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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.
Sun et al. (Fri,) studied this question.