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Computer-aided Characterization of Fundamental Limits of Coded Caching with Linear Coding

Niccolò Brembilla, Yinbin Ma, Pietro Belotti, Federico Malucelli, Daniela Tuninetti

TL;DR

Results seem to indicate that small, structured demand subsets combined with minimal common information constructions may be sufficient to characterize optimal tradeoffs under linear coding, and prove the optimality of some achievable memory-load tradeoff points under the constraint of linear coding placement and delivery.

Abstract

Inspired by prior work by Tian and by Cao and Xu, this paper presents an efficient computer-aided framework to characterize the fundamental limits of coded caching systems under the constraint of linear coding. The proposed framework considers non-Shannon-type inequalities which are valid for representable polymatroids (and hence for linear codes), and leverages symmetric structure and problem-specific constraints of coded caching to reduce the complexity of the linear program. The derived converse bounds are tighter compared to previous known analytic methods, and prove the optimality of some achievable memory-load tradeoff points under the constraint of linear coding placement and delivery. These results seem to indicate that small, structured demand subsets combined with minimal common information constructions may be sufficient to characterize optimal tradeoffs under linear coding.

Computer-aided Characterization of Fundamental Limits of Coded Caching with Linear Coding

TL;DR

Results seem to indicate that small, structured demand subsets combined with minimal common information constructions may be sufficient to characterize optimal tradeoffs under linear coding, and prove the optimality of some achievable memory-load tradeoff points under the constraint of linear coding placement and delivery.

Abstract

Inspired by prior work by Tian and by Cao and Xu, this paper presents an efficient computer-aided framework to characterize the fundamental limits of coded caching systems under the constraint of linear coding. The proposed framework considers non-Shannon-type inequalities which are valid for representable polymatroids (and hence for linear codes), and leverages symmetric structure and problem-specific constraints of coded caching to reduce the complexity of the linear program. The derived converse bounds are tighter compared to previous known analytic methods, and prove the optimality of some achievable memory-load tradeoff points under the constraint of linear coding placement and delivery. These results seem to indicate that small, structured demand subsets combined with minimal common information constructions may be sufficient to characterize optimal tradeoffs under linear coding.

Paper Structure

This paper contains 21 sections, 4 theorems, 16 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation and Known Results
  3. Methodology for LP reduction
  4. Exploitation of Symmetries
  5. Constraint Pruning
  6. Variable Merging
  7. Demand Types
  8. Further Considerations from Decoding Constraints
  9. Integration of Common Information (CI)
  10. Numerical Results
  11. 4U4F: Optimality at $M=1$
  12. 5U5F: Optimality at $M=1$
  13. 6U6F: Optimality at ${\mathsf M}=2$ and Near-Optimality at ${\mathsf M}=1$
  14. Remarks
  15. Conclusion
  16. ...and 6 more sections

Key Result

Theorem 2.1

The lower convex envelope of the following points is achievable, for every $t \in [0:{\mathsf K}]$. Furthermore, the YMA scheme is optimal under the constraint of uncoded placement; and to within a multiplicative factor of two otherwise.

Figures (6)

  • Figure 1: A symmetric example on ${\mathsf K}={\mathsf N}=3$. Permutation on $X_\star$ refers to tian2018symmetry.
  • Figure 2: 4U4F case: achievability is derived from \ref{['eq:performanceYMA']}-\ref{['eq:performanceGV']}. \ref{['p:t4t3']} proves that YMA scheme is optimal at ${\mathsf M}=1$.
  • Figure 3: 5U5F case: achievability is derived from \ref{['eq:performanceYMA']}-\ref{['eq:performanceGV']}. \ref{['p:t5t3']} proves that YMA scheme is optimal at ${\mathsf M}=1$.
  • Figure 4: 6U6F case: achievability from \ref{['eq:performanceYMA']}-\ref{['eq:performanceGV']}. Tradeoffs \ref{['p:t6t3']} and \ref{['p:t6t4']} prove that YMA scheme is optimal at ${\mathsf M}=2$, and appear to get close to optimality at ${\mathsf M}=1$.
  • Figure 5: 7U7F case: achievability from \ref{['eq:performanceYMA']}-\ref{['eq:performanceGV']}.
  • ...and 1 more figures

Theorems & Definitions (4)

  • Theorem 2.1: YMA scheme yu2017exactyu2018characterizing
  • Theorem 2.2: Gomez scheme gomez2018fundamental
  • Theorem 2.3: yu2018characterizing
  • Theorem 2.4: yu2018characterizing