Polyhedral computational geometry for averaging metric phylogenetic trees

This paper investigates the computational geometry relevant to calculations of the Frechet mean and variance for probability distributions on the phylogenetic tree space of Billera, Holmes and Vogtmann, using the theory of probability measures on spaces of nonpositive curvature developed by Sturm. We show that the combinatorics of geodesics with a specified fixed endpoint in tree space are determined by the location of the varying endpoint in a certain polyhedral subdivision of tree space. The variance function associated to a finite subset of tree space has a fixed $C^\infty$ algebraic formula within each cell of the corresponding subdivision, and is continuously differentiable in the interior of each orthant of tree space. We use this subdivision to establish two iterative methods for producing sequences that converge to the Frechet mean: one based on Sturm's Law of Large Numbers, and another based on descent algorithms for finding optima of smooth functions on convex polyhedra. We present properties and biological applications of Frechet means and extend our main results to more general globally nonpositively curved spaces composed of Euclidean orthants.Comment: 43 pages, 6 figures; v2: fixed typos, shortened Sections 1 and 5, added counter example for polyhedrality of vistal subdivision in general CAT(0) cubical complexes; v1: 43 pages, 5 figure

The Complexity of Reliability Computations in Planar and Acyclic Graphs

We show that the problem of computing source-sink reliability is NP-hard, in fact # P-complete, even for undirected and acyclic directed source-sink planar graphs having vertex degree at most three. Thus the source-sink reliability problem is unlikely to have an efficient algorithm, even when the graph can be laid out on a rectilinear grid

A new approach to solving three combinatorial enumeration problems on planar graphs

The purpose of this paper is to show how the technique of delta-wye graph reduction provides an alternative method for solving three enumerative function evaluation problems on planar graphs. In particular, it is shown how to compute the number of spanning trees and perfect matchings, and how to evaluate energy in the Ising spin glass model of statistical mechanics. These alternative algorithms require O(n2) arithmetic operations on an n-vertex planar graph, and are relatively easy to implement

Sudoku: Strategy versus Structure

Sudoku puzzles, and their variants, have become extremely popular in the last decade, and can now be found daily in most major U.S. newspapers. In addition to the countless books of Sudoku puzzles, there are many guides to Sudoku strategy and logic. (Some good references are the books, and the web pages. The reader is also directed to these for explanations of some of the terms mentioned throughout this discussion.) The purpose of this paper is to relate a common class of strategies, used to solve the vast majority of Sudoku puzzles, to the formulation of Sudoku puzzles as assignment problems and as linear programs. In particular, we give a simple characterization of this class, using a well-known graph theorem, and show further how the ability of this set of strategies to solve a Sudoku puzzle also implies that the solution can be represented as the unique nonnegative solution to a system of linear equations. These results provide excellent applications of principles commonly presented in introductory classes infinite mathematics and combinatorial optimization, and point as well to some interesting open research problems in the area

Two-path subsets: Efficient counting and applications to performability analysis

The problem of computing performability probabilities in stochastic PERT and flow networks is studied when the network is minimally designed to withstand any two component failures. Polynomial-time algorithms to compute performability when the network is planar - the nonplanar versions being NP-hard - solve related two-path subset problems. Given an acyclic graph with weights on the arcs, the algorithms compute the total weight of all subsets of arcs that are contained in (1) two source-sink paths, or (2) two are-disjoint source-sink paths. A polynomial algorithm is given for (1), and for (2) in the case where the graph is a source-sink planar k-flow graph, that is, cdge-minimal with respect to supporting k units of flow

On finding two-connected subgraphs in planar graphs

We consider a basic problem in the design of reliable networks, namely, that of finding a minimum-wight 2-connected subgraph spanning a given set K of vertices in a planar graph. We show a relationship between this problem and that of finding shortest trails and cycles enclosing K, and derive a polynomial-time algorithm in the case when the vertices of K all lie on the same face