Algorithmic complexity of Greenberg’s conjecture

Algorithmic proof of the epsilon constant conjecture

In this paper we will algorithmically prove the global epsilon constant conjecture for all Galois extensions L/Q of degree at most 15. In fact, we will obtain a slightly more general result whose proof is based on an algorithmic proof of the local epsilon constant conjecture for Galois extensions E/Qp of small degree. To this end we will present an efficient algorithm for the computation of loc...

On the algorithmic aspects of Hedetniemi’s conjecture

We present a polynomial algorithm, implicit in the work of ElZahar and Sauer, which inputs a 3-colouring of a categorical product of two graphs and outputs a 3-colouring of one of the factors. We raise a question about the existence of polynomial algorithms for colouring the vertices of some graphs in terms of intrinsic succint description of the vertices rather than in terms of the (exponentia...

Algorithmic Folding Complexity

How do we most quickly fold a paper strip (modeled as a line) to obtain a desired mountain-valley pattern of equidistant creases (viewed as a binary string)? Define the folding complexity of a mountain-valley string as the minimum number of simple folds required to construct it. We first show that the folding complexity of a length-n uniform string (all mountains or all valleys), and hence of a...

Algorithmic “Kolmogorov” Complexity

The information content or complexity of an object can be measured by the length of its shortest description. For instance the string ‘01010101010101010101010101010101’ has the short description “16 repetitions of 01”, while ‘11001000011000011101111011101100’ presumably has no simpler description other than writing down the string itself. More formally, the Algorithmic “Kolmogorov” Complexity (...

Algorithmic Randomness and Complexity