How to linearize your cluster

The DeepSeek approach presents an innovative method for constructing a network flow that optimizes transaction subsets for maximum weight under specific conditions, which is particularly intriguing for those interested in its application to blockchain and cryptocurrency transactions. This technique involves transforming the feerate into a weight by utilizing a lambda variable ((\lambda)), thereby enabling the optimization of the subset (C) based on the weight function (f-\lambda s), where (f) represents the feerate, and (s) signifies the size. The unique aspect of this methodology is its reliance on network flow mechanics, specifically the max flow/min cut theorem, to identify a subset of transactions ((C)) that maximizes the given weight function under the constraint that (C) is a closure.

A closure in this context is achieved through the formulation of the problem as a min-cut scenario, where original edges within the network are assigned infinite capacity to ensure they are never cut. This clever setup guarantees that the solution, or the min-cut found, represents a subset (C) with the desired properties—maximizing the weight by accounting for both included nodes with negative weight and excluded nodes with positive weight. The introduction of (\lambda) as a means to convert feerate into a weight allows for the total weight to be summed up directly, offering a straightforward way to optimize the ratio indirectly rather than tackling it head-on.

Furthermore, the approach utilizes bisection on different values of (\lambda) to find the subset (C) that yields the largest feerate. This process does not arrange breakpoints in any order but instead finds the optimal improvement within the selected interval. After identifying the optimal subset within one interval, the next interval is chosen for analysis, continuing the process iteratively. This aspect of the DeepSeek method highlights its efficiency and precision in optimizing transaction subsets based on feerate, showcasing a practical application of network flow principles in financial technology solutions. For more detailed insights into the workings and implications of this approach, refer to the DeepSeek paper.