Carving-width and contraction trees for tensor networks

We study the problem of finding contraction orderings on tensor networks for physical simulations using a syncretic abstract data type, the $\mathit{\text{contraction-tree}}$, and explain its connection to temporal and spatial measures of tensor contraction computational complexity (nodes express time; arcs express space). We have implemented the Ratcatcher of Seymour and Thomas for determining the carving-width of planar networks, in order to offer experimental evidence that this measure of spatial complexity makes a generally effective heuristic for limiting their total contraction time.