# Covering a tree with rooted subtrees

## TL;DR

This paper studies a complex tree covering problem related to multiple traveling salesmen, proves its NP-hardness even under restrictions, and introduces an FPT algorithm based on a novel ILP structure.

## Contribution

It establishes the NP-hardness of the problem under certain constraints and develops an FPT algorithm using a new tree-fold ILP framework.

## Key findings

- Problem remains NP-hard with constant tree height and edge weights.
- Introduces a fixed-parameter tractable algorithm for the problem.
- Extends the FPT results for n-fold integer programming to tree-fold structures.

## Abstract

We consider the multiple traveling salesman problem on a weighted tree. In this problem there are $m$ salesmen located at the root initially. Each of them will visit a subset of vertices and return to the root. The goal is to assign a tour to every salesman such that every vertex is visited and the longest tour among all salesmen is minimized. The problem is equivalent to the subtree cover problem, in which we cover a tree with rooted subtrees such that the weight of the maximum weighted subtree is minimized. The classical machine scheduling problem can be viewed as a special case of our problem when the given tree is a star. We observe that, the problem remains NP-hard even if tree height and edge weight are constant, and present an FPT algorithm for this problem parameterized by the largest tour length. To achieve the FPT algorithm, we show a more general result. We prove that, integer linear programming that has a tree-fold structure is in FPT, which extends the FPT result for the $n$-fold integer programming by Hemmecke, Onn and Romanchuk.