Pliable Index Coding via Conflict-Free Colorings of Hypergraphs

• Published in: IEEE transactions on information theory Vol. 70; no. 6; pp. 3903 - 3921
• Main Authors: Krishnan, Prasad, Mathew, Rogers, Kalyanasundaram, Subrahmanyam
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.06.2024; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Apexes; Codes; Color coding; Coloring; conflict-free coloring; Encoding; Graph theory; Graphs; hypergraph coloring; Index coding; Indexes; Parameters; pliable index coding; Polynomials; Receivers; Servers; Symbols; Upper bound; Upper bounds
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2024.3355416

We present a hypergraph coloring based approach to pliable index coding (PICOD). We represent the given PICOD problem using a hypergraph consisting of <inline-formula> <tex-math notation="LaTeX">m </tex-math></inline-formula> messages as vertices and the request-sets of the <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula> clients as hyperedges. A conflict-free coloring of a hypergraph is an assignment of colors to its vertices so that each hyperedge contains a uniquely colored vertex. We show that various parameters arising out of conflict-free colorings (and some new variants) of the PICOD hypergraph result in new upper bounds for the optimal PICOD length. Using these new upper bounds, we show the existence of single-request PICOD schemes with length <inline-formula> <tex-math notation="LaTeX">O(\log ^{2}\Gamma) </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">\Gamma </tex-math></inline-formula> is the maximum number of hyperedges overlapping with any hyperedge. For the <inline-formula> <tex-math notation="LaTeX">t </tex-math></inline-formula>-request PICOD scenario, we show the existence of PICOD schemes of length <inline-formula> <tex-math notation="LaTeX">\max (O(\log \Gamma \log m), O(t \log m)) </tex-math></inline-formula>, under some mild conditions on the graph parameters. These results improve upon earlier work in general. We also show that our achievable lengths in the <inline-formula> <tex-math notation="LaTeX">t </tex-math></inline-formula>-request case are asymptotically optimal, up to a multiplicative factor of <inline-formula> <tex-math notation="LaTeX">\log t </tex-math></inline-formula>. Our existence results are accompanied by randomized constructive algorithms, which have complexity polynomial in the parameters of the PICOD problem, in expectation or with high probability.