Covering a tree with rooted subtrees – parameterized and approximation algorithms

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 provide approximation and parameterized algorithms for this problem. We first present a PTAS (Polynomial Time Approximation Scheme). We then 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 first formulate the problem as an integer linear program having a certain "tree-fold" structure. Then we show that an ILP with such a structure is FPT, which is a generalization of an earlier FPT result for n-fold integer programming by Hemmecke, Onn and Romanchuk [5]. This extension of n-fold ILP may be of independent interest. Keywords: Approximation schemes; Fixed Parameter Tractable; Integer Programming; Scheduling IntroductionWe consider the multiple traveling salesmen problem on a given tree T = (V, E). In this problem there is a root r ∈ V where all the m salesmen are initially located. There is a weight w e ∈ Z + associated with each edge e ∈ E, which is the time consumed by a salesman if he passes this edge. Each salesman starts at r, travels a subset of the vertices and returns to r. The goal is to determine the tours traveled by each salesman such that every vertex is visited by some salesman, and the makespan, i.e., the time when the last salesman returns to r, is minimized.We observe that the tour of every salesman is actually a subtree rooted at r, and the total traveling time of each salesman is exactly twice the total weight

A Framework for Vehicle Routing Approximation Schemes in Trees

“…The difficulty with this approach is in showing that this type of grouping of small segments into large segments can be done without incurring too much error. As did [4], we also observed that assuming each subtree with length less than OP T to be covered by a single tour increases the makespan by O( OP T ), where OP T is the optimum makespan. The presented argument of Theorem 2.1 [4] (which their PTAS requires) depends on the following argument, though they use different terminology: Let T v denote the subtree rooted at v. The input can be safely modified 1 so that there exists a set of vertices CR such that the subtrees rooted at vertices in CR collectively partition the leaves of the tree and such that CR has the following properties:…”

“…In this paper, we improve this to a PTAS. Although a recent paper of Chen and Marx [4] also claimed to present a PTAS, in Appendix A we show that their result is incorrect and cannot be salvaged using the authors' proposed techniques. Additionally, we compare their approach to our own and describe how we successfully overcome the challenges where their approach fell short.…”

“…In fact, the crux of the problem is highlighted by the absence of such a set CR: How can the input be simplified so as to balance error with runtime given that coverage choices are not independent? Our approach uses similar intuition to that of [4], but manages to successfully address the critical challenges of the problem where prior attempts have fallen short.…”

Evaluating and Tuning n -fold Integer Programming

“…The generated batch is a cartesian product of all possible choices of the parameters. p_s A list of integers, by default [5,6,7,8,9,10,11,12,13]. For each number ∈ p_s, we compute the first primes and randomly pick a subset of size k of them as the processing times p 1 ≤ · · · ≤ p k .…”

“…Hemmecke, Onn, and Romanchuk [17] prove the following. Recently, algorithmic breakthroughs in stringology [23], computational social choice [24], scheduling [6,19,22], etc., were achieved by applying this algorithm and its subsequent non-trivial improvements.…”