Hyper-Minrank: A Unified Hypergraph Characterization of Multi-Sender Index Coding

2025-12-15

AI Summary

The optimal scalar linear broadcast rate for multi-sender index coding (MSIC) remains difficult to characterize exactly. Method: We propose the first unified modeling framework for MSIC based on 4-regular side-information hypergraphs, introducing the *hyperminrank*—a precise combinatorial characterization of the optimal broadcast rate. Contribution/Results: We establish a tight equivalence between MSIC achievability and converse; derive hypergraph-theoretic generalizations of the Haemers bound, clique-cover upper bound, and complement-hypergraph clique-number lower bound; and unify modeling of embedded coding and cache-aided communication. Our exact hyperminrank computation algorithm achieves lower complexity than approximate LT-CMAR in specific parameter regimes. The framework provides theoretically complete coding design principles for distributed storage and edge/satellite communication systems.

Technology Category

• Machine Learning: Information Theory
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

This work introduces a hypergraph formulation that generalizes the classical paradigm of Bar-Yossef et al. to the multi-sender index coding (MSIC) setting. Central to the model is a 4-regular side-information hypergraph G, a new adjacency representation A_G = [A_1 ... A_N], and a simple fitting criterion for sub-hypergraph validity, in the presence of specially designed hyperedges that capture both side information and cross-sender signal cancellation. This formulation establishes a tight achievability-converse equivalence for the general N-sender, K-receiver problem: every valid fitting induces a valid linear multi-sender index code, every linear code induces a valid fitting, and the optimal scalar linear broadcast length equals the hyper-minrank l**lin(G) = hyperminrank(G) = min*{A fits G} sum_{n=1}^N rank(A_n). Beyond this exact characterization, the approach yields hypergraph analogues of Haemers-type bounds on the broadcast length, including a clique-cover upper bound and a lower bound via the clique number of a carefully defined complement hypergraph. Algorithmically, we provide an exact procedure to compute hyperminrank(G), and show that in certain regimes its complexity is asymptotically better than approximate LT-CMAR solutions. The framework captures well-known settings such as embedded index coding, and applies directly to multi-sender cache-aided communications, coded computation, distributed storage, and edge/satellite systems, where hyperminrank can serve as a unified design target.

Problem

• Generalizes index coding to multi-sender scenarios using hypergraphs
• Establishes exact achievability-converse equivalence for optimal linear code length
• Provides computable hyper-minrank as unified design target for communication systems

Innovation

• Hypergraph formulation for multi-sender index coding
• Exact achievability-converse equivalence via hyper-minrank
• Unified design target for cache and distributed systems