On Distance Coloring

Call a connected undirected graph (d,c)-colorable if there is a vertex coloring using at most c colors such that no two vertices of distance d or less have the same color. It is well known that (1,2)-colorability is decidable in linear time, but (1,c)-colorability for c greater than or equal to 3 is NP-complete. Sharp (2007) shows that for fixed d greater than or equal to 2, the (d,c)-colorability problem is solvable in linear time for c less than or equal to 3d/2 and NP-complete otherwise. In this note we give an alternative construction that improves the upper time bound as a function of d for the case c less than or equal to 3d/2. The construction entails a generalization of the notion of tree decomposition and bounded treewidth (Robertson and Seymour 1986) to arbitrary overlay graphs, not just trees, which may be of independent interest

The Danger of Testing by Selecting Controlled Subsets, with Applications to Spoken-Word Recognition

When examining the effects of a continuous variable 'x' on an outcome 'y', a researcher might choose to dichotomize on 'x', dividing the population into two sets—low 'x' and high 'x'—and testing whether these two subpopulations differ with respect to 'y'. Dichotomization has long been known to incur a cost in statistical power, but there remain circumstances in which it is appealing: an experimenter might use it to control for confounding covariates through subset selection, by carefully choosing a subpopulation of Low and a corresponding subpopulation of High that are balanced with respect to a list of control variables, and then comparing the subpopulations’ 'y' values. This “divide, select, and test” approach is used in many papers throughout the psycholinguistics literature, and elsewhere. Here we show that, despite the apparent innocuousness, these methodological choices can lead to erroneous results, in two ways. First, if the balanced subsets of Low and High are selected in certain ways, it is possible to conclude a relationship between 'x' and 'y' not present in the full population. Specifically, we show that previously published conclusions drawn from this methodology—about the effect of a particular lexical property on spoken-word recognition—do not in fact appear to hold. Second, if the balanced subsets of Low and High are selected randomly, this methodology frequently fails to show a relationship between 'x' and 'y' that is present in the full population. Our work uncovers a new facet of an ongoing research effort: to identify and reveal the implicit freedoms of experimental design that can lead to false conclusions

Extensive Copy-Number Variation of Young Genes across Stickleback Populations

MM received funding from the Max Planck innovation funds for this project. PGDF was supported by a Marie Curie European Reintegration Grant (proposal nr 270891). CE was supported by German Science Foundation grants (DFG, EI 841/4-1 and EI 841/6-1). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript

Evolutionary history of human Plasmodium vivax revealed by genome-wide analyses of related ape parasites

Wild-living African apes are endemically infected with parasites that are closely related to human Plasmodium vivax, a leading cause of malaria outside Africa. This finding suggests that the origin of P. vivax was in Africa, even though the parasite is now rare in humans there. To elucidate the emergence of human P. vivax and its relationship to the ape parasites, we analyzed genome sequence data of P. vivax strains infecting six chimpanzees and one gorilla from Cameroon, Gabon, and Cote d'Ivoire. We found that ape and human parasites share nearly identical core genomes, differing by only 2% of coding sequences. However, compared with the ape parasites, human strains of P. vivax exhibit about 10-fold less diversity and have a relative excess of nonsynonymous nucleotide polymorphisms, with site-frequency spectra suggesting they are subject to greatly relaxed purifying selection. These data suggest that human P. vivax has undergone an extreme bottleneck, followed by rapid population expansion. Investigating potential host-specificity determinants, we found that ape P. vivax parasites encode intact orthologs of three reticulocyte-binding protein genes (rbp2d, rbp2e, and rbp3), which are pseudogenes in all human P. vivax strains. However, binding studies of recombinant RBP2e and RBP3 proteins to human, chimpanzee, and gorilla erythrocytes revealed no evidence of host-specific barriers to red blood cell invasion. These data suggest that, from an ancient stock of P. vivax parasites capable of infecting both humans and apes, a severely bottlenecked lineage emerged out of Africa and underwent rapid population growth as it spread globally

Distance coloring

Abstract. Given a graph G = (V, E), a (d, k)-coloring is a function from the vertices V to colors {1, 2,..., k} such that any two vertices within distance d of each other are assigned different colors. We determine the complexity of the (d, k)-coloring problem for all d and k, and enumerate some interesting properties of (d, k)-colorable graphs. Our main result is the discovery of a dichotomy between polynomial and NP-hard instances: for fixed d ≥ 2, the distance coloring problem is polynomial time for k ≤ ⌊ 3d 2 ⌋ and NP-hard for k> ⌊ 3

Birthdays, Broadcasts, and Boolean Algebras: Probabilistic Boolean Algebras and Applications

In the area of extremal finite set theory there are many combinatorial results concerning the selection of m k-element sets. This type of set selection can also be viewed as a boolean algebra. In this paper we consider a probabilistic construction of this boolean algebra, concentrating on the structure and properties such an algebra may form, particularly the structure of the algebra's atoms. The results are then applied to a generalization of the popular birthday problem, where the event of interest is now whether all selected sets have a unique element; we find an upper bound on the probability of this event. We also extend the definition of the generalized birthday problem to model content protection protocols. While these protocols are widely used in digital media rights management, they are insufficiently analyzed due to a lack of such an underlying model. We focus on the event that revoking the rights of multiple pirate users inadvertently causes the rights of other, authorized users to be unjustly revoked; we give an exact formula for the probability of this event

Thoughts on the Competitive Ratio

An algorithm for a typical combinatorial optimization problem is given static input, and finds a single solution. Incremental optimization is one framework for handling problems that require solutions to be built up over time due to evolving constraints. An incremental algorithm is given a sequence of input, and finds a sequence of solutions that build incrementally while adapting to the changes in the input. Online algorithms also take in a sequence of input and produce incremental solutions; unlike their incremental counterparts, however, they are not given the input sequence in advance. Thus two factors can affect the competitive ratio of an online problem: that it does not know the input sequence in advance, and that it makes irreversible decisions. Both factors contribute to an algorithm’s performance, but the competitive ratio fails to discern which is more significant. Incremental problems are only affected by the second factor, however, thus we can use incremental results to better understand online algorithms and improve their performance