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

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.