Exponential Time Complexity of Weighted Counting of Independent Sets

We consider weighted counting of independent sets using a rational weight x: Given a graph with n vertices, count its independent sets such that each set of size k contributes x^k. This is equivalent to computation of the partition function of the lattice gas with hard-core self-repulsion and hard-core pair interaction. We show the following conditional lower bounds: If counting the satisfying assignments of a 3-CNF formula in n variables (#3SAT) needs time 2^{\Omega(n)} (i.e. there is a c>0 such that no algorithm can solve #3SAT in time 2^{cn}), counting the independent sets of size n/3 of an n-vertex graph needs time 2^{\Omega(n)} and weighted counting of independent sets needs time 2^{\Omega(n/log^3 n)} for all rational weights x\neq 0. We have two technical ingredients: The first is a reduction from 3SAT to independent sets that preserves the number of solutions and increases the instance size only by a constant factor. Second, we devise a combination of vertex cloning and path addition. This graph transformation allows us to adapt a recent technique by Dell, Husfeldt, and Wahlen which enables interpolation by a family of reductions, each of which increases the instance size only polylogarithmically.Comment: Introduction revised, differences between versions of counting independent sets stated more precisely, minor improvements. 14 page

Snacking characteristics and patterns and their associations with diet quality and BMI in the Childhood Obesity Prevention and Treatment Research Consortium

Objective: To describe snacking characteristics and patterns in children and examine associations with diet quality and BMI. Design: Children's weight and height were measured. Participants/adult proxies completed multiple 24 h dietary recalls. Snack occasions were self-identified. Snack patterns were derived for each sample using exploratory factor analysis. Associations of snacking characteristics and patterns with Healthy Eating Index-2010 (HEI-2010) score and BMI were examined using multivariable linear regression models. Setting: Childhood Obesity Prevention and Treatment Research (COPTR) Consortium, USA: NET-Works, GROW, GOALS and IMPACT studies. Participants: Predominantly low-income, racial/ethnic minorities: NET-Works (n 534, 2-4-year-olds); GROW (n 610, 3-5-year-olds); GOALS (n 241, 7-11-year-olds); IMPACT (n 360, 10-13-year-olds).Results: Two snack patterns were derived for three studies: a meal-like pattern and a beverage pattern. The IMPACT study had a similar meal-like pattern and a dairy/grains pattern. A positive association was observed between meal-like pattern adherence and HEI-2010 score (P for trend < 0-01) and snack occasion frequency and HEI-2010 score (β coefficient (95 % CI): NET-Works, 0-14 (0-04, 0-23); GROW, 0-12 (0-02, 0-21)) among younger children. A preference for snacking while using a screen was inversely associated with HEI-2010 score in all studies except IMPACT (β coefficient (95 % CI): NET-Works, -3-15 (-5-37, -0-92); GROW, -2-44 (-4-27, -0-61); GOALS, -5-80 (-8-74, -2-86)). Associations with BMI were almost all null. Conclusions: Meal-like and beverage patterns described most children's snack intake, although patterns for non-Hispanic Blacks or adolescents may differ. Diets of 2-5-year-olds may benefit from frequent meal-like pattern snack consumption and diets of all children may benefit from decreasing screen use during eating occasions

A low-memory algorithm for finding short product representations in finite groups

We describe a space-efficient algorithm for solving a generalization of the subset sum problem in a finite group G, using a Pollard-rho approach. Given an element z and a sequence of elements S, our algorithm attempts to find a subsequence of S whose product in G is equal to z. For a random sequence S of length d log_2 n, where n=#G and d >= 2 is a constant, we find that its expected running time is O(sqrt(n) log n) group operations (we give a rigorous proof for d > 4), and it only needs to store O(1) group elements. We consider applications to class groups of imaginary quadratic fields, and to finding isogenies between elliptic curves over a finite field.Comment: 12 page

Efficient Reconstruction of Metabolic Pathways by Bidirectional Chemical Search

One of the main challenges in systems biology is the establishment of the metabolome: a catalogue of the metabolites and biochemical reactions present in a specific organism. Current knowledge of biochemical pathways as stored in public databases such as KEGG, is based on carefully curated genomic evidence for the presence of specific metabolites and enzymes that activate particular biochemical reactions. In this paper, we present an efficient method to build a substantial portion of the artificial chemistry defined by the metabolites and biochemical reactions in a given metabolic pathway, which is based on bidirectional chemical search. Computational results on the pathways stored in KEGG reveal novel biochemical pathways

Perspective: Dietary Biomarkers of Intake and Exposure - Exploration with Omics Approaches

While conventional nutrition research has yielded biomarkers such as doubly labeled water for energy metabolism and 24-h urinary nitrogen for protein intake, a critical need exists for additional, equally robust biomarkers that allow for objective assessment of specific food intake and dietary exposure. Recent advances in high-throughput MS combined with improved metabolomics techniques and bioinformatic tools provide new opportunities for dietary biomarker development. In September 2018, the NIH organized a 2-d workshop to engage nutrition and omics researchers and explore the potential of multiomics approaches in nutritional biomarker research. The current Perspective summarizes key gaps and challenges identified, as well as the recommendations from the workshop that could serve as a guide for scientists interested in dietary biomarkers research. Topics addressed included study designs for biomarker development, analytical and bioinformatic considerations, and integration of dietary biomarkers with other omics techniques. Several clear needs were identified, including larger controlled feeding studies, testing a variety of foods and dietary patterns across diverse populations, improved reporting standards to support study replication, more chemical standards covering a broader range of food constituents and human metabolites, standardized approaches for biomarker validation, comprehensive and accessible food composition databases, a common ontology for dietary biomarker literature, and methodologic work on statistical procedures for intake biomarker discovery. Multidisciplinary research teams with appropriate expertise are critical to moving forward the field of dietary biomarkers and producing robust, reproducible biomarkers that can be used in public health and clinical research