Performance Bounds on Pliable Index Coding Using Absent Receivers

TL;DR

The paper tackles the problem of bounding the optimal broadcast rate in pliable index coding when some receivers are absent. It introduces decoding chains with skipped messages and an algorithm to construct acyclic subgraphs, connecting pliable IC bounds to index coding MAIS-type bounds. The authors derive a general lower bound beta_q ≥ m − |S| and provide explicit, tight bounds for several classes, including up to four absent receivers and various nested-absent structures, with achievability via uncoded, cyclic, and some linear codes. Collectively, these results reveal how the pattern and number of absent receivers critically shape the achievable rate and furnish a practical toolkit for analyzing non-symmetric pliable IC instances.

Abstract

We characterise bounds on the optimal broadcast rate for a few classes of pliable-index-coding instances. Unlike the majority of currently solved instances, which belong to a special class where all receivers with a certain side-information cardinality are either present or absent, we consider more general instances without this constraint. We devise a novel algorithm that constructs a decoding chain by iteratively adding a message that can be decoded by a receiver whose side information is already in the chain. If the decoding chain cannot proceed due to the absence of a receiver with the required messages, we skip a message by adding it to the chain regardless. We prove that a lower bound on the optimal broadcast rate is a function of the number of skipped messages, across all possible decoding choices of the receivers and any realisation of the algorithm for each decoding choice. While this result is not computationally feasible in isolation, it serves as a basis for deriving explicit lower bounds on the broadcast rate for specific classes of pliable-index-coding instances. These lower bounds depend on the number of absent receivers or the pattern of their side-information sets. Specifically, we explicitly characterise the optimal broadcast rate for instances with up to and including four absent receivers with any side-information pattern, as well as instances where the side-information sets are nested in particular ways.

Figures (7)

• Figure 1: A realisation of Algorithm \ref{['algo:2']} in Example \ref{['ex:realisation-of-algorithm']}, where each row indicates an iteration, the coloured nodes are $C$ after the iteration, and the red nodes are $S$
• Figure 2: Pliable-index-coding instance $\mathcal{P}_1$ in Examples \ref{['ex:condition-1']} and \ref{['ex2']}
• Figure 3: Perfectly 3-nested absent receivers, where there are seven messages (noted by circles), four partitions, and seven absent receivers
• Figure 4: Slightly imperfectly $3$-nested absent receivers, formed by replacing one absent receiver with a present receiver of a smaller side-information set in the perfectly $3$-nested setup depicted in Figure \ref{['fig:nested']}
• Figure 5: $1$-truncated $3$-nested absent receivers, formed by retaining only the upper four absent receivers in the perfectly $3$-nested setup depicted in Figure \ref{['fig:nested']}

Theorems & Definitions (36)

• Example 1
• Remark 1
• Lemma 1
• Lemma 2
• Definition 1
• Lemma 3
• Lemma 4
• Lemma 5
• Example 2
• Theorem 1