The SBP Algorithm for Maximizing Revenue in Online Dial-a-Ride

TL;DR

This paper analyzes the online Dial-a-Ride problem with revenue maximization, establishing bounds on algorithm performance and proving the optimal competitive ratio of the SBP algorithm in various graph settings.

Contribution

It proves the exact competitive ratio of the SBP algorithm for weighted graphs and uniform revenues, and establishes a lower bound for online algorithms in revenue maximization.

Findings

The competitive ratio of SBP on weighted graphs is exactly 5.

SBP has a competitive ratio of 4 when revenues are uniform.

No deterministic online algorithm can guarantee more than twice the offline revenue.

Abstract

In the Online-Dial-a-Ride Problem (OLDARP) a server travels through a metric space to serve requests for rides. We consider a variant where each request specifies a source, destination, release time, and revenue that is earned for serving the request. The goal is to maximize the total revenue earned within a given time limit. We prove that no non-preemptive deterministic online algorithm for OLDARP can be guaranteed to earn more than twice the revenue earned by an optimal offline solution. We then investigate the \textsc{segmented best path} ($SBP$) algorithm of~\cite{atmos17} for the general case of weighted graphs. The previously-established lower and upper bounds for the competitive ratio of $SBP$ are 4 and 6, respectively, under reasonable assumptions about the input instance. We eliminate the gap by proving that the competitive ratio is 5 (under the same reasonable assumptions). We also prove that when revenues are uniform, $SBP$ has competitive ratio 4. Finally, we provide a competitive analysis of $SBP$ on complete bipartite graphs.