Recommended by Peer Community In Evolutionary Biology (http://dx.doi.org/10.24072/pci.evolbiol.100037)This article has been peer-reviewed and recommended by Peer Community In EvolutionaryBiology (http://dx.doi.org/10.24072/pci.evolbiol.100037).Lateral gene transfers between ancient species contain information about the relative timingof species diversification. Specifically, the ancestors of a donor species must have existedbefore the descendants of the recipient species. Hence, the detection of a transfer event can betranslated into a time constraint between nodes of a phylogeny if the donor and recipient canbe identified. When a set of transfers is detected by interpreting the phylogenetic discordancebetween gene trees and a species tree, the set of all deduced time constraints can be usedto rank the species tree, i.e. order totally its internal nodes. Unfortunately lateral genetransfer detection is challenging and current methods produce a significant proportion offalse positives. As a result, often, no ranking of the species tree is compatible with thefull set of time constraints deduced from predicted transfers. Here we propose a method,implemented in a software called MaxTiC (Maximum Time Consistency), which takes asinput a species tree and a series of (possibly inconsistent) time constraints between its internalnodes, weighted by confidence scores. MaxTiC outputs a ranked species tree compatible witha subset of constraints with maximum cumulated confidence score. We extensively tested themethod on simulated datasets, under a wide range of conditions that we compare to measureson biological datasets. In most conditions the obtained ranked tree is very close to the realone, confirming the potential of dating the history of life with transfers by maximizing timeconsistency. MaxTiC is freely available, distributed along with a documentation and severalexamples: https://github.com/ssolo/ALE/tree/master/maxtic

We introduce the class of bi-arc digraphs, and show they coincide with the class of digraphs that admit a conservative semi-lattice polymorphism, i.e., a min ordering. Surprisingly this turns out to be also the class of digraphs that admit totally symmetric conservative polymorphisms of all arities. We give an obstruction characterization of, and a polynomial time recognition algorithm for, this class of digraphs. The existence of a polynomial time algorithm was an open problem due to Bagan, Durand, Filiot, and Gauwin.We also discuss a generalization to k-arc digraphs, which has a similar obstruction characterization and recognition algorithm.When restricted to undirected graphs, the class of bi-arc digraphs is included in the previously studied class of bi-arc graphs. In particular, restricted to reflexive graphs, bi-arc digraphs coincide precisely with the well known class of interval graphs. Restricted to reflexive digraphs, they coincide precisely with the class of adjusted interval digraphs, and restricted to bigraphs, they coincide precisely with the class of two directional ray graphs. All these classes have been previously investigated as analogues of interval graphs. We believe that, in a certain sense, bi-arc digraphs are the most general digraph version of interval graphs with nice algorithms and characterizations.

Abstract. We consider a special case of the generalized minimum spanning tree problem (GMST) and the generalized travelling salesman problem (GTSP) where we are given a set of points inside the integer grid (in Euclidean plane) where each gride cell is 1 × 1. In the MST version of the problem, the goal is to find a minimum tree that contains exactly one point from each non-empty grid cell (cluster). Similarly, in the TSP version of the problem, the goal is to find a minimum weight cycle containing one point from each non-empty grid cell. We give a (1 + 4 √ 2 + ) and (1.5 + 8 √ 2 + )-approximation algorithm for these two problems in the described setting, respectively. Our motivation is based on the problem posed in [7] for a constant approximation algorithm. The authors designed a PTAS for the more special case of the GMST where non-empty cells are connected end dense enough. However, their algorithm heavily relies on this connectivity restriction and is unpractical. Our results develop the topic further.

Given two (di)graphs G, H and a cost function c : V (G) × V (H) → Q ≥0 ∪ {+∞}, in the minimum cost homomorphism problem, MinHOM(H), we are interested in finding a homomorphism f :The complexity of exact minimization of this problem is well understood [34], and the class of digraphs H, for which the MinHOM(H) is polynomial time solvable is a small subset of all digraphs.In this paper, we consider the approximation of MinHOM within a constant factor. In terms of digraphs, MinHOM(H) is not approximable if H contains a digraph asteroidal triple (DAT). We take a major step toward a dichotomy classification of approximable cases. We give a dichotomy classification for approximating the MinHOM(H) when H is a graph (i.e. symmetric digraph). For digraphs, we provide constant factor approximation algorithms for two important classes of digraphs, namely bi-arc digraphs (digraphs with a conservative semi-lattice polymorphism or min-ordering), and k-arc digraphs (digraphs with an extended min-ordering). Specifically, we show that:• Dichotomy for Graphs: MinHOM(H) has a 2|V (H)|-approximation algorithm if graph H admits a conservative majority polymorphims (i.e. H is a bi-arc graph), otherwise, it is inapproximable;In conclusion, we show the importance of these results and provide insights for achieving a dichotomy classification of approximable cases. Our constant factors depend on the size of H. However, the implementation of our algorithms provides a much better approximation ratio. It leaves open to investigate a classification of digraphs H, where MinHOM(H) admits a constant factor approximation algorithm that is independent of |V (H)|.

A maximal ε-near perfect matching is a maximal matching which covers at least (1 − ε)|V (G)| vertices. In this paper, we study the number of maximal near perfect matchings in generalized quasirandom and dense graphs. We provide tight lower and upper bounds on the number of ε-near perfect matchings in generalized quasirandom graphs. Moreover, based on these results, we provide a deterministic polynomial time algorithm that for a given dense graph G of order n and a real number ε > 0, returns either a conclusion that G has no ε-near perfect matching, or a positive non-trivial number such that the number of maximal εnear perfect matchings in G is at least n n . Our algorithm uses algorithmic version of Szemerédi Regularity Lemma, and has O(f (ε)n 5/2 ) time complexity. Here f (•) is an explicit function depending only on ε.