[Paper Review] Optimal Linear Broadcast Rates of the Two-Sender Unicast Index Coding Problem with Fully-Participated Interactions

Chinmayananda Arunachala, Vaneet Aggarwal+1 more|arXiv (Cornell University)|Aug 29, 2018

Cooperative Communication and Network Coding17 references4 citations

TL;DR

This paper establishes optimal linear broadcast rates and explicit code constructions for the two-sender unicast index coding problem under fully-participated interactions, where all senders collectively possess all messages and receivers have side-information. The optimal rate is derived from the optimal rates of three associated single-sender sub-problems, providing tight lower bounds for related partially-participated interaction cases and enabling scalable solutions for multi-sender index coding systems.

ABSTRACT

The two-sender unicast index coding problem consists of finding optimal coded transmissions from the two senders which collectively know the messages demanded by all the receivers. Each receiver demands a unique message. One important class of this problem consists of the message sets at the senders and the side-information at the receivers satisfying \emph{fully-participated interactions}. This paper provides optimal linear broadcast rates and corresponding code constructions for all the possible cases of the two-sender unicast index coding problem with fully-participated interactions. The optimal linear broadcast rate and the corresponding code for the two-sender problem are given in terms of those of the three single-sender unicast problems associated with the two-sender problem. Optimal linear broadcast rates of two-sender problems with fully-participated interactions provide lower bounds for the optimal linear broadcast rates of many related two-sender problems with \emph{partially-participated interactions}. Proof techniques used to obtain the results for the two-sender problem are shown to be useful in obtaining the results for some cases of the multi-sender unicast index coding problem.

Motivation & Objective

• To determine the optimal linear broadcast rate and corresponding code for all cases of the two-sender unicast index coding problem (TUICP) under fully-participated interactions.
• To express the optimal rate of the two-sender problem in terms of the optimal rates of three constituent single-sender unicast index coding problems.
• To provide tight lower bounds on the optimal linear broadcast rates for related two-sender problems with partially-participated interactions.
• To extend the proof techniques to multi-sender index coding problems, enabling sub-optimal but structured solutions when exact optimality is hard to achieve.
• To establish a framework that reduces the complexity of solving two-sender TUICP to solving simpler single-sender problems.

Proposed method

• The problem is decomposed into three vertex-induced sub-digraphs of the side-information digraph, corresponding to message sets available at sender 1, sender 2, and both senders.
• The optimal linear broadcast rate for the two-sender problem is computed as the sum of the maximum of the optimal rates of the individual sub-digraphs: $\beta^{l}_{t}(\mathcal{D},\mathcal{P}) = \max\{\beta^{l}_{t}(\mathcal{D}_1), \beta^{l}_{t}(\mathcal{D}_{\{1,2\}})\} + \max\{\beta^{l}_{t}(\mathcal{D}_2), \beta^{l}_{t}(\mathcal{D}_{\{1,2\}})\} + \beta^{l}_{t}(\mathcal{D}_{\{1,3\}}) + \beta^{l}_{t}(\mathcal{D}_{\{2,3\}}) + \beta^{l}_{t}(\mathcal{D}_{\{1,2,3\}})$.
• Code construction is performed by combining optimal linear codes from each of the three single-sender sub-problems, ensuring decodability at all receivers using their side-information.
• The interaction digraph is used to classify the problem into cases based on message availability and side-information structure, with fully-participated interactions defined as those where all message interactions are fully shared.
• Proof techniques based on sub-problem decomposition are extended to multi-sender settings, providing sub-optimal but structured solutions when exact optimality is intractable.
• The framework leverages the fact that optimal linear rates for fully-participated cases serve as lower bounds for partially-participated variants with the same sub-problem structure.

Experimental results

Research questions

• RQ1 What is the optimal linear broadcast rate for the two-sender unicast index coding problem when all interactions are fully participated?
• RQ2 How can the optimal rate of the two-sender problem be expressed in terms of the optimal rates of three single-sender sub-problems?
• RQ3 Can the proof techniques used for fully-participated interactions be generalized to derive bounds or solutions for partially-participated interaction cases?
• RQ4 What is the structure of the optimal linear code for two-sender TUICP under fully-participated interactions?
• RQ5 How do the results for fully-participated interactions provide lower bounds for more general two-sender TUICP instances?

Key findings

• The optimal linear broadcast rate for any two-sender unicast index coding problem with fully-participated interactions is given by the sum of the maximum rates from the three single-sender sub-problems, as formalized in the equation $\beta^{l}_{t}(\mathcal{D},\mathcal{P}) = \max\{\beta^{l}_{t}(\mathcal{D}_1), \beta^{l}_{t}(\mathcal{D}_{\{1,2\}})\} + \max\{\beta^{l}_{t}(\mathcal{D}_2), \beta^{l}_{t}(\mathcal{D}_{\{1,2\}})\} + \beta^{l}_{t}(\mathcal{D}_{\{1,3\}}) + \beta^{l}_{t}(\mathcal{D}_{\{2,3\}}) + \beta^{l}_{t}(\mathcal{D}_{\{1,2,3\}})$.
• For the example with 10 messages and fully-participated interactions, the optimal linear broadcast rate is $\beta^{l}_{t}(\mathcal{D},\mathcal{P}) = 5$, with $\beta^{l}_{t}(\mathcal{D}_{\{1,2\}}) = 2$ and all others equal to 1.
• Optimal linear codes are constructed by combining the optimal codes of the three single-sender sub-problems, such as $\mathcal{C}_{\{1,2\}} = ({\bf{x}}_4, {\bf{x}}_8)$ and $\mathcal{C}_{\{2,3\}} = {\bf{x}}_5 \oplus {\bf{x}}_6$.
• The results provide non-trivial lower bounds on the optimal linear broadcast rates for two-sender problems with partially-participated interactions that share the same sub-problem structure.
• The proof techniques are extendable to multi-sender settings, offering a framework to derive sub-optimal solutions when exact optimality is difficult to achieve.
• The framework reduces the complexity of solving two-sender TUICP to solving three single-sender unicast index coding problems, significantly simplifying analysis and code design.