Optimal Cuts and Partitions in Tree Metrics in Polynomial Time

TL;DR

This paper introduces a polynomial-time dynamic programming algorithm for optimal partitions in tree-induced shortest path metrics, resolving complexity questions in one-dimensional geometric settings and extending to bounded treewidth graphs.

Contribution

It provides the first efficient algorithm for optimal partitions in tree metrics and extends the approach to bounded treewidth graph metrics.

Findings

Polynomial-time algorithm for tree metric partitions

Resolution of complexity status in 1D geometric metrics

Extension to bounded treewidth graph metrics

Abstract

We present a polynomial time dynamic programming algorithm for optimal partitions in the shortest path metric induced by a tree. This resolves, among other things, the exact complexity status of the optimal partition problems in one dimensional geometric metric settings. Our method of solution could be also of independent interest in other applications. We discuss also an extension of our method to the class of metrics induced by the bounded treewidth graphs.