An APX for the Maximum-Profit Routing Problem with Variable Supply

TL;DR

This paper introduces a new approximation algorithm for the Maximum-Profit Routing Problem with Variable Supply, extending previous models by allowing supply quantities to increase over time, and improves existing algorithms under specific conditions.

Contribution

It presents a novel approximation algorithm for MPRP-VS with a logarithmic factor in T and improves upon prior algorithms for MPRP under certain scenarios.

Findings

Developed a 5.5 log T (1+ε) (1+1/(1+√m))^2 approximation algorithm.

Extended the MPRP model to include linearly increasing supply quantities.

Achieved improvements over previous algorithms in specific cases.

Abstract

In this paper, we study the Maximum-Profit Routing Problem with Variable Supply (MPRP-VS). This is a more general version of the Maximum-Profit Public Transportation Route Planning Problem, or simply Maximum-Profit Routing Problem (MPRP), introduced in \cite{Armaselu-PETRA}. In this new version, the quantity $q_i(t)$ supplied at site $i$ is linearly increasing in time $t$, as opposed to \cite{Armaselu-PETRA}, where the quantity is constant in time. Our main result is a $5.5 \log{T} (1 + \epsilon) (1 + \frac{1}{1 + \sqrt{m}})^2$ approximation algorithm, where $T$ is the latest time window and $m$ is the number of vehicles used. In addition, we improve upon the MPRP algorithm in \cite{Armaselu-PETRA} under certain conditions.