Pseudoflow Algorithm

Both solve the same maximum closure problem with the same theoretical complexity. Pseudoflow achieves better practical performance on large, sparse graphs typical of mining block models—often 5-50x faster depending on graph structure. Both are optimal; Pseudoflow is simply more efficient in implementation.

Yes. The algorithm has been parallelized on multi-core CPUs and GPUs. The key is parallel push operations on independent high-labeled nodes. Contention on shared labels and excess variables can limit speedup on very large shared-memory systems, but GPUs show promise for specialized designs.

True cycles in the precedence graph usually indicate modeling errors (e.g., block A must be above B, and B above A simultaneously). Strongly-connected components should be merged or one edge removed. If cycles represent valid but circular dependencies (rare in mining), reformulate as a regular maximum-flow problem instead of closure.

Pseudoflow uses integer or fixed-point arithmetic for capacities, making it robust to floating-point rounding. If block values are floats, they should be scaled to integers (e.g., multiply by 1000 and round) before running the algorithm to avoid cumulative rounding errors.

Negative weights (costs) are handled natively. The algorithm finds the maximum sum of weights including profitable ore and necessary costs. Nodes with net-negative value are automatically excluded from the optimal closure.