How to linearize your cluster

The discourse reveals a significant breakthrough in solving the problem of finding the highest-feerate topologically-valid subset, which has been a subject of study for some time. The revelation that this challenge can be addressed in polynomial time comes from revisiting literature where Gallo, Grigoriadis, and Tarjan demonstrated in 1989 an efficient solution through their work titled "A FAST PARAMETRIC MAXIMUM FLOW ALGORITHM AND APPLICATIONS". Published in SIAM J. COMPUT., this paper introduces an algorithm capable of addressing the maximum-ratio closure problem, albeit with a distinct directionality in graph arrows, in $O(nm \log (n^2/m))$ time. This discovery is pivotal as it underscores the feasibility of processing larger clusters through a methodologically complex yet efficient approach.

The solution hinges on transforming the given graph into a flow network with capacities modifiable by a parameter $\lambda$, representing the target feerate. Through iterative calculations over varying $\lambda$ values to find the minimum cut, the algorithm achieves optimal performance remarkably within the same asymptotic time as a single minimum cut would require. This efficiency is attributed to the adaptation of the Goldberg-Tarjan push-relabel algorithm, allowing continuous operation across successive $\lambda$ values and proving that only $O(n)$ breakpoints are necessary for min-cut calculation under certain conditions.

Despite the promising nature of this algorithm, practical implementation poses a challenge due to the absence of readily available open-source versions tailored to the exact requirements. However, a potential starting point for adaptation lies in a repository hosted on GitHub, which offers C++ algorithms designed for closely related problems. These algorithms not only promise faster practical performance but also come with the flexibility afforded by the MIT license, making them suitable candidates for modification to fit the specified needs. Alternatively, crafting a new implementation of the push-relabel parameterized min-cut algorithm from the ground up represents a viable, albeit significantly more complex, route.