Approximation algorithms for some extensions of the maximum profit routing problem

In this paper, we consider extensions of the Maximum-Profit Public Transportation Route Planning Problem, or simply Maximum-Profit Routing Problem (MPRP), intro-duced in Armaselu and Daescu (Approximation algorithms for the maximum profit pick-up problem with time windows and capacity constraint, 2016. arXiv:1612.01038, Interactive assisting framework for maximum profit routing in public transportation in smart cities, PETRA, 13-16, 2017). Specifically, we consider MPRP with Time-Variable Supply (MPRP-VS), in which the quantity qi(t) supplied at site i is linearly increasing in time t, as opposed to the original MPRP problem, where the quantity is ( )2 constant in time. For MPRP-VS, our main result is a 5.5 log T(1 + is an element of) (1 + 1/1+root m)(2) approximation algorithm, where T is the latest time window and m is the number of vehicles used. We also study the MPRP with Multiple Vehicles per Site, in which a site may be visited by a vehicle multiple times, which can have 2 flavors: with quantities fixed in time (MPRP-M), and with time-variable quantities (MPRP-MVS). Our algorithmic solution to MPRP-VS can also improve upon the MPRP algorithm in Armaselu and Daescu (2017) under certain conditions. In addition, we simulate the MPRP-VS algorithm on a few benchmark, real-world, and synthetic instances.