Constructing majority-rule supertrees

<p>Abstract</p> <p>Background</p> <p>Supertree methods combine the phylogenetic information from multiple partially-overlapping trees into a larger phylogenetic tree called a supertree. Several supertree construction methods have been proposed to date, but most of these are not designed with any specific properties in mind. Recently, Cotton and Wilkinson proposed extensions of the majority-rule consensus tree method to the supertree setting that inherit many of the appealing properties of the former.</p> <p>Results</p> <p>We study a variant of one of Cotton and Wilkinson's methods, called majority-rule (+) supertrees. After proving that a key underlying problem for constructing majority-rule (+) supertrees is NP-hard, we develop a polynomial-size exact integer linear programming formulation of the problem. We then present a data reduction heuristic that identifies smaller subproblems that can be solved independently. While this technique is not guaranteed to produce optimal solutions, it can achieve substantial problem-size reduction. Finally, we report on a computational study of our approach on various real data sets, including the 121-taxon, 7-tree Seabirds data set of Kennedy and Page.</p> <p>Conclusions</p> <p>The results indicate that our exact method is computationally feasible for moderately large inputs. For larger inputs, our data reduction heuristic makes it feasible to tackle problems that are well beyond the range of the basic integer programming approach. Comparisons between the results obtained by our heuristic and exact solutions indicate that the heuristic produces good answers. Our results also suggest that the majority-rule (+) approach, in both its basic form and with data reduction, yields biologically meaningful phylogenies.</p

The median function on distributive semilattices

AbstractA median of a k-tuple π=(x1,…,xk) of elements of a finite metric space (X,d) is an element x for which ∑i=1kd(x,xi) is minimum. The function m with domain the set of all k-tuples with k>0 and defined by m(π)={x:x is a median of π} is called the median function on X. Continuing with the program of characterizing m on various metric spaces, this paper presents a characterization of the median function on distributive semilattices endowed with the standard lattice metric