Complexity of Grundy coloring and its variants

The Grundy number of a graph is the maximum number of colors used by the greedy coloring algorithm over all vertex orderings. In this paper, we study the computational complexity of GRUNDY COLORING, the problem of determining whether a given graph has Grundy number at least $k$. We also study the variants WEAK GRUNDY COLORING (where the coloring is not necessarily proper) and CONNECTED GRUNDY COLORING (where at each step of the greedy coloring algorithm, the subgraph induced by the colored vertices must be connected). We show that GRUNDY COLORING can be solved in time $O^*(2.443^n)$ and WEAK GRUNDY COLORING in time $O^*(2.716^n)$ on graphs of order $n$. While GRUNDY COLORING and WEAK GRUNDY COLORING are known to be solvable in time $O^*(2^{O(wk)})$ for graphs of treewidth $w$ (where $k$ is the number of colors), we prove that under the Exponential Time Hypothesis (ETH), they cannot be solved in time $O^*(2^{o(w\log w)})$. We also describe an $O^*(2^{2^{O(k)}})$ algorithm for WEAK GRUNDY COLORING, which is therefore \fpt for the parameter $k$. Moreover, under the ETH, we prove that such a running time is essentially optimal (this lower bound also holds for GRUNDY COLORING). Although we do not know whether GRUNDY COLORING is in \fpt, we show that this is the case for graphs belonging to a number of standard graph classes including chordal graphs, claw-free graphs, and graphs excluding a fixed minor. We also describe a quasi-polynomial time algorithm for GRUNDY COLORING and WEAK GRUNDY COLORING on apex-minor graphs. In stark contrast with the two other problems, we show that CONNECTED GRUNDY COLORING is \np-complete already for $k=7$ colors.Comment: 24 pages, 7 figures. This version contains some new results and improvements. A short paper based on version v2 appeared in COCOON'1

Time-approximation trade-offs for inapproximable problems

In this paper we focus on problems which do not admit a constant-factor approximation in polynomial time and explore how quickly their approximability improves as the allowed running time is gradually increased from polynomial to (sub-)exponential. We tackle a number of problems: For Min Independent Dominating Set, Max Induced Path, Forest and Tree, for any r(n), a simple, known scheme gives an approximation ratio of r in time roughly rn/r. We show that, for most values of r, if this running time could be significantly improved the ETH would fail. For Max Minimal Vertex Cover we give a nontrivial √r-approximation in time 2n/r. We match this with a similarly tight result. We also give a log r-approximation for Min ATSP in time 2n/r and an r-approximation for Max Grundy Coloring in time rn/r. Furthermore, we show that Min Set Cover exhibits a curious behavior in this superpolynomial setting: for any δ > 0 it admits an mδ-approximation, where m is the number of sets, in just quasi-polynomial time. We observe that if such ratios could be achieved in polynomial time, the ETH or the Projection Games Conjecture would fail. © Édouard Bonnet, Michael Lampis and Vangelis Th. Paschos; licensed under Creative Commons License CC-BY

A lower bound for approximating the Grundy number

Graphs and Algorithm

Grundy Distinguishes Treewidth from Pathwidth

Structural graph parameters, such as treewidth, pathwidth, and clique-width, are a central topic of study in parameterized complexity. A main aim of research in this area is to understand the "price of generality" of these widths: as we transition from more restrictive to more general notions, which are the problems that see their complexity status deteriorate from fixed-parameter tractable to intractable? This type of question is by now very well-studied, but, somewhat strikingly, the algorithmic frontier between the two (arguably) most central width notions, treewidth and pathwidth, is still not understood: currently, no natural graph problem is known to be W-hard for one but FPT for the other. Indeed, a surprising development of the last few years has been the observation that for many of the most paradigmatic problems, their complexities for the two parameters actually coincide exactly, despite the fact that treewidth is a much more general parameter. It would thus appear that the extra generality of treewidth over pathwidth often comes "for free". Our main contribution in this paper is to uncover the first natural example where this generality comes with a high price. We consider Grundy Coloring, a variation of coloring where one seeks to calculate the worst possible coloring that could be assigned to a graph by a greedy First-Fit algorithm. We show that this well-studied problem is FPT parameterized by pathwidth; however, it becomes significantly harder (W[1]-hard) when parameterized by treewidth. Furthermore, we show that Grundy Coloring makes a second complexity jump for more general widths, as it becomes para-NP-hard for clique-width. Hence, Grundy Coloring nicely captures the complexity trade-offs between the three most well-studied parameters. Completing the picture, we show that Grundy Coloring is FPT parameterized by modular-width.Comment: To be published in proceedings of ESA 202

The effects of endurance exercise on lipoproteins in Yucatan miniature swine

The effects of chronic, endurance exercise on the concentration of plasma lipids and on the composition and metabolism of plasma lipoproteins were examined in Yucatan miniature swine following more than two years of intensive exercise training. The exercise protocol employed produced significant training effects, as indicated by enhanced cardiovascular and aerobic capacities. Lipoprotein lipase and hepatic triglyceride lipase activities were also significantly increased in the exercised group. Neither total plasma cholesterol nor triglycerides differed between the two groups. Total high-density lipoprotein cholesterol (HDL) was not significantly greater in the exercised swine; however, exercise was associated with a shift in the distribution of cholesterol from the HDL-three to the HDL-two subfraction. The significant rise in HDL-two cholesterol was strongly correlated with the rise in lipoprotein lipase. Compositional analyses of the major lipoprotein classes indicated several exercise-induced shifts, especially in cholesteryl ester, free cholesterol and triglyceride content. These studies also clearly demonstrated the presence of heterogenous low-density lipoproteins (LDL) which differed in lipid composition. Significant increases in the total lipid mass of LDL-one and LDL-two were observed in the exercised swine, perhaps reflecting the demonstrated alterations in lipoprotein lipase. Exercise had little effect on the fractional catabolic rates or production rates of either LDL subclass; hence, control and exercise data were pooled in order to better assess the kinetic characteristics of LDL subclasses in the swine model. It appears that LDL-one is the more metabolically active subclass since its fractional catabolic rate and production rate were significantly higher than those of LDL-two. Exercise also appeared to have little effect on the mass of Apoproteins E or C, or any of their isoforms. Furthermore, the Apoprotein E/C, C-II/C-III and E/C-III ratios were not influenced by exercise. In toto, the results of these studies suggest that physical exercise is associated with subtle but significant changes in plasma lipids and lipoproteins, which are consistent with a reduced risk or coronary heart disease

TAIR: A transonic airfoil analysis computer code

The operation of the TAIR (Transonic AIRfoil) computer code, which uses a fast, fully implicit algorithm to solve the conservative full-potential equation for transonic flow fields about arbitrary airfoils, is described on two levels of sophistication: simplified operation and detailed operation. The program organization and theory are elaborated to simplify modification of TAIR for new applications. Examples with input and output are given for a wide range of cases, including incompressible, subcritical compressible, and transonic calculations

A spatial approach to culture, agriculture and political economy in the late nineteenth-century corn-belt

This work highlights the institutional nature of spatial relationships that restricted cropping choices and patterned both the landscape and work behaviors of farm families in the late nineteenth-century Illinois corn-belt. It accomplishes this through a intra regional comparison of agricultural production statistics among native-stock and German-American farmers and a study of their varied cultural approaches to gendered labor and the built environment

Establishing Ligand Mediated RNA Folding of Translational Riboswitches as Genetic Regulators using Single Molecule Microscopy.

Riboswitches are non-coding RNA regulatory elements located primarily in the 5’ UTR of bacterial messenger RNAs. Most commonly, they regulate gene expression of the downstream gene through transcriptional attenuation or through translational initiation inhibition in response to changing concentrations of a cellular signal. Riboswitches are composed of a ligand binding aptamer domain and an expression platform which confers the genetic decision. One of the most challenging questions remaining in the riboswitch field is how a ligand binding event can confer a large-scale conformational change, and how this change can effect a genetic decision. In this study, we demonstrate that two structurally similar transcriptional (Bsu) and translational (Tte) preQ1 riboswitches adopt similar pre-folded ensembles in the absence of ligand using a combination of single molecule techniques, molecular dynamics simulations and NMR. This result is in contrast to previous studies which suggested a largely unfolded and a loose pseudoknot for the transcriptional and translational riboswitch, respectively. Gō-model simulations of the two aptamers suggest that the ligand binds late (Bsu) and early (Tte) relative to pseudoknot folding, suggesting the two riboswitches tend to fold via conformational selection and induced fit, respectively. Finally, we show through the rational design of a single nucleotide swap distal from the ligand binding pocket that we find to predictably control the aptamers’ pre-folded states and their ligand binding affinities. Additionally, we have developed a single molecule footprinting assay to probe the accessibility of the ribosomal binding site of two translational riboswitches as a function of ligand concentration. The feasibility of the assay is demonstrated using the V. vulnificus adenine riboswitch, in which we use a fluorophore labeled nucleic acid oligo to mimic the 3’ end of the 16S rRNA. Finally, we probe the accessibility of the Shine-Dalgarno sequence of the Tte preQ1 riboswitch in the context of its full-length mRNA, which contains two overlapping open reading frames. Our results provide a novel single molecule assay which can be extended to study other biologically relevant molecules, and provide, for the first time, a molecular basis for the occlusion of the Shine-Dalgarno sequence by translational riboswitches to effect gene regulation.PHDChemistryUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/99792/1/rinaldia_1.pd