Local Search Element Decomposition Method to Solve Integer and m-Integer Problems

Abstract

Integer problems are non-deterministic polynomial-time-hard and can be solved in polynomial time. However, the solution to large problems requires an unreasonably long time. This paper highlights the significance of developing and modifying the local search element decomposition method (LSEDCM) for accurately and promptly solving general integer and m-integer problems. The study involves two parts: creating and developing the algorithm by applying the LSEDCM and modifying the LSEDCM, genetic algorithm, and ant system. Results obtained using this algorithm were then compared with the exact and approximated solutions in terms of their accuracy, number of steps, and solution time. It was found that general integer and m-integer problems involving 2–30000 variables could be effectively solved. For the general integer problem, the average difference from the exact solution was 0.37%; it took 91.28% fewer steps and 29.29% less time to solve the problem. The average difference from the approximated solutions for the modified genetic algorithm (MGA) and modified ant system (MAS) was 1.47 and 2.16%, respectively. For an m-integer problem, the average difference from the exact solution was 0.57%; it took 90.60% fewer steps and 49.33% lesser time to solve the problem. The average difference from the approximated solutions for the MGA and MAS was 1.58 and 2.69%, respectively. Compared with approximated solutions, the LSEDCM obtained the best solution for the large problems. It is thus proven that the LSEDCM could be used to solve large integer problems in a shorter time compared with m-integer problems, which are more complex and intrinsically require a longer time to obtain a solution.