A variant of multitask n-vehicle exploration problem: maximizing every processors average profit

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.

💡 Research Summary

The paper introduces a novel variant of the multitask n‑vehicle exploration problem in which all vehicles belonging to the same group must converge on a common destination. This requirement is reinterpreted as a “maximum average profit” problem for a set of processors (or machines) that must execute a collection of tasks. Each task i is characterized by a processing time t_i and a profit p_i. When a subset S_j of tasks is assigned to processor j, the average profit of that processor is defined as A_j = (∑{i∈S_j} p_i) / (∑{i∈S_j} t_i). The objective is to partition the n tasks among k processors so that all A_j are equal and as large as possible.

To formalize this, the authors define a new “fractional partition” (FP) problem. FP differs from the classic integer partition problem because the partition quality is measured by a ratio rather than by a sum. Two decision versions are considered: (1) the α‑feasibility problem, which asks whether a given target average α can be achieved, and (2) the maximum average profit (MAP) problem, which seeks the optimal α*.

Complexity analysis proceeds in two stages. First, when the number of processors k is a fixed constant, the authors construct a polynomial‑time reduction from 3‑SAT to the α‑feasibility problem, thereby proving that MAP is NP‑hard in this regime. The reduction maps Boolean variables and clauses to tasks with carefully chosen time and profit values so that achieving an average of at least α corresponds exactly to satisfying the original formula.

Second, for the general case where k is part of the input, the authors prove strong NP‑hardness by reducing from the strongly NP‑complete 3‑Partition problem. In this reduction each element of the 3‑Partition instance becomes a task, the total time and profit are set so that the target average α equals 1, and a feasible fractional partition exists if and only if the original 3‑Partition instance admits a perfect partition. Consequently, MAP is strongly NP‑hard without any restriction on k.

Despite these hardness results, the paper presents a pseudo‑polynomial dynamic‑programming (DP) algorithm that solves both FP and MAP when the input numbers are integers of moderate magnitude. The DP maintains a two‑dimensional table DP