Counting Popular Matchings in House Allocation Problems

We study the problem of counting the number of popular matchings in a given instance. A popular matching instance consists of agents A and houses H, where each agent ranks a subset of houses according to their preferences. A matching is an assignment of agents to houses. A matching M is more popular than matching M' if the number of agents that prefer M to M' is more than the number of people that prefer M' to M. A matching M is called popular if there exists no matching more popular than M. McDermid and Irving gave a poly-time algorithm for counting the number of popular matchings when the preference lists are strictly ordered. We first consider the case of ties in preference lists. Nasre proved that the problem of counting the number of popular matching is #P-hard when there are ties. We give an FPRAS for this problem. We then consider the popular matching problem where preference lists are strictly ordered but each house has a capacity associated with it. We give a switching graph characterization of popular matchings in this case. Such characterizations were studied earlier for the case of strictly ordered preference lists (McDermid and Irving) and for preference lists with ties (Nasre). We use our characterization to prove that counting popular matchings in capacitated case is #P-hard

Mask formulas for cograssmannian Kazhdan-Lusztig polynomials

We give two contructions of sets of masks on cograssmannian permutations that can be used in Deodhar's formula for Kazhdan-Lusztig basis elements of the Iwahori-Hecke algebra. The constructions are respectively based on a formula of Lascoux-Schutzenberger and its geometric interpretation by Zelevinsky. The first construction relies on a basis of the Hecke algebra constructed from principal lower order ideals in Bruhat order and a translation of this basis into sets of masks. The second construction relies on an interpretation of masks as cells of the Bott-Samelson resolution. These constructions give distinct answers to a question of Deodhar.Comment: 43 page

Popular matchings in the marriage and roommates problems

Popular matchings have recently been a subject of study in the context of the so-called House Allocation Problem, where the objective is to match applicants to houses over which the applicants have preferences. A matching M is called popular if there is no other matching M′ with the property that more applicants prefer their allocation in M′ to their allocation in M. In this paper we study popular matchings in the context of the Roommates Problem, including its special (bipartite) case, the Marriage Problem. We investigate the relationship between popularity and stability, and describe efficient algorithms to test a matching for popularity in these settings. We also show that, when ties are permitted in the preferences, it is NP-hard to determine whether a popular matching exists in both the Roommates and Marriage cases

Triggering the stringent response: signals responsible for activating (p) ppGpp synthesis in bacteria

The stringent response is a conserved bacterial stress response mechanism that allows bacteria to respond to nutritional challenges. It is mediated by the alarmones pppGpp and ppGpp, nucleotides that are synthesized and hydrolyzed by members of the RSH superfamily. Whilst there are key differences in the binding targets for (p)ppGpp between Gram-negative and Gram-positive bacterial species, the transient accumulation of (p)ppGpp caused by nutritional stresses results in a global change in gene expression in all species. The RSH superfamily of enzymes is ubiquitous throughout the bacterial kingdom, and can be split into three main groups: the long-RSH enzymes; the small alarmone synthetases (SAS); and the small alarmone hydrolases (SAH). Despite the prevalence of these enzymes, there are important differences in the way in which they are regulated on a transcriptional and post-translational level. Here we provide an overview of the diverse regulatory mechanisms that are involved in governing this crucial signalling network. Understanding how the RSH superfamily members are regulated gives insights into the varied important biological roles for this signalling pathway across the bacteria

The impact of the stringent response on TRAFAC GTPases and prokaryotic ribosome assembly

Many facets of ribosome biogenesis and function, including ribosomal RNA (rRNA) transcription, 70S assembly and protein translation, are negatively impacted upon induction of a nutrient stress-sensing signalling pathway termed the stringent response. This stress response is mediated by the alarmones guanosine tetra- and penta-phosphate ((p)ppGpp), the accumulation of which leads to a massive cellular response that slows growth and aids survival. The 70S bacterial ribosome is an intricate structure, with assembly both complex and highly modular. Presiding over the assembly process is a group of P-loop GTPases within the TRAFAC (Translation Factor Association) superclass that are crucial for correct positioning of both early and late stage ribosomal proteins (r-proteins) onto the rRNA. Often described as ‘molecular switches’, members of this GTPase superfamily readily bind and hydrolyse GTP to GDP in a cyclic manner that alters the propensity of the GTPase to carry out a function. TRAFAC GTPases are considered to act as checkpoints to ribosome assembly, involved in binding to immature sections in the GTP-bound state, preventing further r-protein association until maturation is complete. Here we review our current understanding of the impact of the stringent response and (p)ppGpp production on ribosome maturation in prokaryotic cells, focusing on the inhibition of (p)ppGpp on GTPase-mediated subunit assembly, but also touching upon the inhibition of rRNA transcription and protein translation

The stringent response and physiological roles of (pp)pGpp in bacteria

The stringent response is a stress signalling system mediated by the alarmones guanosine tetraphosphate (ppGpp) and guanosine pentaphosphate (pppGpp) in response to nutrient deprivation. Recent research highlights the complexity and broad range of functions that these alarmones control. This Review provides an update on our current understanding of the enzymes involved in ppGpp, pppGpp and guanosine 5′-monophosphate 3′-diphosphate (pGpp) (collectively (pp)pGpp) turnover, including those shown to produce pGpp and its analogue (pp)pApp. We describe the well-known interactions with RNA polymerase as well as a broader range of cellular target pathways controlled by (pp)pGpp, including DNA replication, transcription, nucleotide synthesis, ribosome biogenesis and function, as well as lipid metabolism. Finally, we review the role of ppGpp and pppGpp in bacterial pathogenesis, providing examples of how these nucleotides are involved in regulating many aspects of virulence and chronic infection

Two codependent routes lead to high-level MRSA

Methicillin-resistant Staphylococcus aureus (MRSA), in which acquisition of mecA [which encodes the cell wall peptidoglycan biosynthesis component penicillin-binding protein 2a (PBP2a)] confers resistance to β-lactam antibiotics, is of major clinical concern. We show that, in the presence of antibiotics, MRSA adopts an alternative mode of cell division and shows an altered peptidoglycan architecture at the division septum. PBP2a can replace the transpeptidase activity of the endogenous and essential PBP2 but not that of PBP1, which is responsible for the distinctive native septal peptidoglycan architecture. Successful division without PBP1 activity requires the alternative division mode and is enabled by several possible chromosomal potentiator (pot) mutations. MRSA resensitizing agents differentially interfere with the two codependent mechanisms required for high-level antibiotic resistance, which provides opportunities for new interventions

One thousand plant transcriptomes and the phylogenomics of green plants

Abstract: Green plants (Viridiplantae) include around 450,000–500,000 species1, 2 of great diversity and have important roles in terrestrial and aquatic ecosystems. Here, as part of the One Thousand Plant Transcriptomes Initiative, we sequenced the vegetative transcriptomes of 1,124 species that span the diversity of plants in a broad sense (Archaeplastida), including green plants (Viridiplantae), glaucophytes (Glaucophyta) and red algae (Rhodophyta). Our analysis provides a robust phylogenomic framework for examining the evolution of green plants. Most inferred species relationships are well supported across multiple species tree and supermatrix analyses, but discordance among plastid and nuclear gene trees at a few important nodes highlights the complexity of plant genome evolution, including polyploidy, periods of rapid speciation, and extinction. Incomplete sorting of ancestral variation, polyploidization and massive expansions of gene families punctuate the evolutionary history of green plants. Notably, we find that large expansions of gene families preceded the origins of green plants, land plants and vascular plants, whereas whole-genome duplications are inferred to have occurred repeatedly throughout the evolution of flowering plants and ferns. The increasing availability of high-quality plant genome sequences and advances in functional genomics are enabling research on genome evolution across the green tree of life