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An Achievable Scheme for the K-user Linear Computation Broadcast Channel

Yinbin Ma, Daniela Tuninetti

TL;DR

This work tackles the problem of efficiently delivering linear function requests over a K-user Linear Computation Broadcast Channel (LCBC) by exploiting users' linear side information. It introduces a subspace-decomposition framework rooted in representable polymatroid spaces and an accompanying linear-programming formulation to optimize multicast opportunities among all users. The scheme generalizes known results for the $2$- and $3$-user LCBC and provides constructive multicast/unicast message design that can be applied to linear coded placement in coded caching with scalar linear function retrieval. These contributions enhance understanding of LCBC limits and offer a principled design method with potential practical impact on distributed computing and caching systems.

Abstract

This paper presents a new achievable scheme for the K-user Linear Computation Broadcast Channel (K-LCBC). A K-LCBC comprises data stored on a server and K users, each aiming to retrieve a desired linear function of the data by leveraging their prior locally available side information in the form of another linear function of the data. The proposed scheme is based on a subspace decomposition derived from representable polymatroid spaces. This decomposition enables the server to effectively design multicast messages that simultaneously benefit multiple users and allow users to eliminate interference using their available side information. This work extends existing results for the 3-LCBC by introducing a linear programming framework to optimize multicast opportunities across an arbitrary number of users. The proposed approach can be used to derive achievable scheme for the K-user coded caching problem with linear coded placement and scalar linear function retrieval, which was our original motivation to investigate the K-LCBC.

An Achievable Scheme for the K-user Linear Computation Broadcast Channel

TL;DR

This work tackles the problem of efficiently delivering linear function requests over a K-user Linear Computation Broadcast Channel (LCBC) by exploiting users' linear side information. It introduces a subspace-decomposition framework rooted in representable polymatroid spaces and an accompanying linear-programming formulation to optimize multicast opportunities among all users. The scheme generalizes known results for the - and -user LCBC and provides constructive multicast/unicast message design that can be applied to linear coded placement in coded caching with scalar linear function retrieval. These contributions enhance understanding of LCBC limits and offer a principled design method with potential practical impact on distributed computing and caching systems.

Abstract

This paper presents a new achievable scheme for the K-user Linear Computation Broadcast Channel (K-LCBC). A K-LCBC comprises data stored on a server and K users, each aiming to retrieve a desired linear function of the data by leveraging their prior locally available side information in the form of another linear function of the data. The proposed scheme is based on a subspace decomposition derived from representable polymatroid spaces. This decomposition enables the server to effectively design multicast messages that simultaneously benefit multiple users and allow users to eliminate interference using their available side information. This work extends existing results for the 3-LCBC by introducing a linear programming framework to optimize multicast opportunities across an arbitrary number of users. The proposed approach can be used to derive achievable scheme for the K-user coded caching problem with linear coded placement and scalar linear function retrieval, which was our original motivation to investigate the K-LCBC.
Paper Structure (16 sections, 4 theorems, 49 equations, 4 figures, 1 table)

This paper contains 16 sections, 4 theorems, 49 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. System Model
  3. General Subspace Decomposition
  4. Achievable scheme for the K-LCBC
  5. Example for Theorem \ref{['thm:achievabilityLCBCK']}
  6. Linear Programming when ${\mathsf K} = 4$
  7. Construction of Multicast Messages
  8. Toy Example
  9. Conclusion
  10. Proof of Lemma \ref{['lem:decompositionK']}
  11. Remark on Lemma \ref{['lem:decompositionK']}
  12. Proof of Lemma \ref{['lem:multicastmessage']}
  13. Step 1
  14. Step 2
  15. Step 3
  16. ...and 1 more sections

Key Result

Lemma 1

yao2024capacity$\{{\mathbf B}_{1(23)}, {\mathbf B}_{2(13)}, {\mathbf B}_{3(12)}\}$ have identical dimension, and they are pairwise linearly independent.

Figures (4)

  • Figure 1: Optimal 2-LCBC scheme, where user 1 wants $A_\star$ and knows $A_1 + B_1$, and where user 2 wants $B_\star$ and knows $A_2 + B_2$.
  • Figure 2: The decomposition of $\langle {\mathbf U}_1\rangle, \langle {\mathbf U}_2\rangle, \langle {\mathbf U}_3\rangle$ into subspaces, labeled by corresponding bases, from yao2024capacity.
  • Figure 3: Coefficients associated to each basis for ${\mathsf K}=3$ users.
  • Figure 4: Generated binary trees reveals dependency relationship among bases.

Theorems & Definitions (5)

  • Lemma 1
  • Lemma 2
  • Theorem 4.1
  • Remark 1
  • Lemma 3