L-infinity optimization to linear spaces and phylogenetic trees

Given a distance matrix consisting of pairwise distances between species, a distance-based phylogenetic reconstruction method returns a tree metric or equidistant tree metric (ultrametric) that best fits the data. We investigate distance-based phylogenetic reconstruction using the $l^\infty$-metric. In particular, we analyze the set of $l^\infty$-closest ultrametrics and tree metrics to an arbitrary dissimilarity map to determine its dimension and the tree topologies it represents. In the case of ultrametrics, we decompose the space of dissimilarity maps on 3 elements and on 4 elements relative to the tree topologies represented. Our approach is to first address uniqueness issues arising in $l^\infty$-optimization to linear spaces. We show that the $l^\infty$-closest point in a linear space is unique if and only if the underlying matroid of the linear space is uniform. We also give a polyhedral decomposition of $\rr^m$ based on the dimension of the set of $l^\infty$-closest points in a linear space.