Constraint Programming is a vast subject area in AI and Operations Research. Some well-known Constraint Satisfaction Problems (CSP) are Sudoku, N-Queens, Map Coloring, Scheduling (of many kinds, e.g. classrooms-to-courses).

In general, CSP are NP-hard but many techniques/algorithms have emerged to tackle such problems. The two basic approaches used are **Search** and **Inference**. There are many variations of each and the two are often mixed together. Arc Consistency 3 (AC-3) is one of best known inference algorithms which uses constraint propagation.

The definitive textbook on CSP is “Constraint Processing” by Dr. Rina Dechter. One interesting algorithm presented in the book is “Bucket Elimination”. Bucket Elimination is closely related to Variable Elimination which is used in a variety of areas and can be considered a general framework for reasoning or inference.

For a CSP, Bucket Elimination gives back a ‘compressed’ representation of the underlying ‘knowledge’ that can be useful for further exploration – beyond just finding all solutions. For example, given a partial variable assignment one easily determine the possible assignments of other variables that would lead to a solution.

Here is an experimental implementation of the Bucket Elimination algorithm in F#:

The implementation provides a DSL to declaratively model finite domain CSPs. The constraint programming ‘language’ (i.e. the DSL) does not currently provide ‘global constraints’ such as “allDifferent” but the language can be easily extended, if needed.

The solution includes samples for N-Queens; Map Coloring and “SEND+MORE=MONEY” problems.

The following code is from the 4-Queens sample:

let domain = intSet [for i in 1..4 -> i]

//variables

let x1 = VarDef(“x1”, domain)

let x2 = VarDef(“x2”, domain)

let x3 = VarDef(“x3”, domain)

let x4 = VarDef(“x4”, domain)

let constraints =

[

//all different

&x1 != &x2

&x2 != &x3

&x3 != &x4

&x1 != &x3

&x2 != &x4

//safe

Abs(&x1 – &x2) != 1I

Abs(&x2 – &x3) != 1I

Abs(&x3 – &x4) != 1I

Abs(&x1 – &x3) != 2I

Abs(&x1 – &x4) != 3I

Abs(&x2 – &x4) != 2I

]

let cs = ConstraintSystem([x1;x2;x3;x4], constraints)

let solSet = solve cs

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