Conference Talk. Bertelsmann Policy Brief 04.2019

Commission President Ursula von der Leyen has announced a two-year Conference on the Future of Europe. Even citizens ought to participate. But how? In order to make participatory democracy a reality, it is essential to avoid only paying lip-service to the idea of participation — and give citizens a real say

Unsatisfiable Linear CNF Formulas Are Large and Complex

We call a CNF formula linear if any two clauses have at most one variable in common. We show that there exist unsatisfiable linear k-CNF formulas with at most 4k^2 4^k clauses, and on the other hand, any linear k-CNF formula with at most 4^k/(8e^2k^2) clauses is satisfiable. The upper bound uses probabilistic means, and we have no explicit construction coming even close to it. One reason for this is that unsatisfiable linear formulas exhibit a more complex structure than general (non-linear) formulas: First, any treelike resolution refutation of any unsatisfiable linear k-CNF formula has size at least 2^(2^(k/2-1))$. This implies that small unsatisfiable linear k-CNF formulas are hard instances for Davis-Putnam style splitting algorithms. Second, if we require that the formula F have a strict resolution tree, i.e. every clause of F is used only once in the resolution tree, then we need at least a^a^...^a clauses, where a is approximately 2 and the height of this tower is roughly k.Comment: 12 pages plus a two-page appendix; corrected an inconsistency between title of the paper and title of the arxiv submissio

Recognising Multidimensional Euclidean Preferences

Euclidean preferences are a widely studied preference model, in which decision makers and alternatives are embedded in d-dimensional Euclidean space. Decision makers prefer those alternatives closer to them. This model, also known as multidimensional unfolding, has applications in economics, psychometrics, marketing, and many other fields. We study the problem of deciding whether a given preference profile is d-Euclidean. For the one-dimensional case, polynomial-time algorithms are known. We show that, in contrast, for every other fixed dimension d > 1, the recognition problem is equivalent to the existential theory of the reals (ETR), and so in particular NP-hard. We further show that some Euclidean preference profiles require exponentially many bits in order to specify any Euclidean embedding, and prove that the domain of d-Euclidean preferences does not admit a finite forbidden minor characterisation for any d > 1. We also study dichotomous preferencesand the behaviour of other metrics, and survey a variety of related work.Comment: 17 page

Non-Autonomous Maximal Regularity for Forms of Bounded Variation

We consider a non-autonomous evolutionary problem $$u' (t)+\mathcal A (t)u(t)=f(t), \quad u(0)=u_0,$$ where $V, H$ are Hilbert spaces such that $V$ is continuously and densely embedded in $H$ and the operator $\mathcal A (t)\colon V\to V^\prime$ is associated with a coercive, bounded, symmetric form $\mathfrak{a}(t,.,.)\colon V\times V \to \mathbb{C}$ for all $t \in [0,T]$. Given $f \in L^2(0,T;H)$, $u_0\in V$ there exists always a unique solution $u \in MR(V,V'):= L^2(0,T;V) \cap H^1(0,T;V')$. The purpose of this article is to investigate when $u \in H^1(0,T;H)$. This property of maximal regularity in $H$ is not known in general. We give a positive answer if the form is of bounded variation; i.e., if there exists a bounded and non-decreasing function $g \colon [0,T] \to \mathbb{R}$ such that \begin{equation*} \lvert\mathfrak{a}(t,u,v)- \mathfrak{a}(s,u,v)\rvert \le [g(t)-g(s)] \lVert u \rVert_V \lVert v \rVert_V \quad (s,t \in [0,T], s \le t). \end{equation*} In that case, we also show that $u(.)$ is continuous with values in $V$. Moreover we extend this result to certain perturbations of $\mathcal A (t)$.Comment: 22 page