Generalized Online Routing: New Competitive Ratios, Resource Augmentation, and Asymptotic Analyses

Abstract

We consider online routing optimization problems where the objective is to minimize the time needed to visit a set of locations under various constraints; the problems are online because the set of locations are revealed incrementally over time. We consider two main problems: (1) the online traveling salesman problem (TSP) with precedence and capacity constraints, and (2) the online TSP with m salesmen. For both problems we propose online algorithms, each with a competitive ratio of 2; for the m-salesmen problem, we show that our result is best-possible. We also consider polynomialtime online algorithms. We then consider resource augmentation, where we give the online servers additional resources to offset the powerful offline adversary advantage. In this way, we address a main criticism of competitive analysis. We consider the cases where the online algorithm has access to faster servers, servers with larger capacities, additional servers, and/or advanced information. We derive improved competitive ratios. We also give lower bounds on the competitive ratios under resource augmentation, which in many cases are tight and lead to best-possible results. Finally, we study online algorithms from an asymptotic point of view. We show that, under general stochastic structures for the problem data, unknown and unused by the online player, the online algorithms are almost surely asymptotically optimal. Furthermore, we provide computational results that show that the convergence can be very fast.

Almost sure asymptotic optimality for online routing and machine scheduling problems

“…These results continue the work in [21] to arguably more complicated online routing problems (routing optimization problems with the latency objective are usually considered more difficult that those with the makespan objective). Furthermore, our asymptotic approach is arguably more realistic than the only other asymptotic analysis for the online TRP (see [8]): We consider the limit as the number of cities (rather than the number of servers) goes to infinity.…”

“…Hiller [19] performs a probabilistic asymptotic competitive analysis of an online Diala-Ride problem on trees. Recently, Jaillet and Wagner [21] have investigated generalizations of the online TSP from an asymptotic point of view and have shown a number of almost sure asymptotic optimality results.…”

A review of dynamic vehicle routing problems

“…For these problems, vehicle routes are redefined in an ongoing fashion, requiring technological support for real-time communication between the vehicles and the decision maker (e.g., mobile phones and global positioning systems). This class of problems are also referred to as online or real time by some authors [77].…”

Online Optimization—An Introduction

“…Their algorithm is generalizable to the case where there are multiple servers with capacities; this generalization is also 2-competitive. Jaillet and Wagner [25] consider the (1) online TSP with precedence and capacity constraints and the (2) online TSP with m salesmen. For both problems, they propose 2-competitive online algorithms (optimal in case of the m-salesmen problem), consider polynomial-time online algorithms, and then consider resource augmentation, where the online servers are given additional resources to offset the powerful offline adversary advantage.…”