Surprised by method - functional method and systems theory

"Der Autor versteht diesen Artikel als einen Beitrag zu der Diskussion, wie Systemtheorie und empirische Forschung kombiniert werden können. Dabei fokussiert er auf die funktionale Methode, die einerseits als die Methode der Systemtheorie behauptet, andererseits aber oft nur beiläufig erwähnt wird - in den späteren Werken Niklas Luhmanns sowie in neueren systemtheoretischen Diskussionen. Der Kern seiner Argumentation ist, dass die funktionale Methode eine wichtige Antriebskraft in der Entwicklung interessanter empirischer Fragen und Analysen sein kann. Der erste Teil des Artikels rekonstruiert hierzu die zentralen Merkmale der funktionalen Methode: Es wird gezeigt, wie mittels der Methode Beobachtungen generiert werden, und es wird die Frage aufgeworfen, für welche Forschungsprobleme die funktionale Methode eine Lösung bieten kann. Im letzten Teil setzt der Autor die funktionale Methode zu den Entwicklungen in Niklas Luhmanns späten Werken in Beziehung, insbesondere zur Theorie der Beobachtung zweiter Ordnung. Allgemeiner Zweck des Beitrags ist es, zentrale Kennzeichen der funktionalen Methode zu rekonstruieren, um zu zeigen, wie sie funktioniert und wo ihre Begrenzungen liegen könnten." (Autorenreferat)"The paper is a contribution to the discussions on how to combine systems theory and empirical research. The paper focuses on functional method, which on the one hand is claimed as the method of systems theory but on the other hand is often only mentioned in passing - in Niklas Luhmann's later works as well as in recent discussions on systems theory. The contention of the paper is that functional method can still be an important driving force in the development of interesting empirical problematics and analyses. The first and major part of the paper is a reconstruction of main characteristics of functional method. It is demonstrated how the method generates observations and the question is raised about which problem(s) the method is a solution to. The second part discusses functional method in relation to Niklas Luhmann's later theoretical developments, especially the theory of second order observation. The overall aim of the paper is to reconstruct central traits of functional method in order to demonstrate how it works, what its function is - and where its limitations might lie." (author's abstract

Finding Even Cycles Faster via Capped k-Walks

In this paper, we consider the problem of finding a cycle of length $2k$ (a $C_{2k}$) in an undirected graph $G$ with $n$ nodes and $m$ edges for constant $k\ge2$. A classic result by Bondy and Simonovits [J.Comb.Th.'74] implies that if $m \ge100k n^{1+1/k}$, then $G$ contains a $C_{2k}$, further implying that one needs to consider only graphs with $m = O(n^{1+1/k})$. Previously the best known algorithms were an $O(n^2)$ algorithm due to Yuster and Zwick [J.Disc.Math'97] as well as a $O(m^{2-(1+\lceil k/2\rceil^{-1})/(k+1)})$ algorithm by Alon et al. [Algorithmica'97]. We present an algorithm that uses $O(m^{2k/(k+1)})$ time and finds a $C_{2k}$ if one exists. This bound is $O(n^2)$ exactly when $m=\Theta(n^{1+1/k})$. For $4$-cycles our new bound coincides with Alon et al., while for every $k>2$ our bound yields a polynomial improvement in $m$. Yuster and Zwick noted that it is "plausible to conjecture that $O(n^2)$ is the best possible bound in terms of $n$". We show "conditional optimality": if this hypothesis holds then our $O(m^{2k/(k+1)})$ algorithm is tight as well. Furthermore, a folklore reduction implies that no combinatorial algorithm can determine if a graph contains a $6$-cycle in time $O(m^{3/2-\epsilon})$ for any $\epsilon>0$ under the widely believed combinatorial BMM conjecture. Coupled with our main result, this gives tight bounds for finding $6$-cycles combinatorially and also separates the complexity of finding $4$- and $6$-cycles giving evidence that the exponent of $m$ in the running time should indeed increase with $k$. The key ingredient in our algorithm is a new notion of capped $k$-walks, which are walks of length $k$ that visit only nodes according to a fixed ordering. Our main technical contribution is an involved analysis proving several properties of such walks which may be of independent interest.Comment: To appear at STOC'1

Near-Optimal Induced Universal Graphs for Bounded Degree Graphs

A graph $U$ is an induced universal graph for a family $F$ of graphs if every graph in $F$ is a vertex-induced subgraph of $U$. For the family of all undirected graphs on $n$ vertices Alstrup, Kaplan, Thorup, and Zwick [STOC 2015] give an induced universal graph with $O\!\left(2^{n/2}\right)$ vertices, matching a lower bound by Moon [Proc. Glasgow Math. Assoc. 1965]. Let $k= \lceil D/2 \rceil$. Improving asymptotically on previous results by Butler [Graphs and Combinatorics 2009] and Esperet, Arnaud and Ochem [IPL 2008], we give an induced universal graph with $O\!\left(\frac{k2^k}{k!}n^k \right)$ vertices for the family of graphs with $n$ vertices of maximum degree $D$. For constant $D$, Butler gives a lower bound of $\Omega\!\left(n^{D/2}\right)$. For an odd constant $D\geq 3$, Esperet et al. and Alon and Capalbo [SODA 2008] give a graph with $O\!\left(n^{k-\frac{1}{D}}\right)$ vertices. Using their techniques for any (including constant) even values of $D$ gives asymptotically worse bounds than we present. For large $D$, i.e. when $D = \Omega\left(\log^3 n\right)$, the previous best upper bound was ${n\choose\lceil D/2\rceil} n^{O(1)}$ due to Adjiashvili and Rotbart [ICALP 2014]. We give upper and lower bounds showing that the size is ${\lfloor n/2\rfloor\choose\lfloor D/2 \rfloor}2^{\pm\tilde{O}\left(\sqrt{D}\right)}$. Hence the optimal size is $2^{\tilde{O}(D)}$ and our construction is within a factor of $2^{\tilde{O}\left(\sqrt{D}\right)}$ from this. The previous results were larger by at least a factor of $2^{\Omega(D)}$. As a part of the above, proving a conjecture by Esperet et al., we construct an induced universal graph with $2n-1$ vertices for the family of graphs with max degree $2$. In addition, we give results for acyclic graphs with max degree $2$ and cycle graphs. Our results imply the first labeling schemes that for any $D$ are at most $o(n)$ bits from optimal