  Mixing by cutting-and-shuffling can be understood and predicted using
dynamical systems based tools and techniques. In existing studies, mixing is
generated by maps that repeat the same cut-and-shuffle process at every
iteration, in a "fixed" manner. However, mixing can be greatly improved by
varying the cut-and-shuffle parameters at each step, using a "variable"
approach. To demonstrate this approach, we show how to optimize mixing by
cutting-and-shuffling on the one-dimensional line interval, known as an
interval exchange transformation (IET). Mixing can be significantly improved by
optimizing variable protocols, especially for initial conditions more complex
than just a simple two-color line interval. While we show that optimal variable
IETs can be found analytically for arbitrary numbers of iterations, for more
complex cutting-and-shuffling systems, computationally expensive numerical
optimization methods would be required. Furthermore, the number of control
parameters grows linearly with the number of iterations in variable systems.
Therefore, optimizing over large numbers of iterations is generally
computationally prohibitive. We demonstrate an ad hoc approach to
cutting-and-shuffling that is computationally inexpensive and guarantees the
mixing metric is within a constant factor of the optimum. This ad hoc approach
yields significantly better mixing than fixed IETs which are known to produce
weak-mixing, because cut pieces never reconnect. The heuristic principles of
this method can be applied to more general cutting-and-shuffling systems.
