Optimal Online Edge Coloring of Planar Graphs with Advice

Using the framework of advice complexity, we study the amount of knowledge about the future that an online algorithm needs to color the edges of a graph optimally, i.e., using as few colors as possible. For graphs of maximum degree $\Delta$, it follows from Vizing's Theorem that $O(m\log \Delta)$ bits of advice suffice to achieve optimality, where $m$ is the number of edges. We show that for graphs of bounded degeneracy (a class of graphs including e.g. trees and planar graphs), only $O(m)$ bits of advice are needed to compute an optimal solution online, independently of how large $\Delta$ is. On the other hand, we show that $\Omega (m)$ bits of advice are necessary just to achieve a competitive ratio better than that of the best deterministic online algorithm without advice. Furthermore, we consider algorithms which use a fixed number of advice bits per edge (our algorithm for graphs of bounded degeneracy belongs to this class of algorithms). We show that for bipartite graphs, any such algorithm must use at least $\Omega(m\log \Delta)$ bits of advice to achieve optimality.Comment: CIAC 201

Randomization can be as helpful as a glimpse of the future in online computation

We provide simple but surprisingly useful direct product theorems for proving lower bounds on online algorithms with a limited amount of advice about the future. As a consequence, we are able to translate decades of research on randomized online algorithms to the advice complexity model. Doing so improves significantly on the previous best advice complexity lower bounds for many online problems, or provides the first known lower bounds. For example, if $n$ is the number of requests, we show that: (1) A paging algorithm needs $\Omega(n)$ bits of advice to achieve a competitive ratio better than $H_k=\Omega(\log k)$, where $k$ is the cache size. Previously, it was only known that $\Omega(n)$ bits of advice were necessary to achieve a constant competitive ratio smaller than $5/4$. (2) Every $O(n^{1-\varepsilon})$-competitive vertex coloring algorithm must use $\Omega(n\log n)$ bits of advice. Previously, it was only known that $\Omega(n\log n)$ bits of advice were necessary to be optimal. For certain online problems, including the MTS, $k$-server, paging, list update, and dynamic binary search tree problem, our results imply that randomization and sublinear advice are equally powerful (if the underlying metric space or node set is finite). This means that several long-standing open questions regarding randomized online algorithms can be equivalently stated as questions regarding online algorithms with sublinear advice. For example, we show that there exists a deterministic $O(\log k)$-competitive $k$-server algorithm with advice complexity $o(n)$ if and only if there exists a randomized $O(\log k)$-competitive $k$-server algorithm without advice. Technically, our main direct product theorem is obtained by extending an information theoretical lower bound technique due to Emek, Fraigniaud, Korman, and Ros\'en [ICALP'09]

The Advice Complexity of a Class of Hard Online Problems

The advice complexity of an online problem is a measure of how much knowledge of the future an online algorithm needs in order to achieve a certain competitive ratio. Using advice complexity, we define the first online complexity class, AOC. The class includes independent set, vertex cover, dominating set, and several others as complete problems. AOC-complete problems are hard, since a single wrong answer by the online algorithm can have devastating consequences. For each of these problems, we show that $\log\left(1+(c-1)^{c-1}/c^{c}\right)n=\Theta (n/c)$ bits of advice are necessary and sufficient (up to an additive term of $O(\log n)$) to achieve a competitive ratio of $c$. The results are obtained by introducing a new string guessing problem related to those of Emek et al. (TCS 2011) and B\"ockenhauer et al. (TCS 2014). It turns out that this gives a powerful but easy-to-use method for providing both upper and lower bounds on the advice complexity of an entire class of online problems, the AOC-complete problems. Previous results of Halld\'orsson et al. (TCS 2002) on online independent set, in a related model, imply that the advice complexity of the problem is $\Theta (n/c)$. Our results improve on this by providing an exact formula for the higher-order term. For online disjoint path allocation, B\"ockenhauer et al. (ISAAC 2009) gave a lower bound of $\Omega (n/c)$ and an upper bound of $O((n\log c)/c)$ on the advice complexity. We improve on the upper bound by a factor of $\log c$. For the remaining problems, no bounds on their advice complexity were previously known.Comment: Full paper to appear in Theory of Computing Systems. A preliminary version appeared in STACS 201

Differences in inflammation and acute phase response but similar genotoxicity in mice following pulmonary exposure to graphene oxide and reduced graphene oxide

We investigated toxicity of 2-3 layered >1 μm sized graphene oxide (GO) and reduced graphene oxide (rGO) in mice following single intratracheal exposure with respect to pulmonary inflammation, acute phase response (biomarker for risk of cardiovascular disease) and genotoxicity. In addition, we assessed exposure levels of particulate matter emitted during production of graphene in a clean room and in a normal industrial environment using chemical vapour deposition. Toxicity was evaluated at day 1, 3, 28 and 90 days (18, 54 and 162 μg/mouse), except for GO exposed mice at day 28 and 90 where only the lowest dose was evaluated. GO induced a strong acute inflammatory response together with a pulmonary (Serum-Amyloid A, Saa3) and hepatic (Saa1) acute phase response. rGO induced less acute, but a constant and prolonged inflammation up to day 90. Lung histopathology showed particle agglomerates at day 90 without signs of fibrosis. In addition, DNA damage in BAL cells was observed across time points and doses for both GO and rGO. In conclusion, pulmonary exposure to GO and rGO induced inflammation, acute phase response and genotoxicity but no fibrosis

Photocatalytic chloride to chlorine conversion by ionic iron in aqueous aerosols: A combined experimental, quantum chemical and chemical equilibrium model study

Aerosol chamber experiments show that the ligand-to-metal charge transfer absorption in iron(III) chlorides can lead to the production of chlorine. Based on this mechanism, the photocatalytic oxidation of chloride in mineral dust-sea spray aerosols was recently shown to be the largest source of chlorine over the North Atlantic. However, there has not been a detailed analysis of the mechanism including the aqueous formation equilibria and the absorption spectra of the iron chlorides; neither has there been an analysis of which iron chloride is the main chromophore. Here we present the results of experiments of photolysis FeCl_3 · 6H_2O in specific wavelength bands, an analysis of the absorption spectra of the title compounds from n=1..4 made using density functional theory, and the results of an aqueous phase model that predicts the abundance of the iron chlorides with changes in pH and ion concentrations. Transition state analysis is used to determine the energy thresholds of the dissociations of the species. Based on a speciation model with conditions extending from dilute water droplet to acidic seawater droplet to brine to salty crust, and the absorption rates and dissociation thresholds, we find that FeCl_2^+ is the most important species for chlorine production for a wide range of conditions. The mechanism was found to be active in the range of 400 to 530 nm with a maximum around 440 nm. We conclude that iron chlorides will form in atmospheric aerosols from the combination of iron(III) cations with chloride and that they will be activated by sunlight, generating chlorine (Cl_2/Cl) from chloride (Cl-), in a process that is catalytic in both chlorine and iron