A push–relabel approximation algorithm for approximating the minimum-degree MST problem and its generalization to matroids

AbstractIn the minimum-degree minimum spanning tree (MDMST) problem, we are given a graph G, and the goal is to find a minimum spanning tree (MST) T, such that the maximum degree of T is as small as possible. This problem is NP-hard and generalizes the Hamiltonian path problem. We give an algorithm that outputs an MST of degree at most 2Δopt (G)+o(Δopt (G)), where Δopt (G) denotes the degree of the optimal tree. This result improves on a previous result of Fischer [T. Fischer, Optimizing the degree of minimum weight spanning trees. Technical Report 14853, Dept. of Computer Science, Cornell University, Ithaca, NY, 1993] that finds an MST of degree at most bΔopt (G)+logbn, for any b>1.The MDMST problem is a special case of the following problem: given a k-ary hypergraph G=(V,E) and weighted matroid M with E as its ground set, find a minimum-cost basis (MCB) T of M such that the degree of T in G is as small as possible. Our algorithm immediately generalizes to this problem, finding an MCB of degree at most k2Δopt (G,M)+O(kkΔopt (G,M)).We use the push–relabel framework developed by Goldberg [A. V. Goldberg, A new max-flow algorithm, Technical Report MIT/LCS/TM-291, Massachusetts Institute of Technology, 1985 (Technical Report)] for the maximum-flow problem. To our knowledge, this is the first use of the push–relabel technique in an approximation algorithm for an NP-hard problem.The MDMST problem is closely connected to the bounded-degree minimum spanning tree (BDMST) problem. Given a graph G and degree bound B on its nodes, the BDMST problem is to find a minimum cost spanning tree among the spanning trees with maximum degree B. Previous algorithms for this problem by Könemann and Ravi [J. Könemann, R. Ravi, A matter of degree: Improved approximation algorithms for degree-bounded minimum spanning trees, SIAM Journal on Computing 31(6) (2002) 1783–1793; J. Könemann, R. Ravi, Primal-dual meets local search: Approximating MST’s with nonuniform degree bounds, in: Proceedings of the Thirty-Fifth ACM Symposium on Theory of Computing, 2003, pp. 389–395] and by Chaudhuri et al. [K. Chaudhuri, S. Rao, S. Riesenfeld, K. Talwar, What would Edmonds do? Augmenting paths and witnesses for bounded degree MSTs, in: Proceedings of APPROX/RANDOM, 2005, pp. 26–39] incur a near-logarithmic additive error in the degree. We give the first BDMST algorithm that approximates both the degree and the cost to within a constant factor of the optimum. These results generalize to the case of nonuniform degree bounds

Sorting and Selection in Posets

Classical problems of sorting and searching assume an underlying linear ordering of the objects being compared. In this paper, we study these problems in the context of partially ordered sets, in which some pairs of objects are incomparable. This generalization is interesting from a combinatorial perspective, and it has immediate applications in ranking scenarios where there is no underlying linear ordering, e.g., conference submissions. It also has applications in reconstructing certain types of networks, including biological networks. Our results represent significant progress over previous results from two decades ago by Faigle and Turán. In particular, we present the first algorithm that sorts a width-w poset of size n with query complexity O(n(w+\log n)) and prove that this query complexity is asymptotically optimal. We also describe a variant of Mergesort with query complexity O(wn log n/w) and total complexity O(w2n log n/w); an algorithm with the same query complexity was given by Faigle and Turán, but no efficient implementation of that algorithm is known. Both our sorting algorithms can be applied with negligible overhead to the more general problem of reconstructing transitive relations. We also consider two related problems: finding the minimal elements, and its generalization to finding the bottom k “levels,” called the k-selection problem. We give efficient deterministic and randomized algorithms for finding the minimal elements with query complexity and total complexity O(wn). We provide matching lower bounds for the query complexity up to a factor of 2 and generalize the results to the k-selection problem. Finally, we present efficient algorithms for computing a linear extension of a poset and computing the heights of all elements

PhylOTU: a high-throughput procedure quantifies microbial community diversity and resolves novel taxa from metagenomic data.

Microbial diversity is typically characterized by clustering ribosomal RNA (SSU-rRNA) sequences into operational taxonomic units (OTUs). Targeted sequencing of environmental SSU-rRNA markers via PCR may fail to detect OTUs due to biases in priming and amplification. Analysis of shotgun sequenced environmental DNA, known as metagenomics, avoids amplification bias but generates fragmentary, non-overlapping sequence reads that cannot be clustered by existing OTU-finding methods. To circumvent these limitations, we developed PhylOTU, a computational workflow that identifies OTUs from metagenomic SSU-rRNA sequence data through the use of phylogenetic principles and probabilistic sequence profiles. Using simulated metagenomic data, we quantified the accuracy with which PhylOTU clusters reads into OTUs. Comparisons of PCR and shotgun sequenced SSU-rRNA markers derived from the global open ocean revealed that while PCR libraries identify more OTUs per sequenced residue, metagenomic libraries recover a greater taxonomic diversity of OTUs. In addition, we discover novel species, genera and families in the metagenomic libraries, including OTUs from phyla missed by analysis of PCR sequences. Taken together, these results suggest that PhylOTU enables characterization of part of the biosphere currently hidden from PCR-based surveys of diversity

EmptyDrops: distinguishing cells from empty droplets in droplet-based single-cell RNA sequencing data.

Droplet-based single-cell RNA sequencing protocols have dramatically increased the throughput of single-cell transcriptomics studies. A key computational challenge when processing these data is to distinguish libraries for real cells from empty droplets. Here, we describe a new statistical method for calling cells from droplet-based data, based on detecting significant deviations from the expression profile of the ambient solution. Using simulations, we demonstrate that EmptyDrops has greater power than existing approaches while controlling the false discovery rate among detected cells. Our method also retains distinct cell types that would have been discarded by existing methods in several real data sets

GATA4 controls regionalization of tissue immunity and commensal-driven immunopathology

There is growing recognition that regionalization of bacterial colonization and immunity along the intestinal tract has an important role in health and disease. Yet, the mechanisms underlying intestinal regionalization and its dysregulation in disease are not well understood. This study found that regional epithelial expression of the transcription factor GATA4 controls bacterial colonization and inflammatory tissue immunity in the proximal small intestine by regulating retinol metabolism and luminal IgA. Furthermore, in mice without jejunal GATA4 expression, the commensal segmented filamentous bacteria promoted pathogenic inflammatory immune responses that disrupted barrier function and increased mortality upon Citrobacter rodentium infection. In celiac disease patients, low GATA4 expression was associated with metabolic alterations, mucosal Actinobacillus, and increased IL-17 immunity. Taken together, these results reveal broad impacts of GATA4-regulated intestinal regionalization on bacterial colonization and tissue immunity, highlighting an elaborate interdependence of intestinal metabolism, immunity, and microbiota in homeostasis and disease.</p

The neuropeptide NMU amplifies ILC2-driven allergic lung inflammation

Type 2 innate lymphoid cells (ILC2s) both contribute to mucosal homeostasis and initiate pathologic inflammation in allergic asthma. However, the signals that direct ILC2s to promote homeostasis versus inflammation are unclear. To identify such molecular cues, we profiled mouse lung-resident ILCs using single-cell RNA sequencing at steady state and after in vivo stimulation with the alarmin cytokines IL-25 and IL-33. ILC2s were transcriptionally heterogeneous after activation, with subpopulations distinguished by expression of proliferative, homeostatic and effector genes. The neuropeptide receptor Nmur1 was preferentially expressed by ILC2s at steady state and after IL-25 stimulation. Neuromedin U (NMU), the ligand of NMUR1, activated ILC2s in vitro, and in vivo co-administration of NMU with IL-25 strongly amplified allergic inflammation. Loss of NMU-NMUR1 signalling reduced ILC2 frequency and effector function, and altered transcriptional programs following allergen challenge in vivo. Thus, NMUR1 signalling promotes inflammatory ILC2 responses, highlighting the importance of neuro-immune crosstalk in allergic inflammation at mucosal surfaces