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

The European public sphere, the media, and support for European integration

The goal of this study is to find out whether the emergence of a European public sphere (EPS) contributes to EU support in its affective and evaluative dimensions. The EPS is conceptualized as an entity resting on Europeanized national public spheres. The media is seen as the most important place where Europeanization manifests itself. As an example of Europeanized public spheres, data on the coverage of the 2004 European Parliament election is used. The data set involves measures on EU salience in the media and tone of the messages, and is analyzed together with individual level data on EU support from the 2004 European Election Study. In contrast to earlier approaches to similar data of the 1999 EP elections, multilevel analysis is used to account for the hierarchical structure of the data. In particular, this study analyzes whether the media exerts an independent effect on evaluative and affective support. Additionally, it focuses on whether the impact of cognitive mobilization and egocentric utilitarianism as individual level predictors for EU support varies depending on the Europeanization of national public spheres. A positive, significant effect was found between the tone of the media context and evaluative support. Cross-level interaction effects revealed that EU salience significantly increases the importance of class affiliation when it comes to affective and evaluative support. Utilitarian considerations are found to differentiate people in their EU support only in the presence of negative EU coverage, but not when coverage is positive. This study comes to the conclusion that an emerging EPS can have a positive effect on support for the EU, resulting in a more critical, but also more alerted public

The European public sphere, the media, and support for European integration

"The goal of this study is to find out whether the emergence of a European public sphere (EPS) contributes to EU support in its affective and evaluative dimensions. The EPS is conceptualized as an entity resting on Europeanized national public spheres. The media is seen as the most important place where Europeanization manifests itself. As an example of Europeanized public spheres, data on the coverage of the 2004 European Parliament election is used. The data set involves measures on EU salience in the media and tone of the messages, and is analyzed together with individual level data on EU support from the 2004 European Election Study. In contrast to earlier approaches to similar data of the 1999 EP elections, multilevel analysis is used to account for the hierarchical structure of the data. In particular, this study analyzes whether the media exerts an independent effect on evaluative and affective support. Additionally, it focuses on whether the impact of cognitive mobilization and egocentric utilitarianism as individual level predictors for EU support varies depending on the Europeanization of national public spheres. A positive, significant effect was found between the tone of the media context and evaluative support. Cross-level interaction effects revealed that EU salience significantly increases the importance of class affiliation when it comes to affective and evaluative support. Utilitarian considerations are found to differentiate people in their EU support only in the presence of negative EU coverage, but not when coverage is positive. This study comes to the conclusion that an emerging EPS can have a positive effect on support for the EU, resulting in a more critical, but also more alerted public." (author's abstract