Approximation Algorithms for the Traveling Repairman and Speeding Deliveryman Problems

TL;DR

This paper introduces polynomial-time approximation algorithms for two variants of the traveling salesman problem with time windows, focusing on maximizing visited locations or minimizing required speedup, with improved ratios on trees and variable window lengths.

Contribution

It presents new approximation algorithms for the traveling repairman and speeding deliveryman problems, including extensions for variable time window lengths and improved ratios on tree graphs.

Findings

Constant-factor approximation algorithms developed.

Improved ratios for tree-structured inputs.

Extensions to handle variable time window lengths.

Abstract

Constant-factor, polynomial-time approximation algorithms are presented for two variations of the traveling salesman problem with time windows. In the first variation, the traveling repairman problem, the goal is to find a tour that visits the maximum possible number of locations during their time windows. In the second variation, the speeding deliveryman problem, the goal is to find a tour that uses the minimum possible speedup to visit all locations during their time windows. For both variations, the time windows are of unit length, and the distance metric is based on a weighted, undirected graph. Algorithms with improved approximation ratios are given for the case when the input is defined on a tree rather than a general graph. The algorithms are also extended to handle time windows whose lengths fall in any bounded range.