## 1 Introduction
Researchers have developed and tested various methods to generate the smallest possible test case set, including the ant colony optimization (ACO) algorithm, simulated annealing (SA) algorithm, genetic algorithm (GA), particle swarm optimization (PSO) algorithm, etc. These metaheuristic search algorithms can efficiently and effectively search within the combinatorial space of all possible test cases and generate a combinatorial test case set [8]. Experimental results show that, compared with other methods, metaheuristic search algorithms can reduce the size of the generated combinatorial test case set to a certain extent. However, a large amount of computation time is still needed by metaheuristic optimization approaches to generate optimal test cases [9]. The basic optimization algorithm has been improved and upgraded based on the needs and characteristics of specific optimization problems, and the strategies for improvement include chaotic mapping [10; 11], Cauchy variation [12], opposition-based learning [13], and so on. Current research related to the application of PSO algorithms to combinatorial tests shows that they have certain advantages in terms of the size of the set of combinatorial test cases generated [14]. In addition, the PSO algorithm has the advantages of easy use, rapid computation, and good robustness. However, the traditional PSO algorithm also has several weaknesses, including a low convergence accuracy, the ease of falling into local optima, premature convergence or stagnation, etc. Intensive research has been performed to improve the performance of the PSO algorithm. However, most of the efforts focus on improving the optimality of the generated combinatorial test case set, with less consideration of the computation time needed for convergence. Therefore, a new PSO algorithm with improved computational efficiency is highly desirable for the efficient generation of optimal combinatorial test case sets for software testing. This paper proposes a new approach for the efficient generation of optimal test case sets for software testing. As the first step of the proposed approach, the cause-effect diagram of the software system under testing is obtained through the if-then-else normal form (INF) specification. In the second step, the reduced ordered binary decision diagram (ROBDD) is utilized to simplify the cause-effect diagram to eliminate the influence of useless causes. Finally, test cases for pairwise testing are generated with an improved PSO algorithm. Experiments show that the proposed approach can reduce the number of test cases generated while ensuring the coverage of pairwise tests, thereby achieving the purpose of cost reduction. The rest of the paper is organized as follows. In Section 2, the work related to cause-effect graphs and PSO algorithms in generating test cases is reviewed. In Section 3, the required basics for the proposed test case generation approach are briefly introduced and the technical details of the approach are presented. In Section 4, the performance of the proposed approach is tested and evaluated on a known case study. Section 5 concludes the paper and provides an outlook for future work.