How to linearize your cluster

In exploring the optimality of transaction selection processes within blockchain mining, one can appreciate the intuitive nature of the "convex hull" concept as applied to this domain. The methodology for choosing transactions based on their fee rate, up until the block reaches its capacity, showcases a straightforward algorithm that tends toward optimality when the last included transaction perfectly complements the block's remaining space. This scenario highlights the potential for achieving an optimal solution without resorting to approximations, given that any deviation from this perfect fit results in minimal error. The error margin, tied directly to the fees associated with unselected transactions and the available block space, remains low especially when transactions are relatively small compared to the block size. Bitcoin Core's approach of maximizing the utilization of available space by adjusting to fill gaps as efficiently as possible further reduces the practical impact of approximation errors.

The discussion also extends into the challenges presented by adhering strictly to transaction chunks, a method which introduces its own form of approximation error. This error becomes particularly noticeable when the remaining block space does not align neatly with the sizes of these transaction chunks. However, the extent of this discrepancy is limited by the maximum chunk size, indicating a bounded, predictable level of approximation. The interaction between parent and child transactions in the selection process inevitably leads to some level of efficiency loss due to the necessity of including less profitable transactions to accommodate more valuable ones subsequently.

There's an acknowledgment of the randomness in how space is filled within blocks, suggesting that optimizing for any specific level of block fullness may be inherently flawed due to the unpredictable distribution of transaction sizes. This perspective hints at the complexity of achieving absolute optimality in transaction selection and the inherent limitations of any algorithmic approach designed to linearize transactions based on profitability and dependency criteria.

Finally, the significance of transaction linearization is put into context, emphasizing its minimal impact on overall mining and eviction processes. In most cases, transaction clusters do not offer multiple viable linearizations, either forming a single sequence by default or rendering the order irrelevant due to the all-or-nothing nature of their selection for inclusion in a block. This observation underlines the fact that while theoretical considerations around optimality and approximation are important, the practical implications for blockchain operation are often negligible, focusing attention on the broader mechanics of blockchain transaction processing and optimization strategies.