E-Graphs as Circuits, and Optimal Extraction via Treewidth

We solve the optimal extraction problem for e-graphs by first showing a connection between e-graphs and cyclic monotone Boolean circuits, then solving the weighted satisfiability problem for such circuits. The solution is a parameterized algorithm based on treewidth. Additionally, we show how the circuit view of e-graphs allows us to apply simplification techniques that are not possible when operating directly on e-graphs. While the core parameterized algorithm may be adapted to work directly on e-graphs, the simplification results show why the circuit view is helpful.