In a phylogenetic network analysis of 160 complete human severe acute respiratory syndrome coronavirus 2 (SARS-Cov-2) genomes, we find three central variants distinguished by amino acid changes, which we have named A, B, and C, with A being the ancestral type according to the bat outgroup coronavirus. The A and C types are found in significant proportions outside East Asia, that is, in Europeans and Americans. In contrast, the B type is the most common type in East Asia, and its ancestral genome appears not to have spread outside East Asia without first mutating into derived B types, pointing to founder effects or immunological or environmental resistance against this type outside Asia. The network faithfully traces routes of infections for documented coronavirus disease 2019 (COVID-19) cases, indicating that phylogenetic networks can likewise be successfully used to help trace undocumented COVID-19 infection sources, which can then be quarantined to prevent recurrent spread of the disease worldwide.

https://www.pnas.org/content/pnas/early/2020/04/07/2004999117.full.pdf

Phylogenetic network analysis of SARS-CoV-2 genomes.

On Steiner trees and minimum spanning trees in hypergraphs

Supergenes are tight clusters of loci that facilitate the co-segregation of adaptive variation, providing integrated control of complex adaptive phenotypes. Polymorphic supergenes, in which specific combinations of traits are maintained within a single population, were first described for 'pin' and 'thrum' floral types in Primula and Fagopyrum, but classic examples are also found in insect mimicry and snail morphology. Understanding the evolutionary mechanisms that generate these co-adapted gene sets, as well as the mode of limiting the production of unfit recombinant forms, remains a substantial challenge. Here we show that individual wing-pattern morphs in the polymorphic mimetic butterfly Heliconius numata are associated with different genomic rearrangements at the supergene locus P. These rearrangements tighten the genetic linkage between at least two colour-pattern loci that are known to recombine in closely related species, with complete suppression of recombination being observed in experimental crosses across a 400-kilobase interval containing at least 18 genes. In natural populations, notable patterns of linkage disequilibrium (LD) are observed across the entire P region. The resulting divergent haplotype clades and inversion breakpoints are found in complete association with wing-pattern morphs. Our results indicate that allelic combinations at known wing-patterning loci have become locked together in a polymorphic rearrangement at the P locus, forming a supergene that acts as a simple switch between complex adaptive phenotypes found in sympatry. These findings highlight how genomic rearrangements can have a central role in the coexistence of adaptive phenotypes involving several genes acting in concert, by locally limiting recombination and gene flow.

/pdf/chromosomal-rearrangements-maintain-a-polymorphic-supergene-1s4h2o7gq3.pdf

Chromosomal rearrangements maintain a polymorphic supergene controlling butterfly mimicry

The Steiner tree problem is one of the most fundamental NP-hard problems: given a weighted undirected graph and a subset of terminal nodes, find a minimum-cost tree spanning the terminals. In a sequence of papers, the approximation ratio for this problem was improved from 2 to the current best 1.55 [Robins,Zelikovsky-SIDMA'05]. All these algorithms are purely combinatorial. A long-standing open problem is whether there is an LP-relaxation for Steiner tree with integrality gap smaller than 2 [Vazirani,Rajagopalan-SODA'99]. In this paper we improve the approximation factor for Steiner tree, developing an LP-based approximation algorithm. Our algorithm is based on a, seemingly novel, iterative randomized rounding technique. We consider a directed-component cut relaxation for the k-restricted Steiner tree problem. We sample one of these components with probability proportional to the value of the associated variable in the optimal fractional solution and contract it. We iterate this process for a proper number of times and finally output the sampled components together with a minimum-cost terminal spanning tree in the remaining graph. Our algorithm delivers a solution of cost at most ln(4) times the cost of an optimal k-restricted Steiner tree. This directly implies a ln(4)+e

/pdf/an-improved-lp-based-approximation-for-steiner-tree-4measlqfh9.pdf

An improved LP-based approximation for steiner tree

The Steiner tree problem is one of the most fundamental NP-hard problems: given a weighted undirected graph and a subset of terminal nodes, find a minimum-cost tree spanning the terminals. In a sequence of papers, the approximation ratio for this problem was improved from 2 to 1.55 [Robins and Zelikovsky 2005]. All these algorithms are purely combinatorial. A long-standing open problem is whether there is an LP relaxation of Steiner tree with integrality gap smaller than 2 [Rajagopalan and Vazirani 1999].In this article we present an LP-based approximation algorithm for Steiner tree with an improved approximation factor. Our algorithm is based on a, seemingly novel, iterative randomized rounding technique. We consider an LP relaxation of the problem, which is based on the notion of directed components. We sample one component with probability proportional to the value of the associated variable in a fractional solution: the sampled component is contracted and the LP is updated consequently. We iterate this process until all terminals are connected. Our algorithm delivers a solution of cost at most ln(4) + e

/pdf/steiner-tree-approximation-via-iterative-randomized-rounding-munhd0nqsg.pdf

Steiner Tree Approximation via Iterative Randomized Rounding

Improved algorithms for the Steiner problem in networks

The Steiner problem in networks is the problem of connecting a set of required vertices in a weighted graph at minimum cost. It is a classical NP-hard problem with many important applications. For this problem we develop, implement and test several new techniques. On the side of lower bounds, we present a hierarchy of linear relaxations and class of new relaxations that are the currently strongest polynomially solvable linear relaxations. On the side of preprocessing techniques, we improve some known reduction tests and introduce powerful new ones. For upper bounds we introduce the successful concept of heuristic reductions. Finally, we integrate these blocks into an exact algorithm. For the exact algorithm and for the different components we present very good computational results on the large benchmark library SteinLib.

/pdf/algorithms-for-the-steiner-problem-in-networks-439uzrvtuf.pdf

Algorithms for the Steiner Problem in networks

A comparison of Steiner tree relaxations

Reduction methods are a key ingredient of the most successful algorithms for the Steiner problem. Whereas classical reduction tests just considered single vertices or edges, recent and more sophisticated tests extend the scope of inspection to more general patterns. In this paper, we present such an extended reduction test, which generalizes different tests in the literature. We use the new approach of combining alternative- and bound-based methods, which substantially improves the impact of the tests. We also present several algorithmic contributions. The experimental results show a large improvement over previous methods using the idea of extension, leading to a drastic speed-up in the optimal solution process and the solution of several previously unsolved benchmark instances.

Tobias Polzin

Papers

On Steiner trees and minimum spanning trees in hypergraphs

Improved algorithms for the Steiner problem in networks

Algorithms for the Steiner Problem in networks

A comparison of Steiner tree relaxations

Extending Reduction Techniques for the Steiner Tree Problem