Pliable Index Coding via Conflict-Free Colorings of Hypergraphs

We present a hypergraph coloring based approach to pliable index coding (PICOD). We represent the given PICOD problem using a hypergraph consisting of $m$ messages as vertices and the request-sets of the $n$ 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 $O(\log ^{2}\Gamma)$ , where $\Gamma $ is the maximum number of hyperedges overlapping with any hyperedge. For the $t$ -request PICOD scenario, we show the existence of PICOD schemes of length $\max (O(\log \Gamma \log m), O(t \log m))$ , 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 $t$ -request case are asymptotically optimal, up to a multiplicative factor of $\log t$ . 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.