TSP with Time Windows and Service Time

TL;DR

This paper studies the online TSP with time windows and service times, characterizing the competitive ratio based on the relationship between laxity and metric space diameter, and providing bounds that improve understanding of online scheduling in metric spaces.

Contribution

It provides a detailed analysis of the competitive ratio for TSP with time windows, depending on laxity and metric properties, including new bounds and an innovative embedding technique.

Findings

Competitive ratio depends on laxity and metric diameter.

Constant competitive algorithms exist for different laxity regimes.

Lower bounds match upper bounds, showing tight analysis.

Abstract

We consider TSP with time windows and service time. In this problem we receive a sequence of requests for a service at nodes in a metric space and a time window for each request. The goal of the online algorithm is to maximize the number of requests served during their time window. The time to traverse an edge is the distance between the incident nodes of that edge. Serving a request requires unit time. We characterize the competitive ratio for each metric space separately. The competitive ratio depends on the relation between the minimum laxity (the minimum length of a time window) and the diameter of the metric space. Specifically, there is a constant competitive algorithm depending whether the laxity is larger or smaller than the diameter. In addition, we characterize the rate of convergence of the competitive ratio to $1$ as the laxity increases. Specifically, we provide a matching lower and upper bounds depending on the ratio between the laxity and the TSP of the metric space (the minimum distance to traverse all nodes). An application of our result improves the lower bound for colored packets with transition cost and matches the upper bound. In proving our lower bounds we use an interesting non-standard embedding with some special properties. This embedding may be interesting by its own.