Competitive Online Transportation Simplified
Stephen Arndt, Benjamin Moseley, Kirk Pruhs, Marc Uetz
TL;DR
This paper addresses online transportation with multiple garages of varying capacities by introducing a relaxed itinerant-car problem and a sequence of algorithms that progressively handle general metrics. It begins with Algorithm A on a power-of-two tree metric, proving a $(2m-2)$-competitive bound via bottleneck matching insights, and then extends to general metric spaces with Algorithm B, achieving $(8m-7)$-competitiveness through MST-based embeddings. A final Algorithm C translates the itinerant-car results back to the original online transportation problem, preserving the competitiveness at $(8m-7)$ and avoiding reliance on MPFS-based decompositions. The approach offers a simpler, more transparent analysis relative to prior star-decomposition methods, contributing a pathway toward the conjectured $(2m-1)$-competitive deterministic algorithm for online transportation.
Abstract
The setting for the online transportation problem is a metric space $M$, populated by $m$ parking garages of varying capacities. Over time cars arrive in $M$, and must be irrevocably assigned to a parking garage upon arrival in a way that respects the garage capacities. The objective is to minimize the aggregate distance traveled by the cars. In 1998, Kalyanasundaram and Pruhs conjectured that there is a $(2m-1)$-competitive deterministic algorithm for the online transportation problem, matching the optimal competitive ratio for the simpler online metric matching problem. Recently, Harada and Itoh presented the first $O(m)$-competitive deterministic algorithm for the online transportation problem. Our contribution is an alternative algorithm design and analysis that we believe is simpler.