This paper extends the single-task n-Vehicle Exploration Problem to Multitask n-Vehicle Exploration Problem (MTNVEP), by combining n-Vehicle Exploration Problem with Job Scheduling Problem. At first, the authors prove that MTNVEP is NP-hard for fixed number of tasks, and it is strongly NP-hard for general number of tasks. Then they propose an improved accurate algorithm with computing time O(n3
 n
 ), which is better than O(n!) as n becomes sufficiently large. Moreover, four heuristic algorithms are proposed. Effectiveness of the heuristic algorithms is illustrated by experiments at last.

Multitask n -vehicle exploration problem: Complexity and algorithm

The research of efficient computation focus on special structures of NP-hard problem instances and request providing reasonable computing cost of instances in polynomial time. Based on the theory of combinatorial optimization, by studying the clusters partition and the clusters complexity measurement in N-vehicle exploration problem, we build a frame of efficient computation and provide an application of tractability for NP-hard problem. Three N-vehicle examples show that when we use efficient computation mechanism on N-vehicle, through polynomial steps of tractability analysis, decision makers can get the computing cost of searching optimal solution before practical calculation.

Research on the Efficient Computation Mechanism-in the Case of N -vehicle Exploration Problem

We discuss a variant of the multi-task n-vehicle exploration problem. Instead of requiring an optimal permutation of vehicles in every group, the new problem requires all vehicles in a group to arrive at the same destination. Given n tasks with assigned consume-time and profit, it may also be viewed as a maximization of every processor’s average profit. Further, we propose a new kind of partition problem in fractional form and analyze its computational complexity. By regarding fractional partition as a special case, we prove that the average profit maximization problem is NP-hard when the number of processors is fixed and it is strongly NPhard in general. At last, a pseudo-polynomial time algorithm for the average profit maximization problem and the fractional partition problem is presented, using the idea of the pseudo-polynomial time algorithm for the classical partition problem.

A variant of multi-task n-vehicle exploration problem: Maximizing every processor’s average profit

Finding the accurate solution for N-vehicle exploration problem is NP-hard in strong sense. In this paper, authors build a linear mixed integer programming model for N-vehicle exploration problem based on its properties. The model is then proved equivalent to the original problem. Given the model, one can apply the already existed methods and algorithms for mixed integer linear programming on N-vehicle exploration problem, which helps to enrich methods for solving N-vehicle exploration problem.

A Linear Mixed Integer Programming Model for N-Vehicle Exploration Problem

Multiple Jeeps Problem with Container Restriction

This paper proposes a method of estimating computational complexity of problem through analyzing its input condition for N-vehicle exploration problem. The N-vehicle problem is firstly formulated to determine the optimal replacement in the set of permutations of 1 to N. The complexity of the problem is factorial of N (input scale of problem). To balance accuracy and efficiency of general algorithms, this paper mentions a new systematic algorithm design and discusses correspondence between complexity of problem and its input condition, other than just putting forward a uniform approximation algorithm as usual. This is a new technique for analyzing computation of NP problems. The method of corresponding is then presented. We finally carry out a simulation to verify the advantages of the method: 1) to decrease computation in enumeration; 2) to efficiently obtain computational complexity for any N-vehicle case; 3) to guide an algorithm design for any N-vehicle case according to its complexity estimated by the method.

A method of estimating computational complexity based on input conditions for N -vehicle problem

The goal of efficient computation is to determine reasonable computing cost in polynomial time by using data structure of instance, and analyze the computing cost of satisfactory solution which can meet user’s requirements. When faced with NP-hard problem, we usually assess computational performance in the worst case. Polynomial algorithm cannot handle with NP-hard problem, so we research on NP-hard problem from efficient computation point of view. The work is intended to fill the blank of computational complexity theory. We focus on the cluster structure of instance data of aircraft range problem. By studying the partition and complexity measurement of cluster, we establish a connection between the aircraft range problem and N-vehicle exploration problem, and construct the efficient computation mechanism for aircraft range problem. The last examples show that the effect is significant when we use efficient computation mechanism on aircraft range problem. Decision makers can calculate the computing cost before actually computing.

Jin-chuan Cui

Papers

A method of estimating computational complexity based on input conditions for N -vehicle problem

Research on the Efficient Computation Mechanism-in the Case of N -vehicle Exploration Problem

A variant of multi-task n-vehicle exploration problem: Maximizing every processor’s average profit

The Efficient Computation of Aircraft Range Problem