Reducing number field defining polynomials: an application to class group computations

In this paper we describe how to compute smallest monic polynomials that define a given number field $\mathbb{K}$. We make use of the one-to-one correspondence between monic defining polynomials of $\mathbb{K}$ and algebraic integers that generate $\mathbb{K}$. Thus, a smallest polynomial corresponds to a vector in the lattice of integers of $\mathbb{K}$ and this vector is short in some sense. The main idea is to consider weighted coordinates for the vectors of the lattice of integers of $\mathbb{K}$. This allows us to find the desired polynomial by enumerating short vectors in these weighted lattices. In the context of the subexponential algorithm of Biasse and Fieker for computing class groups, this algorithm can be used as a precomputation step that speeds up the rest of the computation. It also widens the applicability of their faster conditional method, which requires a defining polynomial of small height, to a much larger set of number field descriptions.

An algorithm for the principal ideal problem in indefinite quaternion algebras

Deciding whether an ideal of a number field is principal and finding a generator is a fundamental problem with many applications in computational number theory. For indefinite quaternion algebras, the decision problem reduces to that in the underlying number field. Finding a generator is hard, and we present a heuristically subexponential algorithm.

TOWARDS MORE EFFICIENT INFECTION AND FIRE FIGHTING

The firefighter problem models the situation where an infection, a computer virus, an idea or fire etc. is spreading through a network and the goal is to save as many as possible nodes of the network through targeted vaccinations. The number of nodes that can be vaccinated at a single time-step is typically one, or more generally O(1). In a non-standard model, the so called spreading model, the vaccinations also spread in contrast to the standard model. Our main results are concerned with general graphs in the spreading model. We provide a very simple exact [Formula: see text]-time algorithm. In the special case of trees, where the standard and spreading model are equivalent, our algorithm is substantially simpler than that exact subexponential algorithm for trees presented in Ref. 2. On the other hand, we show that the firefighter problem on weighted directed graphs in the spreading model cannot be approximated within a constant factor better than 1 − 1/e unless NP ⊆ DTIME (nO( log log n)). We also present several results in the standard model. We provide approximation algorithms for planar graphs in case when at least two vaccinations can be performed at a time-step. We also derive trade-offs between approximation factors for polynomial-time solutions and the time complexity of exact or nearly exact solutions for instances of the firefighter problem for the so called directed layered graphs.