Worst-case gap between greedy lattice animals and optimal transport

Characterize the worst-case ratio between the optimal target water level \(\kappa(v)\) and the greedy-lattice-animal lower bound \(\mathrm{GLA}(v)\), under fixed barrel capacity and a fixed number of vertices.

Background

For every finite water-transport instance, the greedy lattice animal value $\mathrm{GLA}(v)$ is a valid lower bound for the optimal value $\kappa(v)$. The paper provides a star-graph construction in which $\kappa(v)$ exceeds $\mathrm{GLA}(v)$ by a factor of order $\log(n)$, showing that the heuristic can be substantially suboptimal. The authors leave open a sharper characterization of how large this discrepancy can be in the worst case, particularly when barrel capacity and the number of vertices are fixed.