Algorithmic and Hardness Results for the Colorful Components Problems

In this paper we investigate the colorful components framework, motivated by applications emerging from comparative genomics. The general goal is to remove a collection of edges from an undirected vertex-colored graph $G$ such that in the resulting graph $G'$ all the connected components are colorful (i.e., any two vertices of the same color belong to different connected components). We want $G'$ to optimize an objective function, the selection of this function being specific to each problem in the framework. We analyze three objective functions, and thus, three different problems, which are believed to be relevant for the biological applications: minimizing the number of singleton vertices, maximizing the number of edges in the transitive closure, and minimizing the number of connected components. Our main result is a polynomial time algorithm for the first problem. This result disproves the conjecture of Zheng et al. that the problem is $NP$-hard (assuming $P \neq NP$). Then, we show that the second problem is $APX$-hard, thus proving and strengthening the conjecture of Zheng et al. that the problem is $NP$-hard. Finally, we show that the third problem does not admit polynomial time approximation within a factor of $|V|^{1/14 - \epsilon}$ for any $\epsilon > 0$, assuming $P \neq NP$ (or within a factor of $|V|^{1/2 - \epsilon}$, assuming $ZPP \neq NP$).Comment: 18 pages, 3 figure

Unfolding symmetric Bogdanov-Takens bifurcations for front dynamics in a reaction-diffusion system

This manuscript extends the analysis of a much studied singularly perturbed three-component reaction-diffusion system for front dynamics in the regime where the essential spectrum is close to the origin. We confirm a conjecture from a preceding paper by proving that the triple multiplicity of the zero eigenvalue gives a Jordan chain of length three. Moreover, we simplify the center manifold reduction and computation of the normal form coefficients by using the Evans function for the eigenvalues. Finally, we prove the unfolding of a Bogdanov-Takens bifurcation with symmetry in the model. This leads to stable periodic front motion, including stable traveling breathers, and these results are illustrated by numerical computations.Comment: 39 pages, 7 figure

Quantifying the land footprint of Germany and the EU using a hybrid accounting model

Footprint analysis reveals the appropriation of land resources from a consumer’s perspective. We here present a novel hybrid land-flow accounting method for the calculation of land footprints, employing a globally consistent top-down approach and combining physical with environmental economic accounting. Physical accounting tracks food products from ‘field to plate’ and non-food from ‘field to industrial use’ using the large harmonized FAO data to track biomass flows and related land use in physical volumes (tons of biomass). Environmental-economic accounting is used to further track non-food commodities in monetary values to final consumption. The hybrid methodology has been applied annually between 1995 and 2010 for 21 regional markets globally and including major economies separately (e.g. USA, China, India). Per capita extents and composition of cropland footprints vary widely across the world. Detailed results for Germany and the EU28 highlight the higher land demand of livestock-based diets compared to crop-based diets, the growing integration in international markets, and the growing importance of the non-food sector since 2000. Today the land footprint of each Germany citizen appropriates on average 2693 m2 cropland (about one half for livestock-based diets, one quarter for crop-based diets and one quarter for non-food products). Additional 1655 m2 of grassland per capita are used for the consumption of ruminant livestock products. Germany is a major and increasing trading partner with current net ‘cropland imports’ of 10.6 Mha. Overall, half of Germany’s 22 Mha cropland footprint relies on domestic cultivation and half on land resources abroad. Albeit large uncertainties in the calculation of grassland footprints, results point towards Germany being a significant net importer of grassland embedded in ruminant livestock products

Coral Population Trajectories, Increased Disturbance and Management Intervention: A Sensitivity Analysis

Coral reefs distant from human population were sampled in the Red Sea and one-third showed degradation by predator outbreaks (crown-of-thorns-starfish = COTS observed in all regions in all years) or bleaching (1998, 2010). Models were built to assess future trajectories. They assumed variable coral types (slow/fast growing), disturbance frequencies (5,10,20 years), mortality (equal or not), and connectivity (un/connected to un/disturbed community). Known disturbances were used to parameterize models. Present and future disturbances were estimated from remote-sensing chlorophyll and temperature data. Simulations and sensitivity analysis suggest community resilience at \u3e20-year disturbance frequency, but degradation at higher frequency. Trajectories move from fast-grower to slow-grower dominance at intermediate disturbance frequency, then again to fast-grower dominance. A similar succession was observed in the field: Acropora to Porites to Stylophora/Pocillopora dominance on shallow reefs, and a transition from large poritids to small faviids on deep reefs. Synthesis and application: Even distant reefs are impacted by global changes. COTS impacts and bleaching were key driver of coral degradation, coral population decline could be reduced if these outbreaks and bleaching susceptibility were managed by maintaining water quality and by other interventions. Just leaving reefs alone, seems no longer a satisfactory option

Review of land flow accounting methods and recommendations for further development

Robust land footprint indicators can potentially extend the consumption-based resource use indicator of the German sustainability strategy, which focuses on abiotic resources including fossil fuels, metals, and construction and industrial minerals and decidedly excludes biotic resources. Various approaches exist for quantifying the land embodied in international trade flows and consumption, i.e. the land footprint. These can be classified into a) environmental-economic accounting approaches, applying input-output analysis and tracking supply chains in monetary values, b) physical accounting approaches, using an accounting framework based on data for production, trade and utilization of agricultural and forestry commodities and tracking supply chains in physical units, and c) hybrid accounting, combining elements from both environmental-economic and physical accounting. The results of recent studies vary widely, indicating a lack of robustness and thus hampering their application in policy making. This report provides an in-depth review of the current state of the art in measuring land footprints. We identify differences in available accounting methods and indicate their shortcomings, which are mainly attributable to the product and supply chain coverage and detail, and biases introduced by the use of monetary flows as a proxy for actual physical flows. We offer options and give clear recommendations for the further development of actual and virtual global biomass and land flow accounting methods, particularly highlighting the advantages of hybrid accounting approaches as a framework for the robust and transparent assessment of land footprints associated with global biomass flows

Trends and driving forces of China's virtual land consumption and trade

Land resources are important for China’s rapid economic development, especially for food and construction. China’s land resources are under tremendous pressures, and therefore land use is increasingly displaced to other parts of the world. This study analyses the evolution and driving forces of China’s land consumption from 1995 to 2015. The main results show that China’s land footprint increased from 8.8% of the global land resources under human use in 1995 to 15.7% in 2015. China’s domestic land resources are mainly used for serving domestic consumption. Moreover, China needs to import virtual land from foreign countries to satisfy 30.8% of its land demand. Among the three land use types of cropland, grassland and forests, grassland had the largest fraction in China’s land footprint from 1995 to 2000, while forest has become the largest one since 2000. China’s virtual land trade experienced a sharp increase in net imports from 9.4E + 04 km2 in 1995 to 3.4E + 06 km2 in 2015. Observing China’s virtual land network by a cluster analysis, this study concludes that China keeps tight relationships with Australia, Japan, Brazil and Korea for its cropland consumption, and Canada, USA, Mexico, Australia, Korea and Japan are relevant for its grassland consumption. In addition, decomposition analysis results show that affluence is the major driving factor for China’s land consumption, while changes in land use intensity could mitigate some of the related effects. Lastly, policy recommendations are proposed so that China can move toward sustainable land management