Tight Information Theoretic Converse Results for Some Pliable Index Coding Problems

This paper studies the Pliable Index CODing problem (PICOD), which models content-type distribution networks. In the PICOD ({t}) problem there are {m} messages, {n} users and each user has a distinct message side information set, as in the classical Index Coding problem (IC). Differently from IC, where each user has a pre-specified set of messages to decode, in the PICOD ({t}) a user is "pliable" and is satisfied if it can decode any {t} messages that are not in its side information set. The goal is to find a code with the shortest length that satisfies all the users. This flexibility in determining the desired message sets makes the PICOD ({t}) behave quite differently compared to the IC, and its analysis even more challenging. This paper mainly focuses on the complete - {S} PICOD ({t}) with {m} messages, where the set {S}\subset [{m}] contains the sizes of the side information sets, and the number of users is {n}=\sum _{s\in {S}}\binom {m} {s} , with no two users having the same side information set. Capacity results are shown for: (i) the consecutive complete- {S} PICOD ({t}) , where {S}=[{s}_{\text {min}}:{s}_{\text {max}}] for some 0 \leqslant {s}_{\text {min}}\leqslant {s}_{\text {max}} \leqslant {m}-{t} , and (ii) the complement-consecutive complete- {S} PICOD ({t})