Title: Does Your Blockchain Need Multidimensional Transaction Fees? Show Chinese title

Title: 你的区块链需要多维交易费用吗？

Abstract: Blockchains have block-size limits to ensure the entire cluster can keep up with the tip of the chain. These block-size limits are usually single-dimensional, but richer multidimensional constraints allow for greater throughput. The potential for performance improvements from multidimensional resource pricing has been discussed in the literature, but exactly how big those performance improvements are remains unclear. In order to identify the magnitude of additional throughput that multi-dimensional transaction fees can unlock, we introduce the concept of an $\alpha$-approximation. A constraint set $C_1$ is $\alpha$-approximated by $C_2$ if every block feasible under $C_1$ is also feasible under $C_2$ once all resource capacities are scaled by a factor of $\alpha$ (e.g., $\alpha =2$ corresponds to doubling all available resources). We show that the $\alpha$-approximation of the optimal single-dimensional gas measure corresponds to the value of a specific zero-sum game. However, the more general problem of finding the optimal $k$-dimensional approximation is NP-complete. Quantifying the additional throughput that multi-dimensional fees can provide allows blockchain designers to make informed decisions about whether the additional capacity unlocked by multidimensional constraints is worth the additional complexity they add to the protocol. Show Chinese abstract

Abstract: 区块链有区块大小限制，以确保整个集群能够跟上链的顶端。这些区块大小限制通常是单维的，但更丰富的多维约束可以实现更高的吞吐量。文献中已经讨论了来自多维资源定价的性能改进潜力，但这些性能改进到底有多大仍然不清楚。为了确定多维交易费用可以解锁的额外吞吐量的规模，我们引入了$\alpha$-近似概念。 如果所有资源容量都按 $\alpha$ 因子缩放后，每个在 $C_1$ 下可行的区块在 $C_2$ 下也是可行的（例如，$\alpha =2$ 对应于将所有可用资源增加一倍），则约束集 $C_1$ 由 $C_2$ 近似为 $\alpha$。 我们证明了最优单维气体度量的$\alpha$-近似值对应于一个特定的零和博弈的值。然而，寻找最优$k$-维近似的更一般性问题属于 NP 完全问题。量化多维费用能提供的额外吞吐量，使区块链设计者能够就解锁多维约束所带来的额外容量是否值得协议增加的复杂性做出明智的决定。