Bounding Generalized Coloring Numbers of Planar Graphs Using Coin Models

We study Koebe orderings of planar graphs: vertex orderings obtained bymodelling the graph as the intersection graph of pairwise internally-disjointdiscs in the plane, and ordering the vertices by non-increasing radii of theassociated discs. We prove that for every $d\in \mathbb{N}$, any such orderinghas $d$-admissibility bounded by $O(d/\ln d)$ and weak $d$-coloring numberbounded by $O(d^4 \ln d)$. This in particular shows that the $d$-admissibilityof planar graphs is bounded by $O(d/\ln d)$, which asymptotically matches aknown lower bound due to Dvo\v{r}\'ak and Siebertz.<br

Computing the Chromatic Number Using Graph Decompositions via Matrix Rank

Computing the smallest number $q$ such that the vertices of a given graph can be properly $q$-colored is one of the oldest and most fundamental problems in combinatorial optimization. The $q$-Coloring problem has been studied intensively using the framework of parameterized algorithmics, resulting in a very good understanding of the best-possible algorithms for several parameterizations based on the structure of the graph. While there is an abundance of work for parameterizations based on decompositions of the graph by vertex separators, almost nothing is known about parameterizations based on edge separators. We fill this gap by studying $q$-Coloring parameterized by cutwidth, and parameterized by pathwidth in bounded-degree graphs. Our research uncovers interesting new ways to exploit small edge separators. We present two algorithms for $q$-Coloring parameterized by cutwidth $cutw$: a deterministic one that runs in time $O^*(2^{\omega \cdot cutw})$, where $\omega$ is the matrix multiplication constant, and a randomized one with runtime $O^*(2^{cutw})$. In sharp contrast to earlier work, the running time is independent of $q$. The dependence on cutwidth is optimal: we prove that even 3-Coloring cannot be solved in $O^*((2-\varepsilon)^{cutw})$ time assuming the Strong Exponential Time Hypothesis (SETH). Our algorithms rely on a new rank bound for a matrix that describes compatible colorings. Combined with a simple communication protocol for evaluating a product of two polynomials, this also yields an $O^*((\lfloor d/2\rfloor+1)^{pw})$ time randomized algorithm for $q$-Coloring on graphs of pathwidth $pw$ and maximum degree $d$. Such a runtime was first obtained by Bj\"orklund, but only for graphs with few proper colorings. We also prove that this result is optimal in the sense that no $O^*((\lfloor d/2\rfloor+1-\varepsilon)^{pw})$-time algorithm exists assuming SETH.Comment: 29 pages. An extended abstract appears in the proceedings of the 26th Annual European Symposium on Algorithms, ESA 201

Analysis of binding heterogeneity

Binding heterogeneity, due to different functional groups on a reactive surface, plays an important role in the binding of small molecules or ions to many adsorbents, both in industrial processes and in natural environments. The binding heterogeneity is described by a distribution of affinity constants since the different functional groups have different affinities for the adsorbing species.Three appraoches are discussed to obtain distribution functions on the basis of adsorption isotherms: the Local Isotherm Approximation (LIA), the Affinity Spectrum (AS) and the Differential Equilibrium Function (DEF). The methods are compared both on the basis of their derivation and on their ability to reproduce (known) distribution functions. All methods discussed need derivatives of the binding function, which are hard to obtain from experimental data. In order to apply the methods to experimental data a smoothing spline routine was adapted for the present problem. The methodology is applied to proton and copper binding to fulvic acids.Analogous to the heterogeneity analysis for binding under equilibrium conditions, a procedure was derived to determine first order rate constant distributions. The newly developed method is called LOcal Decay function Approximation (LODA). Also here an adapted smoothing spline routine is used to apply the method to experimental data. The method is illustrated by copper dissociation data from estuarine humic material.Finally it is shown how on the basis of the obtained distribution function a suitable model can be chosen for the description and prediction of binding or dissociation data

Makespan Scheduling of Unit Jobs with Precedence Constraints in O(1.995n)O(1.995^n) time

In a classical scheduling problem, we are given a set of $n$ jobs of unit length along with precedence constraints and the goal is to find a schedule of these jobs on $m$ identical machines that minimizes the makespan. This problem is well-known to be NP-hard for an unbounded number of machines. Using standard 3-field notation, it is known as $P|\text{prec}, p_j=1|C_{\max}$. We present an algorithm for this problem that runs in $O(1.995^n)$ time. Before our work, even for $m=3$ machines the best known algorithms ran in $O^\ast(2^n)$ time. In contrast, our algorithm works when the number of machines $m$ is unbounded. A crucial ingredient of our approach is an algorithm with a runtime that is only single-exponential in the vertex cover of the comparability graph of the precedence constraint graph. This heavily relies on insights from a classical result by Dolev and Warmuth (Journal of Algorithms 1984) for precedence graphs without long chains.Comment: 26 pages, 7 figure

On problems as hard as CNF-SAT

The field of exact exponential time algorithms for non-deterministic polynomial-time hard problems has thrived since the mid-2000s. While exhaustive search remains asymptotically the fastest known algorithm for some basic problems, non-trivial exponential time algorithms have been found for a myriad of problems, including GRAPH COLORING, HAMILTONIAN PATH, DOMINATING SET, and 3-CNF-SAT. In some instances, improving these algorithms further seems to be out of reach. The CNF-SAT problem is the canonical example of a problem for which the trivial exhaustive search algorithm runs in time O(2(n)), where n is the number of variables in the input formula. While there exist non-trivial algorithms for CNF-SAT that run in time o(2(n)), no algorithm was able to improve the growth rate 2 to a smaller constant, and hence it is natural to conjecture that 2 is the optimal growth rate. The strong exponential time hypothesis (SETH) by Impagliazzo and Paturi [JCSS 2001] goes a little bit further and asserts that, for every epsilon < 1, there is a (large) integer k such that k-CNF-SAT cannot be computed in time 2(epsilon n). In this article, we show that, for every epsilon < 1, the problems HITTING SET, SET SPLITTING, and NAE-SAT cannot be computed in time O(2(epsilon n)) unless SETH fails. Here n is the number of elements or variables in the input. For these problems, we actually get an equivalence to SETH in a certain sense. We conjecture that SETH implies a similar statement for SET COVER and prove that, under this assumption, the fastest known algorithms for STEINER TREE, CONNECTED VERTEX COVER, SET PARTITIONING, and the pseudo-polynomial time algorithm for SUBSET SUM cannot be significantly improved. Finally, we justify our assumption about the hardness of SET COVER by showing that the parity of the number of solutions to SET COVER cannot be computed in time O(2(epsilon n)) for any epsilon < 1 unless SETH fails

Connecting Terminals and 2-Disjoint Connected Subgraphs

Given a graph $G=(V,E)$ and a set of terminal vertices $T$ we say that a superset $S$ of $T$ is $T$-connecting if $S$ induces a connected graph, and $S$ is minimal if no strict subset of $S$ is $T$-connecting. In this paper we prove that there are at most ${|V \setminus T| \choose |T|-2} \cdot 3^{\frac{|V \setminus T|}{3}}$ minimal $T$-connecting sets when $|T| \leq n/3$ and that these can be enumerated within a polynomial factor of this bound. This generalizes the algorithm for enumerating all induced paths between a pair of vertices, corresponding to the case $|T|=2$. We apply our enumeration algorithm to solve the {\sc 2-Disjoint Connected Subgraphs} problem in time $O^*(1.7804^n)$, improving on the recent $O^*(1.933^n)$ algorithm of Cygan et al. 2012 LATIN paper.Comment: 13 pages, 1 figur

Nutrient retention efficiencies in integrated multi-trophic aquaculture

One of the bottlenecks for commercial implementation of integrated multi-trophic aquaculture (IMTA) is the difficulty in quantifying its environmental performance. We reviewed a large body of literature to determine the variability in nutrient dynamics within different IMTA systems (open sea-cages, land-based flow-through and recirculating aquaculture systems), with the aim to provide a generic framework to quantify nutrient retention efficiencies in integrated aquaculture systems. Based on the eco-physiological requirements of the cultured species, as well as the response of “extractive” species to waste from “fed” species, the maximum retention efficiency was defined for a conceptual four-species marine IMTA system (fish–seaweed–bivalve–deposit feeder). This demonstrated that 79%–94% of nitrogen, phosphorus and carbon supplied with fish feed could theoretically be retained. In practice, however, various biological and environmental factors may limit retention efficiencies and thereby influence the bioremediation of IMTA systems. These biological (waste production, stoichiometry in nutrient requirements) and environmental (temporal and spatial connectivity) factors were therefore evaluated against the theoretical reference frame and showed that efficiencies of 45%–75% for closed systems and 40%–50% for open systems are more realistic. This study is thereby the first to provide quantitative estimates for nutrient retention across IMTA systems, demonstrating that a substantial fraction of nutrients released from fish culture units can be retained by extractive species and subsequently harvested. Furthermore, by adapting this framework to the design and the condition prevailing for a specific IMTA system, it becomes a generic tool to analyse the system's bioremediation potential and explore options for further improvement.publishedVersio

Coke formation in the oxidative dehydrogenation of ethylbenzene to styrene by TEOM

A packed bed microbalance reactor setup (TEOM-GC) is used to investigate the formation of coke as a function of time-on-stream on gamma-Al2O3 and 3P/SiO2 catalyst samples under different conditions for the ODH reaction of ethylbenzene to styrene. All samples show a linear correlation of the styrene selectivity and yield with the initial coverage of coke. The COX production increases with the coverage of coke. On the 3 wt% P/SiO2 sample, the initial coke build-up is slow and the coke deposition rate increases with time. On alumina-based catalyst samples, a fast initial coke build-up takes place, decreasing with time-on-stream, but the amount of coke does not stabilize. A higher O-2 : EB feed ratio results in more coke, and a higher temperature results in less coke. This coking behaviour of Al2O3 can be described by existing "monolayer-multilayer" models. Further, the coverage of coke on the catalyst varies with the position in the bed. For maximal styrene selectivity, the optimal coverage of coke should be sufficient to convert all O-2, but as low as possible to prevent selectivity loss by COX production. This is in favour of high temperature and low O-2 : EB feed ratios. The optimal coke coverage depends in a complex way on all the parameters: temperature, the O-2 : EB feed ratio, reactant concentrations, and the type of starting material.</p

Varied Length Stokes Shift BODIPY-Based Fluorophores for Multicolor Microscopy

Multicolor microscopy tools necessary to localize and visualize the complexity of subcellular systems are limited by current fluorophore technology. While commercial fluorophores cover spectral space from the ultraviolet to the near infrared region and are optimized for conventional bandpass based fluorescence microscopy, they are not ideal for highly multiplexed fluorescence microscopy as they tend to have short Stokes shifts, restricting the number of fluorophores that can be detected in a single sample to four to five. Herein, we synthesized a library of 95 novel boron-dipyrromethene (BODIPY)- based fluorophores and screened their photophysical, optical and spectral properties for their utility in multicolor microscopy. A subset of our BODIPY-based fluorophores yielded varied length Stokes shifts probes, which were used to create a five-color image using a single excitation with confocal laser scanning microscopy for the first time. Combining these novel fluorophores with conventional fluorophores could facilitate imaging in up to nine to ten colors using linear unmixing based microscopy approaches