Content uploaded by Igor Rivin
Author content
All content in this area was uploaded by Igor Rivin on Nov 19, 2015
Content may be subject to copyright.
The n-Queens Problem
The American Mathematical Monthly
... In 1874, Pauls [10,11] proved that Q(n) > 0 for every n ≥ 4. In 1918, Pólya [12] showed that T (n) > 0 if and only if n ≡ 1 or 5 (mod 6). In 1994, Rivin, Vardi, and Zimmerman [13] conjectured that log Q(n) = Θ(n log(n)) and that log T (n) = Θ(n log(n)) for n ≡ 1, 5 (mod 6). In 2017, Luria [8] showed that T (n) ≤ ((1 + o(1))ne −3 ) n and that there exists a constant α > 1.587 such that Q(n) ≤ ((1 + o(1))ne −α ) n for all n. ...
... Bowtell and Keevash [2] also proved that T (n) ≥ ((1 − o(1))ne −3 ) n for n ≡ 1, 5 (mod 6), thereby giving an asymptotic solution to the toroidal n-queens problem. These results, combined with Luria's result, completely settled the conjecture by Rivin, Vardi, and Zimmerman [13]. Furthermore, Simkin [16] improved the bounds for the classical n-queens problem, showing that there exists a constant 1.94 < α < 1.9449 such that Q(n) = ((1 ± o(1))ne −α ) n . ...
In 1967, Klarner proposed a problem concerning the existence of reflecting n-queens configurations. The problem considers the feasibility of placing n mutually non-attacking queens on the reflecting chessboard, an chessboard with a "reflecting strip" of squares added along one side of the board. A queen placed on the reflecting chessboard can attack the squares in the same row, column, and diagonal, with the additional feature that its diagonal path can be reflected via the reflecting strip. Klarner noted the equivalence of this problem to a number theory problem proposed by Slater, which asks: for which n is it possible to pair up the integers 1 through n with the integers n+1 through 2n such that no two of the sums or differences of the n pairs of integers are the same. We prove the existence of reflecting n-queens configurations for all sufficiently large n, thereby resolving both Slater's and Klarner's questions for all but a finite number of integers.
... The permanent was previously considered by Rivin and Zabih to compute Q(n) and T (n) [7]. Similarly, in 1874 Gunther used the determinant to construct solutions to the n-queens problem for small values of n [1]. ...
In this paper, we derive simple closed-form expressions for the n-queens problem and three related problems in terms of permanents of (0,1) matrices. These formulas are the first of their kind. Moreover, they provide the first method for solving these problems with polynomial space that has a nontrivial time complexity bound. We then show how a closed-form for the number of Latin squares of order n follows from our method. Finally, we prove lower bounds. In particular, we show that the permanent of Schur's complex valued matrix is a lower bound for the toroidal semi-queens problem, or equivalently, the number of transversals in a cyclic Latin square.
... The eight-queens puzzle is the problem of placing eight chess queens mutually non-attacking on an 8 × 8 chessboard [48]. Thus, a solution requires that no two queens share the same row, column, or diagonal. ...
BDDs are representations of a Boolean expression in the form of a directed acyclic graph. BDDs are widely used in several fields, particularly in model checking and hardware verification. There are several implementations for BDD manipulation, where each package differs depending on the application. This paper presents HermesBDD: a novel multi-core and multi-platform binary decision diagram package focused on high performance and usability. HermesBDD supports a static and dynamic memory management mechanism, the possibility to exploit lock-free hash tables, and a simple parallel implementation of the If-Then-Else procedure based on a higher-level wrapper for threads and futures. HermesBDD is completely written in C++ with no need to rely on external libraries and is developed according to software engineering principles for reliability and easy maintenance over time. We provide experimental results on the n-Queens problem, the de-facto SAT solver benchmark for BDDs, demonstrating a significant speedup of 18.73x over our non-parallel baselines, and a remarkable performance boost w.r.t. other state-of-the-art BDDs packages.
... 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] toward the n-queens problem and its toroidal version which was introduced by Pólya in 1918. ...
An n-queens configuration is a placement of n mutually non-attacking queens on an n×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.
... 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. ...
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. ...
An n-queens configuration is a placement of n mutually non-attacking queens on an 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). ...
Let D be a digraph, possibly with loops. A queen labeling of D is a bijective function such that, for every pair of arcs in E(D), namely (u, v) and we have (i) and (ii) . Similarly, if the two conditions are satisfied modulo , 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 -product was introduced in 2008 as a generalization of the Kronecker product and since then, many relations among labelings have been established using the -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 -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. ...
... This construction is taken from [2] and firstly introduced by Polya in [7]. Obviously, if arrangements (A 1 , A 2 , . . . ...
Using modular arithmetic of the ring we obtain a new short solution to the problem of existence of at least one solution to the N-Queens problem on an 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.
The N-Queens problem is relevant in Artificial Intelligence (AI); the solution methodology has been used in different computational intelligent approaches. Max Bezzel proposed the problem in 1848 for eight queens in 8 8 chessboard. After that, the formulation was modified to an N-Queens problem in a chessboard. There are several ways of posing the problem and algorithms to solve it. We describe two commonly used mathematical models that handle the position of queens and restrictions. The first and easiest way is to find one combination that satisfies the solution. The second model uses a more compact notation to represent the queen’s potions. This generic problem has been solved with many different algorithms. However, there is no comparison of the performance among the methods. In this work, a comparison of performance for different problem sizes is presented. We tested the Backtracking, Branch and Bound, and Linear Programming algorithms for a different number of queens, reaching 17. In addition, we present statistical comparative experimental results of the different methods.
ResearchGate has not been able to resolve any references for this publication.