On the Complexity of Digraph Colourings and Vertex Arboricity

It has been shown by Bokal et al. that deciding 2-colourability of digraphs is an NP-complete problem. This result was later on extended by Feder et al. to prove that deciding whether a digraph has a circular $p$-colouring is NP-complete for all rational $p>1$. In this paper, we consider the complexity of corresponding decision problems for related notions of fractional colourings for digraphs and graphs, including the star dichromatic number, the fractional dichromatic number and the circular vertex arboricity. We prove the following results: Deciding if the star dichromatic number of a digraph is at most $p$ is NP-complete for every rational $p>1$. Deciding if the fractional dichromatic number of a digraph is at most $p$ is NP-complete for every $p>1, p \neq 2$. Deciding if the circular vertex arboricity of a graph is at most $p$ is NP-complete for every rational $p>1$. To show these results, different techniques are required in each case. In order to prove the first result, we relate the star dichromatic number to a new notion of homomorphisms between digraphs, called circular homomorphisms, which might be of independent interest. We provide a classification of the computational complexities of the corresponding homomorphism colouring problems similar to the one derived by Feder et al. for acyclic homomorphisms.Comment: 21 pages, 1 figur

Forecasting economic activity in Germany: how useful are sentiment indicators?

We analyze four economic sentiment indicators for the German economy regarding their ability to forecast economic activity. Using cross correlations and Granger causality tests we find that the ifo business expectations (ifo), the Purchasing Managers Index (PMI) and the ZEW Indicator of Economic Sentiment (ZEW) lead the yearon-year growth rate of German industrial production by five months. Taking into account the publication lag of industrial production this lead is even larger. On the contrary, the European Commission?s Economic Sentiment Indicator (ESIN) does not exhibit a lead but rather seems to coincide or even lag economic activity. Analyzing lead/lag structures among the indicators we find that the ZEW indicator leads the ifo business expectations significantly by one month and that the latter has a onemonth lead over the PMI. Out-of-sample forecast evaluations suggest that both ifo and ZEW provide the best forecasts for industrial production among the three indicators ifo, PMI and ZEW. It is found that the ZEW indicator performs better than the ifo and PMI over the whole sample (Jan. 1994 – Mar. 2002) and especially over horizons from six to twelve months. The ifo expectations predict better at shorter horizons (up to three months) and is superior to the ZEW and PMI indicator when a shorter sample (Jan. 1998 – Mar. 2002) is regarded. --leading indicators,Germany,ifo,zew,PMI,ESIN

Unternehmens- versus Analystenbefragungen – Zum Prognosegehalt von ifo-Geschäftserwartungen und ZEW-Konjunkturerwartungen

In der vorliegenden Arbeit untersuchen wir die Eignung der ifo-Geschäftserwartungen und der ZEW-Konjunkturerwartungen als Frühindikatoren für die deutsche Industrieproduktion. Anhand von Granger-Kausalitätstests wird gezeigt, dass die auf Umfragen unter Finanzanalysten basierenden ZEW-Konjunkturerwartungen einen signifikanten einmonatigen Vorlauf vor den ifo-Geschäftserwartungen haben. Wir führen dies auf unterschiedliche Erwartungsbildungen der beiden Gruppen zurück. Die ZEW-Konjunkturerwartungen erlauben Prognosen der Industrieproduktion für 3 bis 12 Monate im voraus, die signifikant besser als eine naive Vergleichsprognose und eine Prognose mit den ifo-Erwartungen sind. Die Vorhersagen mit Hilfe der ifo-Geschäftserwartungen sind hingegen auf sehr kurze Sicht besser. Mit Hilfe von Encompassing-Tests wird gezeigt, dass für einen Horizont von drei bis sechs Monaten eine Kombination beider Indikatoren bessere Prognoseergebnisse liefert als die Verwendung der jeweiligen Indikatoren alleine. Eine Analyse der Richtungsprognosen zeigt, dass diese mit den beiden Erwartungsindikatoren erheblich besser möglich sind, als mit der naiven Prognose oder einer Zufallsprognose

Forecasting Economic Activity in Germany – How Useful are Sentiment Indicators?

We analyze four economic sentiment indicators for the German economy regarding their ability to forecast economic activity. Using cross correlations and Granger causality tests we find that the ifo business expectations (ifo), the Purchasing Managers Index (PMI) and the ZEW Indicator of Economic Sentiment (ZEW) lead the year-on-year growth rate of German industrial production by five months. Taking into account the publication lag of industrial production this lead is even larger. On the contrary, the European Commission’s Economic Sentiment Indicator (ESIN) does not exhibit a lead but rather seems to coincide or even lag economic activity. Analyzing lead/lag structures among the indicators we find that the ZEW indicator leads the ifo business expectations significantly by one month and that the latter has a onemonth lead over the PMI. Out-of-sample forecast evaluations suggest that both ifo and ZEW provide the best forecasts for industrial production among the three indicators ifo, PMI and ZEW. It is found that the ZEW indicator performs better than the ifo and PMI over the whole sample (Jan. 1994 – Mar. 2002) and especially over horizons from six to twelve months. The ifo expectations predict better at shorter horizons (up to three months) and is superior to the ZEW and PMI indicator when a shorter sample (Jan. 1998 – Mar. 2002) is regarded

Exchange Rate Pass-Through to Consumer Prices: A European Perspective

We study the pass-through of exchange rate changes to consumer prices for the euro area by estimating vector error correction models for Germany, France, Italy, the Netherlands and Spain. Using the weights of the Harmonized Index of Consumer Prices (HICP) we compute a weighted average of the country results for the euro area. We find that in response to a ten percent depreciation of the euro nominal effective exchange rate index, the HICP tends to increase by 0,4 percent after 12 months. The total effect amounts to 0,8 percent and the adjustment of consumer prices is completed after three years

Space-resolved dynamics of a tracer in a disordered solid

The dynamics of a tracer particle in a glassy matrix of obstacles displays slow complex transport as the free volume approaches a critical value and the void space falls apart. We investigate the emerging subdiffusive motion of the test particle by extensive molecular dynamics simulations and characterize the spatio-temporal transport in terms of two-time correlation functions, including the time-dependent diffusion coefficient as well as the wavenumber-dependent intermediate scattering function. We rationalize our findings within the framework of critical phenomena and compare our data to a dynamic scaling theory.Comment: 10 pages, 7 figures, submitted to Journal of Non-Crystalline Solid

Unternehmens- versus Analystenbefragungen: Zum Prognosegehalt von ifo-Geschäftserwartungen und ZEW-Konjunkturerwartungen

In der vorliegenden Arbeit untersuchen wir die Eignung der ifo-Geschäftserwartungen und der ZEW-Konjunkturerwartungen als Frühindikatoren für die deutsche Industrieproduktion. Anhand von Granger-Kausalitätstests wird gezeigt, dass die auf Umfragen unter Finanzanalysten basierenden ZEW-Konjunkturerwartungen einen signifikanten einmonatigen Vorlauf vor den ifo-Geschäftserwartungen haben. Wir führen dies auf unterschiedliche Erwartungsbildungen der beiden Gruppen zurück. Die ZEW-Konjunkturerwartungen erlauben Prognosen der Industrieproduktion für 3 bis 12 Monate im voraus, die signifikant besser als eine naive Vergleichsprognose und eine Prognose mit den ifo-Erwartungen sind. Die Vorhersagen mit Hilfe der ifo-Geschäftserwartungen sind hingegen auf sehr kurze Sicht besser. Mit Hilfe von Encompassing-Tests wird gezeigt, dass für einen Horizont von drei bis sechs Monaten eine Kombination beider Indikatoren bessere Prognoseergebnisse liefert als die Verwendung der jeweiligen Indikatoren alleine. Eine Analyse der Richtungsprognosen zeigt, dass diese mit den beiden Erwartungsindikatoren erheblich besser möglich sind, als mit der naiven Prognose oder einer Zufallsprognose. --Konjunkturerwartungen,Prognosemodelle,Prognosevergleich,Frühindikatoren

Coloring Drawings of Graphs

We consider face-colorings of drawings of graphs in the plane. Given a multi-graph $G$ together with a drawing $\Gamma(G)$ in the plane with only finitely many crossings, we define a face-$k$-coloring of $\Gamma(G)$ to be a coloring of the maximal connected regions of the drawing, the faces, with $k$ colors such that adjacent faces have different colors. By the $4$-color theorem, every drawing of a bridgeless graph has a face-$4$-coloring. A drawing of a graph is facially $2$-colorable if and only if the underlying graph is Eulerian. We show that every graph without degree 1 vertices admits a $3$-colorable drawing. This leads to the natural question which graphs $G$ have the property that each of its drawings has a $3$-coloring. We say that such a graph $G$ is facially $3$-colorable. We derive several sufficient and necessary conditions for this property: we show that every $4$-edge-connected graph and every graph admitting a nowhere-zero $3$-flow is facially $3$-colorable. We also discuss circumstances under which facial $3$-colorability guarantees the existence of a nowhere-zero $3$-flow. On the negative side, we present an infinite family of facially $3$-colorable graphs without a nowhere-zero $3$-flow. On the positive side, we formulate a conjecture which has a surprising relation to a famous open problem by Tutte known as the $3$-flow-conjecture. We prove our conjecture for subcubic and for $K_{3,3}$-minor-free graphs.Comment: 24 pages, 17 figure

Exponential Adoption of Battery Electric Cars

The adoption of battery electric vehicles (BEVs) may significantly reduce greenhouse gas emissions caused by road transport. However, there is wide disagreement as to how soon battery electric vehicles will play a major role in overall transportation. Focusing on battery electric passenger cars, we here analyze BEV adoption across 17 individual countries, Europe, and the World, and consistently find exponential growth trends. Modeling-based estimates of future adoption given past trends suggests system-wide adoption substantially faster than typical economic analyses have proposed so far. For instance, we estimate the majority of passenger cars in Europe to be electric by about 2031. Within regions, the predicted times of mass adoption are largely insensitive to model details. Despite significant differences in current electric fleet sizes across regions, their growth rates consistently indicate fast doubling times of approximately 15 months, hinting at radical economic and infrastructural consequences in the near future