Computational Improvements in the Substitution Method for Bounding Percolation Thresholds

The substitution method has proven to be an effective tool for bounding the critical probability of a variety of percolation models. Nevertheless, until recently substitution method calculations have been done by hand. This has severely restricted the size of computationally feasible substitution regions. We examine the computational problems posed by the substitution method, beginning with an analysis of the component calculations. We seek to better understand the nature of the computational problem, hoping that better understanding will lead to improvements. Our goal is a little different from that of most algorithmic investigations. Since the substitution method constitutes a proof, there is little reason to perform a particular computation more than once. We use each speed improvement to attempt a new, larger computation that will lead to tighter bounds on the critical probability. Our major results can be grouped into two categories: recognition of links between substitution method calculations and well-known results in other areas of mathematics, and the development of novel algorithms to exploit special structure. The links to other areas of mathematics include the use of 2-attached graphs to compute partition polynomials, the use of network flow techniques to prove stochastic ordering on a partially ordered set, and the use of the Lerchs and Grossmann algorithm to address the parametric problem more efficiently. The novel techniques we introduce are the use of symmetry to reduce the size of the calculation, and the use of non-crossing partitions. The exploitation of symmetry often makes the difference between a calculation that is feasible versus one that is