On the complexity of computing the capacity of codes that avoid forbidden difference patterns

TL;DR

This paper investigates the computational complexity of determining the capacity of codes avoiding certain forbidden difference patterns, providing new bounds, algorithms, and complexity results, including NP-hardness and extremal norm existence.

Contribution

It introduces a new family of bounds for approximating capacity, a polynomial-time algorithm for positivity detection, and proves NP-hardness for extended symbol sets.

Findings

New bounds enable exponential-time approximation of capacity.

Polynomial-time algorithm determines if capacity is positive.

NP-hardness established for extended symbol sets.

Abstract

We consider questions related to the computation of the capacity of codes that avoid forbidden difference patterns. The maximal number of $n$-bit sequences whose pairwise differences do not contain some given forbidden difference patterns increases exponentially with $n$. The exponent is the capacity of the forbidden patterns, which is given by the logarithm of the joint spectral radius of a set of matrices constructed from the forbidden difference patterns. We provide a new family of bounds that allows for the approximation, in exponential time, of the capacity with arbitrary high degree of accuracy. We also provide a polynomial time algorithm for the problem of determining if the capacity of a set is positive, but we prove that the same problem becomes NP-hard when the sets of forbidden patterns are defined over an extended set of symbols. Finally, we prove the existence of extremal norms for the sets of matrices arising in the capacity computation. This result makes it possible to apply a specific (even though non polynomial) approximation algorithm. We illustrate this fact by computing exactly the capacity of codes that were only known approximately.