Limits of computational reduction for compatibility testing

Determine how far the computational cost of solving the compatibility problem for conditional distributions can be reduced in high-dimensional settings, beyond the tractability limitations established for succinct arithmetic-circuit representations.

Background

The introduction discusses the difficulty of testing compatibility by enumerating all values of large discrete spaces. Although prior work has proposed more efficient algorithms, the paper identifies the remaining issue as determining the extent to which this computational cost can be reduced. The paper addresses one succinct representation—tractable arithmetic circuits—and proves strong complexity lower bounds for that setting, but it does not resolve the broader question of how much more efficient compatibility testing might be under other representations or structural restrictions.