相关论文: Finding First and Most-Beautiful Queens by Integer…
Number the cells of a (possibly infinite) chessboard in some way with the numbers 0, 1, 2, ... Consider the cells in order, placing a queen in a cell if and only if it would not attack any earlier queen. The problem is to determine the…
We consider the problem of placing $n$ nonattacking queens on a symmetric $n \times n$ Toeplitz matrix. As in the $N$-queens Problem on a chessboard, two queens may attack each other if they share a row or a column in the matrix. However,…
Block-structured integer linear programs (ILPs) play an important role in various application fields. We address $n$-fold ILPs where the matrix $\mathcal{A}$ has a specific structure, i.e., where the blocks in the lower part of…
Integer linear programming (ILP) encompasses a very important class of optimization problems that are of great interest to both academia and industry. Several algorithms are available that attempt to explore the solution space of this class…
Due to their importance in practice, dominating set problems in graphs have been greatly studied in past and different formulations of these problems are presented in literature. This paper's focus is on two problems: weakly convex…
Some preliminary results are reported on the equivalence of any n-queens problem with the roots of a Boolean valued quadratic form via a generic dimensional reduction scheme. It is then proven that the solutions set is encoded in the…
Cutting plane methods are a fundamental approach for solving integer linear programs (ILPs). In each iteration of such methods, additional linear constraints (cuts) are introduced to the constraint set with the aim of excluding the previous…
In this paper, we show how a resource allocation problem can be solved through Integer Linear Programming (ILP). A detailed illustrative example is presented, together with an exhaustive overview of the mathematical model. The size of the…
In his list of open problems, Martin Erickson described a certain game: "Two players alternately put queens on an n x n chess board so that each new queen is not in range of any queen already on the board (the color of the queens is…
Integer Linear Programming (ILP) has a broad range of applications in various areas of artificial intelligence. Yet in spite of recent advances, we still lack a thorough understanding of which structural restrictions make ILP tractable.…
We introduce a two player game on an n x n chessboard where queens are placed by alternating turns on a chessboard square whose availability is determined by the number of queens already on the board which can attack that square modulo two.…
Python currently is the dominant language in the field of Machine Learning but is often criticized for being slow to perform certain tasks. In this report, we use the well-known $N$-queens puzzle as a benchmark to show that once compiled…
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…
We apply to the $n\times n$ chessboard the counting theory from Part I for nonattacking placements of chess pieces with unbounded straight-line moves, such as the queen. Part I showed that the number of ways to place $q$ identical…
This paper addresses the challenge of determining optimal cut-offs for a set of n items with m scores to maximize distinguishability. The term distinguishability is defined as the fraction of item pairs assigned to different buckets, where…
We propose a new exact approach for solving integer linear programming (ILP) problems which we will call projective splitting algorithms (PSAs). Unlike classical methods for solving ILP problems, PSAs conduct the search for the optimal…
Floor planning is an important and difficult task in architecture. When planning office buildings, rooms that belong to the same organisational unit should be placed close to each other. This leads to the following NP-hard mathematical…
We introduce QUEENS, a derivative chess problem based on the classical n-queens problem. We prove that QUEENS is NP-complete, with respect to polynomial-time reductions.
While quantum computing proposes promising solutions to computational problems not accessible with classical approaches, due to current hardware constraints, most quantum algorithms are not yet capable of computing systems of practical…
The No-Three-In-Line problem asks for the maximum number of points that can be placed on an n by n grid with no three collinear, representing a famous problem in combinatorial geometry. While classical methods like Integer Linear…