# Generalized Alignment Chain: Improved Converse Results for Index Coding

## TL;DR

This paper introduces a generalized alignment chain for index coding, leading to tighter converse bounds that outperform existing bounds like MAIS, especially for large problems, and extends these results to multi-server scenarios.

## Contribution

It generalizes the alignment chain concept to derive improved converse bounds for index coding, applicable to single and multi-server cases, and identifies instances requiring non-Shannon inequalities.

## Key findings

- New bounds are always as tight as MAIS and can outperform it.
- The bounds are computationally practical for large problems.
- Identifies smaller instances where non-Shannon inequalities are necessary.

## Abstract

In this paper, we study the information-theoretic converse for the index coding problem. We generalize the definition for the alignment chain, introduced by Maleki et al., to capture more flexible relations among interfering messages at each receiver. Based on this, we derive improved converse results for the single-server index coding problem. Compared to the maximum acyclic induced subgraph (MAIS) bound, the new bounds are always as tight and can strictly outperform the MAIS bound. They can also be useful for large problems, where the generally tighter polymatroidal bound is computationally impractical. We then extend these new bounds to the multi-server index coding problem. We also present a separate, but related result where we identify a smaller single-server index coding instance, compared to those identified in the literature, for which non-Shannon-type inequalities are necessary to give a tighter converse.