Hyper-Minrank: A Unified Hypergraph Characterization of Multi-Sender Index Coding
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908713382576128 |
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| author | Khalesi, Ali Elia, Petros |
| author_facet | Khalesi, Ali Elia, Petros |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_13615 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hyper-Minrank: A Unified Hypergraph Characterization of Multi-Sender Index Coding Khalesi, Ali Elia, Petros Information Theory Combinatorics 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. |
| title | Hyper-Minrank: A Unified Hypergraph Characterization of Multi-Sender Index Coding |
| topic | Information Theory Combinatorics |
| url | https://arxiv.org/abs/2512.13615 |