Graph-Theoretic Bounds for Non-Linear Function Computation Broadcast

2026-10-06

AI Summary

This study addresses the problem of minimizing the transmission rate required for lossless recovery of all demands in function computation broadcast with nonlinear functions. We propose a broadcast graph framework that jointly models source distributions, side information, and user demands, enabling nonlinear coding to transcend conventional linear limitations. Building upon Körner’s characteristic graph theory and operation block analysis, we construct achievable schemes applicable to an arbitrary number of users. By combining independent set methods with genie-aided techniques, we derive tight lower bounds and multi-letter response profile bounds. The results precisely characterize the optimal rates for specific scenarios, demonstrating that nonlinear coding strictly outperforms the best linear schemes, while quantifying the corresponding additive gap.

Abstract

This work studies non-linear function computation broadcast (NFCB), in which a sender with access to $N$ datasets $(X_1,\dots,X_N)$ broadcasts a common message to $K$ users, each possessing side information and requesting a function of the datasets. The goal is to minimize the rate required for asymptotically lossless recovery of all demands. We introduce a broadcast graph that jointly captures the source distribution, side information, and demanded functions. Using Körner's characteristic-graph framework, we develop an achievable scheme for arbitrary $K$, general source distributions, and general finite-field demands, including linear, non-separable, and non-linear functions, without restricting the encoding or decoding operations to be linear. We also present a side-information-assisted independent-set scheme and characterize the optimal graph-based achievable rate for compatible functions. For the converse, we derive a multi-letter response-profile bound that strengthens a basic side-information converse, zero-error and asymptotically lossless clique-entropy bounds based on the operational block broadcast graph, and a genie-aided lower bound. For binary NFCB with $N=K$ and side information $X_i$ at user $i$, we characterize the optimal rate in several special cases and bound the worst-case and average additive gaps between the proposed achievable rate and the genie-aided converse for $K=3$ and $K=4$. Finally, three-user examples with Boolean and linear demands illustrate the proposed bounds and show that non-linear encoding can strictly outperform the best scalar and vector linear schemes.

Problem

• Non-linear function computation broadcast
• Broadcast rate minimization
• Side information
• Graph-theoretic bounds
• Asymptotically lossless recovery

Innovation

• Non-linear function computation broadcast
• Broadcast graph
• Characteristic-graph framework
• Genie-aided bound
• Non-linear encoding