A growing interest in using forest biomass for bioenergy generation may stimulate intensive harvesting scenarios in Germany. We calculated and compared nutrient exports of conventional stem only (SO), whole tree without needles (WT excl. needles), and whole tree (WT) harvesting in two medium aged Norway spruce (Picea abies L. Karst.) and Scots pine (Pinus sylvestris L.) stands differing in productivity, and related them to soil nutrient pools and fluxes at the study sites. We established allometric biomass functions for each aboveground tree compartment and analyzed their nutrient contents. We analyzed soil nutrient stocks, estimated weathering rates, and obtained deposition and seepage data from nearby Level II stations. WT (excl. needles) and WT treatments cause nutrient losses 1.5 to 3.6 times higher than SO, while the biomass gain is only 1.18 to 1.25 in case of WT (excl. needles) and 1.28 to 1.30 in case of WT in the pine and spruce stand, respectively. Within the investigated 25-year period, WT harvesting would cause exports of N, K+, Ca2+, and Mg2+ of 6.6, 8.8, 5.4, and 0.8 kg·ha−1 in the pine stand and 13.9, 7.0, 10.6, and 1.8 kg·ha−1 in the spruce stand annually. The relative impact of WT and WT (excl. needles) on the nutrient balance is similar in the pine and spruce stands, despite differences in stand productivities, and thus the absolute amount of nutrients removed. In addition to the impact of intensive harvesting, both sites are characterized by high seepage losses of base cations, further impairing the nutrient budget. While intensive biomass extraction causes detrimental effects on many key soil ecological properties, our calculations may serve to implement measures to improve the nutrient balance in forested ecosystems.

Diese Arbeit nutzt und verfeinert Methoden der Komplexitätstheorie, um mit diesen die Komplexität dünner Instanzen zu untersuchen. Dazu gehören etwa Graphen mit wenigen Kanten oder Formeln mit wenigen Bedingungen beschränkter Weite. Dabei ergeben sich zwei natürliche Fragestellungen: (a) Gibt es einen effizienten Algorithmus, der beliebige Instanzen eines NP-schweren Problems auf äquivalente, dünne Instanzen reduziert? (b) Gibt es einen Algorithmus, der dünne Instanzen NP-schwerer Probleme bedeutend schneller löst als allgemeine Instanzen gelöst werden können? Wir formalisieren diese Fragen für verschiedene Probleme und zeigen, dass positive Antworten jeweils zu komplexitätstheoretischen Konsequenzen führen, die als unwahrscheinlich gelten. Frage (a) wird als Kommunikation modelliert, in der zwei Akteure kooperativ eine NP-schwere Sprache entscheiden möchten und dabei möglichst wenig kommunizieren. Unter der komplexitätstheoretischen Annahme, dass coNP keine Teilmenge von NP/poly ist, erhalten wir aus unseren Ergebnissen erstaunlich scharfe untere Schranken für interessante Parameter aus verschiedenen Teilgebieten der theoretischen Informatik. Im Speziellen betrifft das die Ausdünnung von Formeln, die Kernelisierung aus der parameterisierten Komplexitätstheorie, die verlustbehaftete Kompression von Entscheidungsproblemen, und die Theorie der probabilistisch verifizierbaren Beweise. Wir untersuchen Fragestellung (b) anhand der Exponentialzeitkomplexität von Zählproblemen. Unter (Varianten) der bekannten Exponentialzeithypothese (ETH) erhalten wir exponentielle untere Schranken für wichtige #P-schwere Probleme: das Berechnen der Zahl der erfüllenden Belegungen einer 2-KNF Formel, das Berechnen der Zahl aller unabhängigen Mengen in einem Graphen, das Berechnen der Permanente einer Matrix mit Einträgen 0 und 1, das Auswerten des Tuttepolynoms an festen Punkten. / In this thesis, we use and refine methods of computational complexity theory to analyze the complexity of sparse instances, such as graphs with few edges or formulas with few constraints of bounded width. Two natural questions arise in this context: (a) Is there an efficient algorithm that reduces arbitrary instances of an NP-hard problem to equivalent, sparse instances? (b) Is there an algorithm that solves sparse instances of an NP-hard problem significantly faster than general instances can be solved? We formalize these questions for different problems and show that positive answers for these formalizations would lead to consequences in complexity theory that are considered unlikely. Question (a) is modeled by a communication process, in which two players want to cooperatively decide an NP-hard language and at the same time communicate as few as possible. Under the complexity-theoretic hypothesis that coNP is not in NP/poly, our results imply surprisingly tight lower bounds for parameters of interest in several areas, namely sparsification, kernelization in parameterized complexity, lossy compression, and probabilistically checkable proofs. We study the question (b) for counting problems in the exponential time setting. Assuming (variants of) the exponential time hypothesis (ETH), we obtain asymptotically tight, exponential lower bounds for well-studied #P-hard problems: Computing the number of satisfying assignments of a 2-CNF formula, computing the number of all independent sets in a graph, computing the permanent of a matrix with entries 0 and 1, evaluating the Tutte polynomial at fixed evaluation points.

A growing interest in using forest biomass for bioenergy generation may stimulate intensive harvesting scenarios in Germany. We calculated and compared nutrient exports of conventional stem only (SO), whole tree without needles (WT excl. needles), and whole tree (WT) harvesting in two medium aged Norway spruce (Picea abies L. Karst.) and Scots pine (Pinus sylvestris L.) stands differing in productivity, and related them to soil nutrient pools and fluxes at the study sites. We established allometric biomass functions for each aboveground tree compartment and analyzed their nutrient contents. We analyzed soil nutrient stocks, estimated weathering rates, and obtained deposition and seepage data from nearby Level II stations. WT (excl. needles) and WT treatments cause nutrient losses 1.5 to 3.6 times higher than SO, while the biomass gain is only 1.18 to 1.25 in case of WT (excl. needles) and 1.28 to 1.30 in case of WT in the pine and spruce stand, respectively. Within the investigated 25-year period, WT harvesting would cause exports of N, K+, Ca2+, and Mg2+ of 6.6, 8.8, 5.4, and 0.8 kg·ha−1 in the pine stand and 13.9, 7.0, 10.6, and 1.8 kg·ha−1 in the spruce stand annually. The relative impact of WT and WT (excl. needles) on the nutrient balance is similar in the pine and spruce stands, despite differences in stand productivities, and thus the absolute amount of nutrients removed. In addition to the impact of intensive harvesting, both sites are characterized by high seepage losses of base cations, further impairing the nutrient budget. While intensive biomass extraction causes detrimental effects on many key soil ecological properties, our calculations may serve to implement measures to improve the nutrient balance in forested ecosystems.