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Does Your Blockchain Need Multidimensional Transaction Fees?

Nir Lavee, Noam Nisan, Mallesh Pai, Max Resnick

TL;DR

It is shown that the $\alpha$-approximation of the optimal single-dimensional gas measure corresponds to the value of a specific zero-sum game, and the more general problem of finding the optimal $k$-dimensional approximation is NP-complete.

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 $α$-approximation. A constraint set $C_1$ is $α$-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 $α$ (e.g., $α=2$ corresponds to doubling all available resources). We show that the $α$-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.

Does Your Blockchain Need Multidimensional Transaction Fees?

TL;DR

It is shown that the -approximation of the optimal single-dimensional gas measure corresponds to the value of a specific zero-sum game, and the more general problem of finding the optimal -dimensional approximation is NP-complete.

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 -approximation. A constraint set is -approximated by if every block feasible under is also feasible under once all resource capacities are scaled by a factor of (e.g., corresponds to doubling all available resources). We show that the -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 -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.

Paper Structure

This paper contains 14 sections, 5 theorems, 23 equations, 2 figures, 5 tables.

Table of Contents

  1. Introduction and Motivation
  2. Model
  3. Results
  4. Optimal $k$-Dimensional Measures
  5. A Partitional Approach
  6. A Factorization Approach
  7. Practical Bounds
  8. Using Distributional Information
  9. Non-congesting Resources
  10. Modeling Assumptions
  11. Appendix
  12. Difficulties with Multidimensional Resources
  13. Proofs
  14. An example for using distributional information of operations

Key Result

Theorem 1

The single-dimensional approximability of an operation-resource matrix $W=(w_{ij})$ with resource capacities $B=(B_j)$ is the exactly the reciprocal of the value of the zero-sum game with utilities $u_{ij} = w_{ij}/(B_j \cdot g_i)$ and where $g_i = max_j w_{ij}/B_j$ is the gas measure achieving this

Figures (2)

  • Figure 1: Two Ethereum-like constraints: the quantity $x$ of one resource (gas) limited by $x \le 30$ and the quantity $y$ of another resource (fractional blobs) limited by $y \le 6$. The best single-dimensional gas measure that captures both constraints is $x+5y \le 30$, losing a factor of 2 in capacity in the worst case.
  • Figure 2: Each row of the Operations-Resource Matrix represents an operation, each column represents a resource, and each entry $w_{ij}$ represents the amount of resource $j$ consumed by operation $i$.

Theorems & Definitions (10)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Definition 3: $k$-Dimensional Gas Measure
  • Theorem 2
  • Theorem 3: k-Dim. Representation via Upper-Bounding Factorization
  • Theorem 4
  • Corollary 5
  • Definition 4: Equal Cardinality Partition (ECP)
  • Claim 6