Optimal Parametric Search for Path and Tree Partitioning

TL;DR

This paper introduces linear-time algorithms for optimal path and tree partitioning by weights, using parametric search to efficiently find partitions that maximize or minimize component weights.

Contribution

It develops new parametric search algorithms that achieve linear time for path and tree partitioning problems, improving efficiency over previous methods.

Findings

Algorithms run in linear time for both path and tree partitioning.

The path algorithm uses a synthetic weighting scheme for efficiency.

The tree algorithm employs a dual-pronged strategy for different structures.

Abstract

We present linear-time algorithms for partitioning a path or a tree with weights on the vertices by removing $k$ edges to maximize the minimum-weight component. We also use the same framework to partition a path with weight on the vertices, removing $k$ edges to minimize the maximum-weight component. The algorithms use the parametric search paradigm, testing candidate values until an optimum is found while simultaneously reducing the running time needed for each test. For path-partitioning, the algorithm employs a synthetic weighting scheme that results in a constant fraction reduction in running time after each test. For tree-partitioning, our dual-pronged strategy makes progress no matter what the specific structure of our tree is.