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More work on constraint programming

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Jip J. Dekker 2021-05-17 20:33:07 +10:00
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5
assets/glossary.tex
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@ -276,6 +276,11 @@
description={},
}
\newglossaryentry{propagator}{
name={propagator},
description={},
}
\newglossaryentry{reification}{
name={reification},
description={},

43
chapters/2_background.tex
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@ -633,9 +633,26 @@ It is clear that when the value \mzninline{a} is known, then the value of
performing an intelligent search by inferring which which values are still
feasible for each \gls{variable} \autocite{rossi-2006-cp}.
To find a solution to a given \gls{csp}, a \gls{cp} \gls{solver} will perform a
depth first search. At each node, the \gls{solver} will try and eliminate any
impossible value using a process called \gls{propagation}. For each
\gls{constraint} the \gls{solver} has a chosen algorithm called a
\gls{propagator}. Triggered by changes in the \glspl{domain} of its
\glspl{variable}, the \gls{propagator} will analyse and prune the any values
that are proven to be inconsistent.
When there are no more triggered \glspl{propagator} and not all \glspl{variable}
have been assigned a value, the \gls{solver} has to make a search decision. It
will bind a \gls{variable} to a value or add a new \gls{constraint}.
\begin{example}%
\label{ex:back-nqueens}
Consider as an example the N-Queens problem. Given a chess board of size
\(n \times n\), find a placement for \(n\) queen chess pieces such that no queen
can attack another. Meaning only one queen per row, one queen per column, and
one queen per diagonal.
The problem can be modelled as a \gls{csp} as follows.
\begin{mzn}
int: n;
@ -646,9 +663,31 @@ feasible for each \gls{variable} \autocite{rossi-2006-cp}.
constraint all_different([q[i] - i | i in 1..n]);
\end{mzn}
Since we know that there can only be one queen per column, the decision in the
model left to make is, for every queen, where in the column the piece is placed.
The \glspl{constraint} in the model model the remaining rules of the problem: no
two queen can be placed in the same row, no two queen can be place in the same
upward diagonal, and no two queens can be place in the same downward diagonal.
This model can be directly used in most \gls{cp} \glspl{solver}, since integer
\glspl{variable} and a \gls{propagator} are common.
When solving the problem, initially no values can be eliminated from the
\glspl{domain} of the \glspl{variable}. The first propagation will happen when
the first queen is placed on the board. The search space will then look like
this:
\jip{insert image of nqueens board with one queen assigned.}
Now that one queen is placed on the board the propagators for the
\mzninline{all_different} constraints can start propagating. They can eliminate
all values that are on the same row or diagonal as the queen placed on the
board.
\jip{insert previous image with all impossible moves marked red.}
\end{example}
\paragraph{Constraint Propagation}
\subsection{Mathematical Programming}%
\label{subsec:back-mip}
@ -737,7 +776,7 @@ can then be rewritten as linear \glspl{constraint} using the \glspl{variable}
integer \glspl{variable} \(q\) are used to represent where the queen is
located in each column. To encode the \mzninline{all_different} constraints,
the \glspl{indicator-var} \(y\) are inserted to reason about the value of
\(q\). The \cref{line:mip-channel} is used to connect the \(q\) and \(y\)
\(q\). The \cref{line:back-mip-channel} is used to connect the \(q\) and \(y\)
\glspl{variable} and make sure that their values correspond.
\Cref{line:back-mip-row} ensures that only one queen is placed in the same
row.