Approximation algorithms for water transport

Establish positive approximation results for the finite-graph water-transport optimization problem, either for arbitrary graphs or for restricted graph classes such as trees.

Background

The paper shows that the water-transport optimization problem is NP-hard and gives examples demonstrating that intuitive heuristics, including greedy lattice-animal strategies and divide-and-conquer approaches, can perform poorly. It therefore asks whether approximation algorithms with provable guarantees can nevertheless be developed for general graphs or at least for structured classes such as trees.

References

However, here also lies the potential for future research: Can one prove any positive results based on approximative algorithms -- if not for general $G$ then at least for a certain class of graphs, such as trees for instance?

— Water transport on finite graphs  (2501.16911 - Vilkas, 28 Jan 2025) in Section 'Observations and open problems'