Edge-weighted Tree Research Articles

Tree Reconstruction from Partial Orders

The problem of constructing trees given a matrix of interleaf distances is motivated by applications in computational evolutionary biology and linguistics. The general problem is to find an edge-weighted tree which most closely approximates (under some norm) the distance matrix. Although the construction problem is easy when the tree exactly fits the distance matrix, optimization problems under all popular criteria are either known or conjectured to be $NP$-complete. In this paper we consider the related problem where we are given a partial order on the pairwise distances and wish to construct (if possible) an edge-weighted tree realizing the partial order. We are particularly interested in partial orders which arise from experiments on triples of species. We will show that the consistency problem is $NP$-hard in general, but that for certain special cases the construction problem can be solved in polynomial time.

On weighted multiway cuts in trees

A min—max theorem is developed for the multiway cut problem of edge-weighted trees. We present a polynomial time algorithm to construct an optimal dual solution, if edge weights come in unary representation. Applications to biology also require some more complex edge weights. We describe a dynamic programming type algorithm for this more general problem from biology and show that our min—max theorem does not apply to it.

Nonpreemptive Ensemble Motion Planning on a Tree

Consider the problem of finding a minimum-cost tour to transport a set of objects between the vertices of an edge-weighted tree by a vehicle that travels along the edges of the tree. The vehicle can carry only one object at a time, and it starts and finishes at the same vertex of the tree. It is shown that if each object must be carried directly from its initial vertex to its destination then finding such a minimum-cost tour is NP-hard. Several fast approximation algorithms are presented for this problem. The fastest runs in O( k + n) time and generates a tour of cost at most 3 2 times the cost of a minimum-cost tour, where n is the number of vertices in the tree, k is the number of objects to be moved. Another runs in O( k + n log β( n, q)) time and generates a tour of cost at most 5 4 times the cost of a minimum-cost tour, where q ≤ min{ k, n} is the number of nontrivial connected components in a related directed graph, and β( n, q) = min{ i | log i n ≤ n/ q}. In addition, a family of polynomial-time approximation algorithms is given, whose ratio of the cost of tour generated to the cost of a minimum-cost tour decreases to at most 1.21363 as the running time increases.