Journal of Optimization

Volume 2017, Article ID 5650364, 8 pages

https://doi.org/10.1155/2017/5650364

## A Genetic Algorithm Based Approach for Solving the Minimum Dominating Set of Queens Problem

^{1}Computer Science Department, Taibah University, Medina, Saudi Arabia^{2}School of Computer Sciences, Universiti Sains Malaysia, Penang, Malaysia

Correspondence should be addressed to Saad Alharbi; moc.liamg@02ibrahlas

Received 20 February 2017; Revised 24 April 2017; Accepted 3 May 2017; Published 4 June 2017

Academic Editor: Gexiang Zhang

Copyright © 2017 Saad Alharbi and Ibrahim Venkat. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In the field of computing, combinatorics, and related areas, researchers have formulated several techniques for the Minimum Dominating Set of Queens Problem (MDSQP) pertaining to the typical chessboard based puzzles. However, literature shows that limited research has been carried out to solve the MDSQP using bioinspired algorithms. To fill this gap, this paper proposes a simple and effective solution based on genetic algorithms to solve this classical problem. We report results which demonstrate that near optimal solutions have been determined by the GA for different board sizes ranging from 8 × 8 to 11 × 11.

#### 1. Introduction

The MDSQP which is otherwise known as the chess covering problem is one of the well-known chessboard problems which has been receiving continued interests by researchers including the computational intelligence domain. In the field of graph theory, specifically the study of dominating sets had addressed this problem even in the early 1970s. In chessboard, cells are organized as rows and columns. In chess layman terms, a queen (say ) placed on a square will dominate all squares associated with the row, column, and two diagonals with reference to . The idea behind this problem is to find the minimum number of queens required to dominate the whole chessboard. Domination here refers to the coverage of all the possible squares being attacked by those queens including the square dominated by the respective queens. Previous researches have been trying to find the domination number for the -queens problem using mathematical and combinatorial approaches [1–8]. Many formulations have been derived and presented in these approaches. It has been observed that limited efforts have been carried out to employ evolutionary algorithms to address the MDSQP. In fact, most of the studies in the literature were devoted to address the original *-*queens problem which is called a nonattacking problem. The results of such studies have demonstrated that evolutionary algorithms are capable of outperforming other principled approaches such as backtracking to solve such problems. However, these studies have focused only on the nonattacking -queens problem, whereas, to the best of our knowledge, the MDSQP has been least considered using bioinspired evolutionary algorithms.

Evolutionary algorithms have been proved to be successful for solving and optimizing a wide range of complex problems including combinatorial ones, such as the one studied here, within reasonable computing time [9]. Generally speaking, evolutionary algorithms can be classified into two main categories which are swarm intelligence based approaches and classical evolutionary approaches. Swarm intelligence (SI) approaches are inspired from the social natural behavior such as animals group behavior when searching for food or avoiding certain threat [10]. Several SI approaches with reference to the literature include Particle Swarm Optimization (PSO) and Ant Colony Optimization (ACO). ACO have been proved to efficiently solve several optimization problems [11]. PSO has also demonstrated good results in optimizing various complex problems such as the supply chain management (e.g., [12]) and shortest path problems [13, 14].

The classical evolutionary algorithms, on the other hand, including genetic algorithms, were successfully adopted in a wide range of optimization problems. Genetic algorithm is one of the most commonly used evolutionary algorithms in the literature which was first proposed in the 1970s [15]. GA was inspired from the natural process of searching and selection process which leads to the survival of the fittest individuals [15]. Researchers in the literature have demonstrated the efficiency of adopting GA in solving various optimization problems such as data mining [16] and network traffic control [17]. In fact, GA have been adopted in the literature solely to solve some problems such as [17, 18] and were hybridized with other approaches for more enhanced results. Commonly, GA is hybridized with local and heuristic search approaches such as Cuckoo search [19] and Tabu search [20]. Furthermore, various researchers hybridized it with other evolutionary approaches like dynamic programming as in [21].

Therefore, this paper intends to fill these gaps and present a bioinspired genetic algorithm (GA) to solve the MDSQP problem by considering board sizes gradually from the typical chessboard up to a board. The proposed approach gradually increments the number of queens placed on the board from just below the lower bounds suggested in the literature until an optimal solution has been determined (i.e., all the squares of the chessboard get dominated). The remaining paper is organized as follows: Section 2 discusses related work; Section 3 formulates the problem; Section 4 presents the evolutionary mechanism of the proposed technique; results are discussed in Section 5 and the paper is finally concluded with insights for future work in Section 6.

#### 2. Related Work

Vast amount of efforts have been devoted to investigate chessboard problems. Problems related to chessboard can mainly be classified to the original -queen problem and the dominating set (i.e., the covering problem). The original -queen problem can be defined as placing queens on chessboard in an optimal way such as none of the queens should attack each other [22, 23]. It was first introduced in 1850 [24]. Numerous researches have been performed to solve this problem using different approaches; most of them adopted mathematical approaches and graph theory such as [25–27]. The study of such a problem may benefit in the understanding of various applications such as traffic control, deadlock prevention, and parallel memory storage [28]. The complexity of the -queen problem is known as exponential where it gets more complex with large values. Various approaches were adopted to solve this problem such as backtracking [29], neural networks [30–32], and evolutionary and optimization algorithms [33–42].

The domination set problem on the other hand has received less attention by computer science researchers. In fact, various researches have been devoted to investigate such a problem but were mainly adopting mathematical models. Burger and Mynhardt pointed out that the queens dominating problem is one of the most difficult chessboard problems [43]. It is commonly defined as finding the minimum numbers of queens required to cover all the squares of chessboard. This number is termed as the domination number and referred by the notation . Researchers have arrived at upper and lower bounds since early 1970s [3]. Many formulations and values were derived and found especially for small values of [3, 43–45]. Table 1 shows some of these values. Figure 1 shows a brief taxonomy of typical chessboard problems. For instance, several studies were conducted to find the domination number of the normal domination [8]. In these studies, it is required to find the minimum queens to be optimally placed on chessboard that will cover all squares in the board. However, many researchers considered the domination set problems under several variations. For instance, many studies were conducted to investigate the independent domination number which can be defined as the number of queens which do not attack each other and cover the entire chessboard [3]. Various values and formulations were derived and proved in a mathematical context. Furthermore, the “diagonal queens domination problem” was frequently stated in the literature. In such a problem, it is required to find the number of minimum queens placed in the main diagonal and cover the entire chessboard [3, 4]. Cockayne has also tried to find the domination number of the queens placed on a single column [3]. Furthermore, many researchers carried out studies to solve the problem of the domination set on a toroidal chessboard [1]. In such studies, it is required to find out the minimum number of queens that would cover the entire squares in the torus. Bozóki et al., on the other hand, shed the light on a different problem called “domination of the rectangular queens graph” where it is required to find the domination number of queens in a rectangular chessboard () in which the number of squares in column and rows are different [46]. They derived lower bounds for small values of and particularly .