A graph G is a max point-tolerance (MPT) graph if each vertex v of G can be mapped to a pointed-interval (I v , p v ) where I v is an interval of R and p v ∈ I v such that uv is an edge of G iff I u ∩ I v ⊇ {p u , p v }. MPT graphs model relationships among DNA fragments in genome-wide association studies as well as basic transmission problems in telecommunications. We formally introduce this graph class, characterize it, study combinatorial optimization problems on it, and relate it to several well known graph classes. We characterize MPT graphs as a special case of several 2D geometric intersection graphs; namely, triangle, rectangle, L-shape, and line segment intersection graphs. We further characterize MPT as having certain linear orders on their vertex set. Our last characterization is that MPT graphs are precisely obtained by intersecting special pairs of interval graphs. We also show that, on MPT graphs, the maximum weight independent set problem can be solved in polynomial time, the coloring problem is NP-complete, and the clique cover problem has a 2-approximation. Finally, we demonstrate several connections to known graph classes; e.g., MPT graphs strictly contain interval graphs and outerplanar graphs, but are incomparable to permutation, chordal, and planar graphs.

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A phylogeny is an unrooted binary tree that represents the evolutionary relationships of a set of $n$ species. Phylogenies find applications in several scientific areas ranging from medical research, to drug discovery, to epidemiology, to systematics, and to population dynamics. In such applications the available information is usually restricted to the leaves of a phylogeny and is represented by molecular data extracted from the analyzed species, such as DNA, RNA, amino acid or codon fragments. On the contrary, the information about the phylogeny itself is generally missing and is determined by solving an optimization problem, called the Phylogeny Estimation Problem (PEP), whose versions depend on the criterion used to select a phylogeny from among plausible alternatives. In this article we investigate a recent version of the PEP, called the \emph{Balanced Minimum Evolution Problem} (BMEP). We present a mixed integer linear programming model\footnote{See the online supplement for codes and data.} to solve exactly instances of the BMEP and develop branching rules and families of valid inequalities to further strengthen the model. Our results give perspective on the mathematics of the BMEP and suggest new directions on the development of future efficient exact approaches to solution of the problem

Molecular phylogenetics studies the hierarchical evolutionary relationships among organisms by means of molecular data. These relationships are typically described by means of weighted trees, or phylogenies, whose leaves represent the observed organisms, internal vertices the intermediate ancestors, and edges the evolutionary relationships between pairs of organisms. Molecular phylogenetics provides several criteria for selecting one phylogeny from among plausible alternatives. Usually, such criteria can be expressed in terms of objective functions, and the phylogenies that optimize them are referred to as optimal. One of the most important criteria is the minimum evolution (ME) criterion, which states that the optimal phylogeny for a given set of organisms is the one whose sum of edge weights is minimal. Finding the phylogeny that satisfies the ME criterion involves solving an optimization problem, called the minimum evolution problem (MEP), which is notoriously N P-Hard. This article offers an overview of the MEP and discusses the different versions of it that occur in the literature.