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Solving the Sudoku Using Integer Programming
A 9 X9 SUDOKU puzzle has the following rules. Each row and column must have the numbers between 1 and 9 and each of the inner boxes must have the numbers between 1 and 9. Each number in each column and row and in each small box must appear only once.
Let’s just set Xijk for all values of I, j and k from 1 to 9 to be 1. If cell (I,j) contains the number k where I, j and k are all between 1 and 9. Here I denotes the ith row and j denotes the jth column and k denotes an integer between 1 and 9. When X134 = 1, it means that cell (1,3) contains the number 4. This would also imply that none of the other elements of the 1st row or 3rd column except cell (1,3) can be equal to 4.
In order to model the SUDOKU, we will use a total of 729 variables.
Let us now formulate algebraically each of the three classes of rules.
Each line must contain a number between 1 and 9 exactly once.
For the first line, this rule would appear as (called “constraint” in integer programming language).
for each I from 1 to 9 and for each k from 1 to 9 (I is a mathematical representation of a counter variable)
sum (Xijk) for all j from 1 to 9 = 1;
Written in verbose form for the 1st line for each number between 1 and 9
X111 + X121 + X131 + X141 + X151 + X161 + X171 + X181 + X191 = 1.
X112 + X122 + X132 + X142 + X152 + X162 + X172 + X182 + X192 = 1.
X113 + X123 + X133 + X143 + X153 + X163 + X173 + X183 + X193 = 1.
X114 + X124 + X134 + X144 + X154 + X164 + X174 + X184 + X194 = 1.
These equations follow for variables beginning with X115 through X119.
Similarly, let’s formulate equations for the rules of each number between 1 and 9 appearing only once in each of the 9 columns.
Written in mathematical notation,
sum X for all j from 1 to 9 (for all I and k between 1 and 9) = 1
Written in detailed form for a few columns for each number between 1 and 9
X111 + X211 + X311 + X411 + X511 + X611 + X711 + X811 + X911 = 1.
X112 + X212 + X312 + X412 + X512 + X612 + X712 + X812 + X912 = 1.
X113 + X213 + X313 + X413 + X513 + X613 + X713 + X813 + X913 = 1.
This must be completed for all other numbers 4 through 9.
X121 + X221 + X321 + X421 + X521 + X621 + X721 + X821 + X921 = 1.
X122 + X222 + X322 + X422 + X522 + X622 + X722 + X822 + X922 = 1.
X123 + X223 + X323 + X423 + X523 + X623 + X723 + X823 + X923 = 1.
This must be completed for all other numbers from 4 to 9.
Let us now represent the small squares ( 3 x 3 ) with a total number of 9 squares.
So in each 3 x 3 box, there must be only one 1 or 2 or 3 or 4 or 5 or 6 or 7 or 8 or 9 etc.,
Cells appear between columns (1-3) and rows (1-3), columns (4-6) and rows (1-3) columns (7-9) rows (1-3). Also for the same set of columns, they appear in rows (4 to 6) and (6 to 9). So let’s formulate the equations for a single small square located between the columns (1 to 3) and the rows (1 to 3). The corresponding decision variables for the number “1” are (9 in total)
X111, X121, X131, X211, X221, X231, X311, X321, X331.
Let’s formulate the equation that this square (3 x 3) contains only one “1”.
So the equation is
X111 + X121+ X131 + X211 +X221+ X231+ X311 + X321 + X331 = 1.
The above equation would imply that only one of these 9 variables or only one of these nine cells can take the value 1.
Similarly, constraints must be formulated for the digit “2”, the digit “3” and so on up to 9.
For integer programming problems in addition to the equations describing the constraints, there should also be integer constraints imposed on each variable so that eventually when the system of equations is solved one gets either a 0 or a 1 as a solution to the variable Xijk .
The geometric equivalent of a linear programming problem with an objective function and some constraints is nothing but a dimensional polyhedron where n represents the number of constraints in the problem. Usually the optimal solution will be found on the vertices of the polytope, also the rules of some methods such as SIMPLEX will require the polytope to be convex so that one can traverse from vertex to vertex along the edges and find the optimal solution.
Imposing integer constraints in addition would mean that the optimal solution will not be on the vertices of the polytope because a solution found on the vertex may not be an integer. So, after taking into account that the optimal solution must be 0 or 1, this will mean that geometrically the solution will be somewhere inside the feasible region of the convex polytope and on one of the many lines coming from l ‘hyperplane equivalent to Xi jk having integer values.
Note that the solution above used 729 decision variables and 81 row constraints. 81 column constraints and 729 small square constraints totaling 901 constraints. There can be many objective functions, but one can formulate an objective function by finding the min of (sum of all 729 variables). One can reduce the number of constraints by finding some redundancy.
These equations above cannot be solved using programming languages such as Visual Basic, Pascal or C. Integer programming problems can be solved using optimization software such as the CPLEX optimizer, the Excel add-in for solving linear programming problems, Lingo, etc.
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