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# The n-Queens Problem

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... complexity of the optimization tasks, and hence, the complexity of the approaches that handle the problem (Chong and Zak 2004). Some factors are the drastic increase of the number of variables involved in the optimization problem (large scale optimization) such as in portfolio optimization problems (Markowitz 1952), the imposition of several constraints in the problem (constrained optimization) such as in the n-queens problem (Rivin et al. 1994), the optimization of more than one objective for the same problem (multi-objective optimization) such as the multi-objective knapsack problem (Bazgan et al. 2009), the setting of evidences in the model so that the cost function depends on some fixed values for some of the variables (evidence optimization) which is common in the industrial areas such as textile optimization depending on characteristics of the textile (Sahani and Linden 2002), or the optimization of a cost function which varies along runtime (dynamic optimization) such as feature subset selection in data streams (Huang et al. 2015), among others. ...
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Many real-world optimization problems involve two different subsets of variables: decision variables, and those variables which are not present in the cost function but constrain the solutions, and thus, must be considered during optimization. Thus, dependencies between and within both subsets of variables must be considered. In this paper, an estimation of distribution algorithm (EDA) is implemented to solve this type of complex optimization problems. A Gaussian Bayesian network is used to build an abstraction model of the search space in each iteration to identify patterns among the variables. As the algorithm is initialized from data, we introduce a new hyper-parameter to control the influence of the initial data in the decisions made during the EDA execution. The results show that our algorithm improves the cost function more than the expert knowledge does.
... The 8-queens problem was first published by German chess composer Max Bezzel in 1848, and attracted the attention of many mathematicians, including Gauss. We refer to the surveys [5,31] for a more detailed account of the history and many related problems. Recently, there has been some exciting progress [6,19,20,32] towards the n-queens problem and its toroidal version which was introduced by Pólya in 1918. ...
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An $n$-queens configuration is a placement of $n$ mutually non-attacking queens on an $n\times n$ chessboard. The $n$-queens completion problem, introduced by Nauck in 1850, is to decide whether a given partial configuration can be completed to an $n$-queens configuration. In this paper, we study an extremal aspect of this question, namely: how small must a partial configuration be so that a completion is always possible? We show that any placement of at most $n/60$ mutually non-attacking queens can be completed. We also provide partial configurations of roughly $n/4$ queens that cannot be completed, and formulate a number of interesting problems. Our proofs connect the queens problem to rainbow matchings in bipartite graphs and use probabilistic arguments together with linear programming duality.
... Theorem 2 [23]. For all m and n for which m ≥ 3 and gcd(n, 6) = 1, it holds that Q(mn) > (Q(m)) n M (n). ...
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Let D be a digraph, possibly with loops. A queen labeling of D is a bijective function $$l:V(G)\longrightarrow \{1,2,\ldots ,|V(G)|\}$$ such that, for every pair of arcs in E(D), namely (u, v) and $$(u',v')$$ we have (i) $$l(u)+l(v)\ne l(u')+l(v')$$ and (ii) $$l(v)-l(u)\ne l(v')-l(u')$$. Similarly, if the two conditions are satisfied modulo $$n=|V(G)|$$, we define a modular queen labeling. There is a bijection between (modular) queen labelings of 1-regular digraphs and the solutions of the (modular) n-queens problem. The $$\otimes _h$$-product was introduced in 2008 as a generalization of the Kronecker product and since then, many relations among labelings have been established using the $$\otimes _h$$-product and some particular families of graphs. In this paper, we study some families of 1-regular digraphs that admit (modular) queen labelings and present a new construction concerning to the (modular) n-queens problem in terms of the $$\otimes _h$$-product, which in some sense complements a previous result due to Pólya.
... A trivial upper bound is n!, because every set of n independent queens can have at most one queen in each row and column. More recently, Luria [20] gave an upper bound on the number solutions of order Oð n n e an Þ, where a [ 1, while previously, Rivin, Vardi, and Zimmerman [32] had provided a lower bound of 2 ffiffiffiffiffiffiffiffiffiffiffiffi ðpÀ1Þ=2 p for prime-length chessboards and also proved that the number of n-queen solutions is greater than 4 n=5 for n a multiple of 5 that is relatively prime to 6. ...
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... This construction is taken from [2] and firstly introduced by Polya in [7]. Obviously, if arrangements (A 1 , A 2 , . . . ...
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Using modular arithmetic of the ring $\mathbb{Z}_{n+1}$ we obtain a new short solution to the problem of existence of at least one solution to the $N$-Queens problem on an $N \times N$ chessboard. It was proved, that these solutions can be represented as the Queen function with the width fewer or equal to 3. It is shown, that this estimate could not be reduced. A necessary and sufficient condition of being a composition of solutions a solution is found. Based on the obtained results we formulate a conjecture about the width of the representation of arbitrary solution. If this conjecture is valid, it entails solvability of the $N$-Queens completion in polynomial time. The connection between the $N$-Queens completion and the Millennium $P$ vs $NP$ Problem is found by the group of mathematicians from Scotland in August 2017.
... For more information about the N queens problem, see [7], [15], and [16]. For more about the N + k queens problem and related problems, see [6], [4], and [5] (Preprints are available online at [3]). ...
... It uses different principles of soccer to solve combinatorial optimization problems. The quality of this technique is demonstrated applying it to four combinatorial problems [23]: Asymmetric traveling salesman problem (ATSP) [24], Vehicle Routing Problem with Backhauls (VRPB) [25], [26], n-Queen Problem (NQP) [27], One-Dimensional Bin Packing Problem (BPP) [28].This algorithm is a promising metaheuristic to solve combinatorial optimization problems [23]. ...
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The Quadratic Assignment Problem (QAP) is a combinatorial optimization problem; it belongs to the class of NP-hard problems. This problem is applied in various fields such as hospital layout, scheduling parallel production lines and analyzing chemical reactions for organic compounds. In this paper we propose an application of Golden Ball algorithm mixed with Simulated Annealing (GBSA) to solve QAP. This algorithm is based on different concepts of football. The simulated annealing search can be blocked in a local optimum due to the unacceptable movements; our proposed strategy guides the simulated annealing search to escape from the local optima and to explore in an efficient way the search space. To validate the proposed approach, numerous simulations were conducted on 64 instances of QAPLIB to compare GBSA with existing algorithms in the literature of QAP. The obtained numerical results show that the GBSA produces optimal solutions in reasonable time; it has the better computational time. This work demonstrates that our proposed adaptation is effective in solving the quadratic assignment problem.
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