  Probabilistic graphical models (PGMs) are powerful tools for solving systems
of complex relationships over a variety of probability distributions. However,
while tree-structured PGMs always result in efficient and exact solutions,
inference on graph (or loopy) structured PGMs is not guaranteed to discover the
optimal solutions. It is in principle possible to convert loopy PGMs to an
equivalent tree structure, but this is usually impractical for interesting
problems due to exponential blow-up. To address this, we developed the
purge-and-merge algorithm. This algorithm iteratively nudges a malleable graph
structure towards a tree structure by selectively merging factors. The merging
process is designed to avoid exponential blow-up by way of sparse structures
from which redundancy is purged as the algorithm progresses. We set up tasks to
test the algorithm on constraint-satisfaction puzzles such as Sudoku,
Fill-a-pix, and Kakuro, and it outperformed other PGM-based approaches reported
in the literature. While the tasks we set focussed on the binary logic of CSP,
we believe the purge-and-merge algorithm could be extended to general PGM
inference.
