How to linearize your cluster

The min-cut algorithm plays a crucial role in optimizing transaction fees by identifying the subset of transactions, referred to as closure $C_1$, that maximizes the net value $Q_1$. This net value is calculated as the difference between the total fee of the transactions within $C_1$ and a product of a predetermined target feerate $\lambda_1$ and the size of $C_1$. The underlying principle of this approach is to select those transactions from the optimal feerate diagram whose individual feerates exceed the target feerate $\lambda_1$. This selection criterion ensures that $Q_1$ achieves its maximum value, as it essentially collects chunks demonstrating a higher slope on the feerate diagram. These segments indicate an increasing disparity in cumulative transaction size, which, after reaching its peak, begins to decrease, signifying a reduction in the marginal benefit of including additional transactions.

The methodology relies on the dynamics of the residual network derived from the max flow algorithm. This network facilitates the exclusion of lower-feerate transactions in two ways: either such transactions are directly connected to the sink, implying they do not contribute sufficiently to the net value $Q_1$, or their connections to the source are severed. The latter occurs because the algorithm prioritizes the flow towards the sink for higher feerate transactions, effectively blocking out lower feerate transactions by redirecting resources away from them. This process ensures that only transactions with a feerate higher than $\lambda_1$ are included, thereby maximizing the overall value obtained from the set of transactions processed.