Discrete Nodal Domain Theorems

We give a detailed proof for two discrete analogues of Courant's Nodal Domain Theorem

A Simple Data-Adaptive Probabilistic Variant Calling Model

Background: Several sources of noise obfuscate the identification of single nucleotide variation (SNV) in next generation sequencing data. For instance, errors may be introduced during library construction and sequencing steps. In addition, the reference genome and the algorithms used for the alignment of the reads are further critical factors determining the efficacy of variant calling methods. It is crucial to account for these factors in individual sequencing experiments. Results: We introduce a simple data-adaptive model for variant calling. This model automatically adjusts to specific factors such as alignment errors. To achieve this, several characteristics are sampled from sites with low mismatch rates, and these are used to estimate empirical log-likelihoods. These likelihoods are then combined to a score that typically gives rise to a mixture distribution. From these we determine a decision threshold to separate potentially variant sites from the noisy background. Conclusions: In simulations we show that our simple proposed model is competitive with frequently used much more complex SNV calling algorithms in terms of sensitivity and specificity. It performs specifically well in cases with low allele frequencies. The application to next-generation sequencing data reveals stark differences of the score distributions indicating a strong influence of data specific sources of noise. The proposed model is specifically designed to adjust to these differences.Comment: 19 pages, 6 figure

Cover-Encodings of Fitness Landscapes

The traditional way of tackling discrete optimization problems is by using local search on suitably defined cost or fitness landscapes. Such approaches are however limited by the slowing down that occurs when the local minima that are a feature of the typically rugged landscapes encountered arrest the progress of the search process. Another way of tackling optimization problems is by the use of heuristic approximations to estimate a global cost minimum. Here we present a combination of these two approaches by using cover-encoding maps which map processes from a larger search space to subsets of the original search space. The key idea is to construct cover-encoding maps with the help of suitable heuristics that single out near-optimal solutions and result in landscapes on the larger search space that no longer exhibit trapping local minima. We present cover-encoding maps for the problems of the traveling salesman, number partitioning, maximum matching and maximum clique; the practical feasibility of our method is demonstrated by simulations of adaptive walks on the corresponding encoded landscapes which find the global minima for these problems.Comment: 15 pages, 4 figure

Convex Cycle Bases

Convex cycles play a role e.g. in the context of product graphs. We introduce convex cycle bases and describe a polynomial-time algorithm that recognizes whether a given graph has a convex cycle basis and provides an explicit construction in the positive case. Relations between convex cycles bases and other types of cycles bases are discussed. In particular we show that if G has a unique minimal cycle bases, this basis is convex. Furthermore, we characterize a class of graphs with convex cycles bases that includes partial cubes and hence median graphs. (authors' abstract)Series: Research Report Series / Department of Statistics and Mathematic

Square Property, Equitable Partitions, and Product-like Graphs

Equivalence relations on the edge set of a graph $G$ that satisfy restrictive conditions on chordless squares play a crucial role in the theory of Cartesian graph products and graph bundles. We show here that such relations in a natural way induce equitable partitions on the vertex set of $G$, which in turn give rise to quotient graphs that can have a rich product structure even if $G$ itself is prime.Comment: 20 pages, 6 figure

Glassy states in a shaken sandbox

Our model of shaken sand, presented in earlier work, has been extended to include a more realistic `glassy' state, i.e., when the sandbox is shaken at very low intensities of vibration. We revisit some of our earlier results, and compare them with our new results on the revised model. Our analysis of the glassy dynamics in our model shows that a variety of ground states is obtained; these fall in two categories, which we argue are representative of regular and irregular packings.Comment: 10 pages. 3 figures. To appear in Proceedings of Research Workshop on "Challenges in Granular Physics" (ICTP, Trieste, August 7-11, 2001). Special issue of Advances in Complex System