# On Upper Bounding Shannon Capacity of Graph Through Generalized Conic Programming

## TL;DR

This paper investigates the potential of generalized conic programming to upper bound the Shannon capacity of graphs, concluding that such approaches cannot surpass existing bounds like the Lovasz number.

## Contribution

It introduces a unified conic programming framework for upper bounding Shannon capacity and proves the limitations of this approach.

## Key findings

- General conic programming cannot improve upon the Lovasz number for Shannon capacity.
- Previous sum-of-squares based bounds are special cases within the conic programming framework.
- The paper establishes fundamental limitations of conic programming methods in this context.

## Abstract

The Shannon capacity of a graph is an important graph invariant in information theory that is extremely difficult to compute. The Lovasz number, which is based on semidefinite programming relaxation, is a well-known upper bound for the Shannon capacity. To improve this upper bound, previous researches tried to generalize the Lovasz number using the ideas from the sum-of-squares optimization. In this paper, we consider the possibility of developing general conic programming upper bounds for the Shannon capacity, which include the previous attempts as special cases, and show that it is impossible to find better upper bounds for the Shannon capacity along this way.