Combinatorial Optimization for Constructing Covering Arrays and Sequence Covering Arrays

Dear Colleagues,

Nowadays, many combinatorial designs have been used that have succeeded in reducing the number of trials needed to obtain certain expected results in many fields, like Software Testing Processes, Business Processes, and Industrial Processes. In general, it is highly desirable to reduce the number of trials while increasing the possibilities of obtaining certain adjustments of a configuration for a system. For instance, in Software Testing Processes, it is advantageous to use the minimum number of tests to exercise at least once: a) all combinations of values from all subsets of a certain size with the total number of input variables of a software component; and b) all sequences in which all subsets of variables of a certain size are fed to a software component.

The combinatorial design used to test combinations of values of variables is the Covering Array (CA) defined as CA(N ; t, k, v), which is a matrix of the size N × k, where N is the number of tests, k is the number of variables, v is the number of possible values of each variable, and t is the size of the desired interaction. The combinatorial design used to exercise the order in which the input variables are fed to a system is called Sequence Covering Array (SCA) defined as SCA(M ; t, k), which is a matrix of the size M xk, where M is the number of possible orders, k is the number of input variables, and t is the size of the subsequences of input variables to be covered at least once.

As an optimization problem: a) the problem of constructing an optimal CA(N ; t, k, v) aims to minimize the value of N , maintaining the coverage of all the combinations of values of each t of the k input variables, at least once (each test in a CA belongs to the set v^k); b) the problem of constructing an optimal SCA(M ; t, k) aims to minimize the value of M, satisfying the coverage of all subsequences for each t of the k input variables (each test in an SCA belongs to the set of k! permutations).

There are few optimal known cases for CAs and SCAs, for instance, CA(v^t; t, k + 1, v) where v is a prime power and SCA(t!; t; t + 1) for t∈ {3, 5, 6}, but in general, the problems of constructing optimal CAs and optimal SCAs are hard combinatorial optimization problems, so it is desirable to use optimization algorithms to construct both combinatorial designs that are nearly optimal ones.

In this Special Issue of Mathematics papers are welcomed that report: a) novel constructions that use greedy, metaheuristic, and/or exact approaches to construct CAs and SCAs; and b) novel applications of CAs and/or SCAs.

Dr. Jose Torres-Jimenez
Guest Editor

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• combinatorial optimization
• covering array
• optimization problems
• constructing covering arrays
• sequence covering arrays