# Storage Capacity as an Information-Theoretic Vertex Cover and the Index Coding Rate

## TL;DR

This paper explores the storage capacity of graphs in distributed storage, relating it to vertex covers and index coding, providing approximation algorithms, exact characterizations for certain graph families, and extending to partial node failure recovery.

## Contribution

It introduces a new information-theoretic perspective on storage capacity, develops approximation schemes, a gadget covering method, and extends the concept to partial failure scenarios.

## Key findings

- Provides 3/2 approximation for planar graphs' storage capacity
- Offers a polynomial-time approximation scheme for index coding rate with constant alphabet size
- Derives exact storage capacity for cycles-with-chords graphs

## Abstract

Motivated by applications in distributed storage, the storage capacity of a graph was recently defined to be the maximum amount of information that can be stored across the vertices of a graph such that the information at any vertex can be recovered from the information stored at the neighboring vertices. Computing the storage capacity is a fundamental problem in network coding and is related, or equivalent, to some well-studied problems such as index coding with side information and generalized guessing games. In this paper, we consider storage capacity as a natural information-theoretic analogue of the minimum vertex cover of a graph. Indeed, while it was known that storage capacity is upper bounded by minimum vertex cover, we show that by treating it as such we can get a 3/2 approximation for planar graphs, and a 4/3 approximation for triangle-free planar graphs. Since the storage capacity is intimately related to the index coding rate, we get a 2 approximation of index coding rate for planar graphs and 3/2 approximation for triangle-free planar graphs. We also show a polynomial time approximation scheme for the index coding rate when the alphabet size is constant. We then develop a general method of "gadget covering" to upper bound the storage capacity in terms of the average of a set of vertex covers. This method is intuitive and leads to the exact characterization of storage capacity for various families of graphs. As an illustrative example, we use this approach to derive the exact storage capacity of cycles-with-chords, a family of graphs related to outerplanar graphs. Finally, we generalize the storage capacity notion to include recovery from partial node failures in distributed storage. We show tight upper and lower bounds on this partial recovery capacity that scales nicely with the fraction of failures in a vertex.