Minimum cost of job assignment in polynomial time by adaptive unbiased filtering and branch-and-bound algorithm with the best predictor

Abstract

• Propose adaptive unbiased (AU) filtering to solve minimum job assignment. • Focus on indirect assignment with proper job-orders and unbiased predictors. • Improve with Latin square of n permutations and n complex mod-functions. • Analyze correctness and time complexity of AU-filtering and deep reduction. • Evaluate performance of AU-filtering, compared to the best of existing methods. The minimum cost of job assignment (Min-JA) is one of the practical NP-hard problems to manage the optimization in science-and-engineering applications. Formally, the optimal solution of the Min-JA can be computed by the branch-and-bound (BnB) algorithm (with the efficient predictor) in O( n !), n = problem size, and O( n 3 ) in the best case but that best case hardly occurs. Currently, metaheuristic algorithms, such as genetic algorithms (GA) and swarm-optimization algorithms, are extensively studied, for polynomial-time solutions. Recently, unbiased filtering (in search-space reduction) could solve some NP-hard problems, such as 0/1-knapsack and multiple 0/1-knapsacks with Latin square (LS) of m -capacity ranking, for the ideal solutions in polynomial time. To solve the Min-JA problem, we propose the adaptive unbiased-filtering (AU-filtering) in O( n 3 ) with a new hybrid (search-space) reduction (of the indirect metaheuristic strategy and the exact BnB). Innovation-and-contribution of our AU-filtering is achieved through three main steps: 1. find 9+ n effective job-orders for the good initial solutions (by the indirect assignment with UP: unbiased predictor), 2. improve top 9-solutions by the indirect improvement of the significant job-orders (by Latin square of n permutations plus n complex mod-functions), and 3. classify objects (from three of the best solutions) for AU-filtering (on large n ) with deep-reduction (on smaller n ’) and repeat (1)-(3) until n ’ < 6, the exact BnB is applied. In experiments, the proposed AU-filtering was evaluated by a simulation study, where its ideal results outperformed the best results of the hybrid swarm-GA algorithm on a variety of 2D datasets ( n ≤ 1000).