[论文解读] Local Partial Clique Covers for Index Coding

Abhishek Agarwal, Arya Mazumdar|arXiv (Cornell University)|Mar 8, 2016

Cooperative Communication and Network Coding参考文献 13被引用 9

一句话总结

本文提出了一种基于广义局部部分团覆盖（LPC）公式的索引编码新型编码方案，统一并改进了基于局部色数和部分团覆盖的先前界限。所提出的基于线性规划的方法在最优广播速率上实现了比现有方法更紧的上界，包括早期尝试结合这些概念的工作。

ABSTRACT

Index coding, or broadcasting with side information, is a network coding problem of most fundamental importance. In this problem, given a directed graph, each vertex represents a user with a need of information, and the neighborhood of each vertex represents the side information availability to that user. The aim is to find an encoding to minimum number of bits (optimal rate) that, when broadcasted, will be sufficient to the need of every user. Not only the optimal rate is intractable, but it is also very hard to characterize with some other well-studied graph parameter or with a simpler formulation, such as a linear program. Recently there have been a series of works that address this question and provide explicit schemes for index coding as the optimal value of a linear program with rate given by well-studied properties such as local chromatic number or partial clique-covering number. There has been a recent attempt to combine these existing notions of local chromatic number and partial clique covering into a unified notion denoted as the local partial clique cover (Arbabjolfaei and Kim, 2014). We present a generalized novel upper-bound (encoding scheme) - in the form of the minimum value of a linear program - for optimal index coding. Our bound also combines the notions of local chromatic number and partial clique covering into a new definition of the local partial clique cover, which outperforms both the previous bounds, as well as beats the previous attempt to combination. Further, we look at the upper bound derived recently by Thapa et al., 2015, and extend their $n$-$\mathsf{GIC}$ (Generalized Interlinked Cycle) construction to $(k,n)$-$\mathsf{GIC}$ graphs, which are a generalization of $k$-partial cliques.

研究动机与目标

• 通过统一局部色数和部分团覆盖概念，开发更紧的最优索引编码广播速率上界。
• 克服先前方案的局限性，这些方案要么性能不佳，要么未能正确整合两种方法。
• 将GIC（广义互连环）构造推广至(k,n)-GIC图以提升适用范围。
• 基于所提出的LP方法，提供基于MDS码和干扰对齐的构造性编码方案。

提出的方法

• 提出局部部分团覆盖的新定义，将局部色数和部分团覆盖整合到统一框架中。
• 开发一个线性规划（LP）公式，通过约束条件最小化广播速率，确保每个用户能利用其侧信息解码所需数据。
• 对LP进行分数松弛，以通过MDS码和生成矩阵的张量积实现高效编码。
• 应用干扰对齐原理，确保用户在存在重叠传输的情况下仍能解码其数据。
• 将GIC构造扩展至(k,n)-GIC图，推广k-部分团以提升性能。
• 从MDS码和递归编码构造生成矩阵G，通过为每个用户分配合适的列以实现所需速率。

实验结果

研究问题

• RQ1 能否开发一个统一框架，同时改进索引编码中局部色数和部分团覆盖的界限？
• RQ2 如何形式化局部色数与部分团覆盖之间的相互作用，以获得更紧的广播速率上界？
• RQ3 能否将GIC构造推广至(k,n)-GIC图，以在性能上超越k-部分团？
• RQ4 所提出的编码方案的可实现速率是多少，与已知界限相比如何？
• RQ5 所提出的基于LP的方案是否能通过MDS码和干扰对齐实现构造性可实现？

主要发现

• 所提出的局部部分团覆盖公式在性能上优于单独的局部色数和部分团覆盖界限。
• 新的基于LP的方案在最优广播速率上实现了比早期统一两种概念尝试更紧的上界。
• LP的分数松弛使基于MDS码和生成矩阵张量积的构造性编码方案成为可能。
• 该方案支持参与多个部分团的用户的递归编码，所需消息长度为分数系数分母的乘积。
• 广义的(k,n)-GIC构造扩展了GIC框架，并在结构化图上支持性能提升。
• 编码方案满足干扰对齐条件，确保每个用户仅通过广播信号和其侧信息即可解码所需数据。