Computing maximum Hamiltonian paths in complete graphs with tree metric

Abstract

We design a linear time algorithm computing the maximum weight Hamiltonian path in a weighted complete graph K T, where T is a given undirected tree. The vertices of K T are nodes of T and weight(i, j) is the distance between i, j in T. The input is the tree T and two nodes u, ν ∈T, the output is the maximum weight Hamiltonian path between these nodes. The size n of the input is the size of T (however the total size of the complete graph K T is quadratic with respect to n). Our algorithm runs in O(n) time. Correctness is based on combinatorics of alternating sequences. The problem has been inspired by a similar (but much simpler) problem in a famous book of Hugo Steinhaus. © 2012 Springer-Verlag Berlin Heidelberg.