I'm trying to formulate the problem of placing $N$ queens on an $N \times N$ chessboard such that no two queens share any row, column, or diagonal.
I managed to define my decision variable as $x[n][n]$, a binary variable indicating if the location is used or not. But I couldn't find a way to write any linear constraints. Any ideas on how to proceed, please?
I will use the following convention :
Any space is referred to by its coordinate $(i,j)$. The bottom left space is $(1,1)$, The bottom right space is $(1,8)$, the top right is $(8,1)$ and top left is $(8,8)$.
Variables
$x_{ij} = 1$ if a queen is on the space $(i,j)$, $0$ otherwise
Problem
$$\max \sum\limits_{i,j} x_{ij}$$
(All the spaces in an increasing diagonal have the same value for $i-j$) $$ \sum\limits_{i,j|j-i = k} x_{ij} \leq 1 \mbox{ for } k\in \{-6, -5, ..., 5, 6\}$$
(all the spaces in a decreasing diagonal have the same value for $i+j$) $$ \sum\limits_{i,j|i+j = k} x_{ij} \leq 1 \mbox{ for } k\in \{2, 3, ..., 14\}$$
$$x_{ij}\in \{0,1\}$$
There are several different formulations for the problem as both a Linear Program and as a Constraint Program. Let's look at the Linear Program first.
Binary Linear Program Formulation:
Let $Q_{ij}$ be a binary variable to denote if a Queen is placed in column $i$, row $j$, and let $n$ denote the board size of a $n\times n$ board. Then, the formulation will go as follows:
$$\max z = 0$$ Subject to:
There should only exist one queen in each row:
$$\sum_{i=1}^n Q_{ij} = 1\quad\forall j\in 1,2,\ldots,n$$
There should only exist one queen in each column:
$$\sum_{j=1}^n Q_{ij} = 1\quad\forall i\in 1,2,\ldots,n$$
Queens shouldn't attack each other on each diagonal: $$\sum_{i-j=k}^nQ_{ij}\le 1,\quad\forall k = 2-n, \ldots,n-2$$ $$\sum_{i+j=k}^nQ_{ij}\le 1,\quad\forall k = 2, \ldots,2n-2$$
Variable Restrictions:
$$Q_{ij}\in\{0,1\}\quad\forall i,j \in 1,\ldots, n$$
Integer Linear Program Formulation:
Let $Q_{i}$ be a integer variable to denote if a Queen of a certain column is placed in row $i$ (since we know all columns will have a Queen), and let $n$ denote the board size of a $n\times n$ board. Then, the formulation will go as follows:
$$\min z = S_1 + S_2$$
Queens cannot be placed in the same rows:
$$Q_i \le Q_j-1 + M b_1 \quad\forall i < j$$ $$Q_i \ge Q_j+1 - M (1-b_1) \quad\forall i < j$$
Queens diagonals must be respected:
$$Q_{i} - Q_{j} \le S_1\quad\forall i,j\in{1,2,\ldots, n}\quad\text{where }i\ne j$$
$$Q_{j} - Q_{i} \le S_2\quad\forall i,j\in{1,2,\ldots, n}\quad\text{where }i\ne j$$
$$S_1 + S_2\le |i - j| + M b_2 \quad\forall i < j$$ $$S_1 + S_2\ge |i - j|+1 - M(1-b_2) \quad\forall i < j$$
Variable Restrictions: $$1\le Q_{i}\le n\quad\forall i \in 1,\ldots, n$$ $$Q_{i}\in\mathbb{Z}^+\quad\forall i \in 1,\ldots, n$$ $$b_1, b_2\in\{0,1\}$$ $$S_1,S_2\ge0$$
$M$ is a pre-chosen integer $n+1$.
Constraint Program Formulation (Not LP)
Let $Q_{i}$ be a integer variable to denote if a Queen of a certain column is placed in row $i$ (since we know all columns will have a Queen), and let $n$ denote the board size of a $n\times n$ board. Then, the formulation will go as follows:
Queens cannot be placed in the same rows:
$$Q_{i} \ne Q_{j}\quad\forall i,j\in{1,2,\ldots, n}\quad\text{where }i\ne j$$
Queens diagonals must be respected:
$$|Q_{i} - Q_{j}| \ne |i - j| \quad\forall i,j\in{1,2,\ldots, n}\quad\text{where }i\ne j$$
Variable Restrictions: $$1\le Q_{i}\le n\quad\forall i \in 1,\ldots, n$$ $$Q_{i}\in\mathbb{Z}^+\quad\forall i \in 1,\ldots, n$$
