Polynomial-time decidability for 3-dimensional SLPS with constant updates

Determine whether the reachability problem in 3-dimensional Simple Linear Path Schemes (SLPS) with constant updates (i.e., all transition labels drawn from a fixed finite set {-u, …, u} independent of the instance size) is solvable in polynomial time.

Background

The authors show how to simulate fixed-magnitude updates by unitary updates at the cost of increasing dimension, and they obtain hardness for unitary SLPS in growing dimension. For fixed dimension, however, this transformation is not cost-free, leaving open the precise complexity of 3-dimensional SLPS with constant updates.

They explicitly flag this as an interesting open problem distinct from the unitary case, arguing that for fixed dimension the “fixed-updates” class may be the more robust target.