degree-four family in which every potentially valid selection has exactly the same mixed volume. Thus, the hardness result does not arise from computing or comparing varying costs.

A separate unresolved optimization problem remains for families where mixed volume genuinely varies. Lower mixed volume can reduce the number of paths followed by a polyhedral homotopy solver, motivating both exact optimization and approximation guarantees. The authors contrast this problem with weighted matroid intersection, which handles additive costs, and with approximation results for mixed volume of a fixed tuple of convex bodies.