A variant of multitask n-vehicle exploration problem: maximizing every processor's average profit

TL;DR

This paper introduces a new variant of the multitask n-vehicle exploration problem focused on maximizing each processor's average profit, analyzing its complexity, and proposing a pseudo-polynomial time solution.

Contribution

It defines a novel fractional partition problem, analyzes its NP-hardness, and provides a pseudo-polynomial time algorithm for the new profit maximization problem.

Findings

Maximizing average profit is NP-hard with fixed processors.

The problem is strongly NP-hard in general.

A pseudo-polynomial time algorithm is developed for both problems.

Abstract

We discuss a variant of multitask n-vehicle exploration problem. Instead of requiring an optimal permutation of vehicles in every group, the new problem asks all vehicles in a group to arrive at a same destination. It can also be viewed as to maximize every processor's average profit, given n tasks, and each task's consume-time and profit. Meanwhile, we propose a new kind of partition problem in fractional form, and analyze its computational complexity. Moreover, by regarding fractional partition as a special case, we prove that the maximizing average profit problem is NP-hard when the number of processors is fixed and it is strongly NP-hard in general. At last, a pseudo-polynomial time algorithm for the maximizing average profit problem and the fractional partition problem is presented, thanks to the idea of the pseudo-polynomial time algorithm for the classical partition problem.