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. Cependant, 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.

Cet article explore les excursions dans le temps et leurs implications.

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