(Pre)solving Killer Sudokus

Killer Sudoku [1] is a variant of Sudoku where an additional constraint is that cells belonging to a cage (the colored regions in the picture) add up to a given value.

Killer Sudoku (from [1])

As in standard Sudokus, we need to fill each cell with a number \(1,\dots,9\), and values in each row, column and \((3 \times 3)\) box must be unique. In this puzzle we don't see that some cells already have given values: we need to fill-in all the cells. The solution for the above puzzle is:

Solution (from [1])

Using the standard decision variables for Sudoku puzzles: \[x_{i,j,k} = \begin{cases} 1 & \text{if cell $(i,j)$ has value $k$} \\ 0 & \text{otherwise}\end{cases}\] we can solve this easily with a MIP model. In the model below we use two data structures:

• a mapping \(\mathit{amap}(a,i,j)\) between area \(a\) (an area is a row, column or box) and cells \((i,j)\): \(\mathit{amap}(a,i,j)\) is \(\mathit{true}\) if cell \((i,j)\) belongs to area \(a\), 
• a similar mapping \(\mathit{cmap}(c,i,j)\) between a cage \(c\) and cell numbers \((i,j)\).

With this we can form the compact model:

Killer Sudoku model

\[\begin{align} & \sum_k \color{DarkRed} x_{i,j,k} = 1 && \forall i,j && \text{One value per cell}\\ & \sum_{i,j| \color{DarkBlue}{\mathit{amap}}(a,i,j)} \color{DarkRed} x_{i,j,k} = 1 && \forall a,k && \text{Unique values in each area} \\ & \sum_{i,j,k| \color{DarkBlue}{\mathit{cmap}}(c,i,j)} \color{DarkBlue} k \cdot \color{DarkRed} x_{i,j,k} = \color{DarkBlue} {\mathit{CageSum}}_c && \forall c && \text{Cell values add up to cage sum} \\ & \color{DarkRed} x_{i,j,k} \in \{0,1\} \\ \end{align}\]

There is no objective in this model: we are just looking for a feasible integer solution.

Implementation

The GAMS model can look like:

set   a     'area'/i1*i9,j1*j9,b1*b9/   i(a)  'rows'/i1*i9/   j(a)  'columns'/j1*j9/   b(a)  'boxes'/b1*b9/   amap(a,i,j) 'mapping area<->cells'->   k     'values'/1*9/ ; * * populate amap * rows, columns, and boxes a.k.a nonets * amap(i,i,j) = yes; amap(j,i,j) = yes; amap(b,i,j) = ord(b) = 3*ceil(ord(i)/3) + ceil(ord(j)/3) - 3; display amap; * * cages * sets   c 'cages'/cage1*cage29/   cmap(c,i,j)  'mapping cage<->cells'-> ; table cageno(i,j) 'cage numbering'       j1 j2 j3 j4 j5 j6 j7 j8 j9   i1   1  1  2  2  2  3  4  5  6   i2   7  7  8  8  3  3  4  5  6   i3   7  7  9  9  3 10 11 11  6   i4  12 13 13  9 14 10 11 15  6   i5  12 16 16 17 14 10 15 15 18   i6  19 16 20 17 14 21 22 22 18   i7  19 20 20 17 23 21 21 24 24   i8  19 25 26 23 23 27 27 24 24   i9  19 25 26 23 28 28 28 29 29 ; cmap(c,i,j) = ord(c)=cageno(i,j); parameter cagesum(c) /   cage1   3,  cage11  20,  cage21  20   cage2  15,  cage12   6,  cage22   6   cage3  22,  cage13  14,  cage23  10   cage4   4,  cage14  17,  cage24  14   cage5  16,  cage15  17,  cage25   8   cage6  15,  cage16  13,  cage26  16   cage7  25,  cage17  20,  cage27  15   cage8  17,  cage18  12,  cage28  13   cage9   9,  cage19  27,  cage29  17   cage10  8,  cage20   6 /; binaryvariable x(i,j,k); variable z 'obj'; equation    cellvalue(i,j) 'one value per cell'    unique(a,k)    'unique values in each area'    cagesums(c)    'summation of cage cells'    obj            'dummy objective' ; cellvalue(i,j).. sum(k, x(i,j,k)) =e= 1; unique(a,k)..    sum(amap(a,i,j),x(i,j,k)) =e= 1; cagesums(c)..    sum((cmap(c,i,j),k), k.val*x(i,j,k)) =e= cagesum(c); obj..            z =e= 0; model killer /all/; solve killer using mip minimizing z; parameter v(i,j) 'values'; v(i,j) = sum(k, k.val*x.l(i,j,k)); option v:0; display v;

For more information on how I populated the set amap see [2]. The results are:



Sudokus are typically solved in the presolver: the MIP solver does not need to do any Simplex iterations or Branch-and-Bound nodes. We see the same thing here. The Cplex solver log shows:

MIP Presolve eliminated 347 rows and 724 columns. MIP Presolve modified 885 coefficients. Aggregator did 6 substitutions. All rows and columns eliminated. Presolve time = 0.03 sec. (5.05 ticks) Proven optimal solution. MIP Solution:            0.000000    (0 iterations, 0 nodes)

CBC is also able to solve this without much sweat:

Calling CBC main solution routine... No integer variables out of 560 objects (560 integer) have costs branch on satisfied N create fake objective Y random cost Y Integer solution of 0 found by feasibility pump after 0 iterations and 0 nodes (0.51 seconds) Search completed - best objective 0, took 0 iterations and 0 nodes (0.54 seconds)

I am not sure why the feasibility pump was invoked here. I suspect CBC's presolver did not remove all variables and equations.

Multiple solutions?

We can easily prove this solution is unique, by adding a cut that forbids this solution, and resolving. I added to the model:

   equation cut;
   cut.. sum((i,j,k),(2*x.l(i,j,k)-1)*x(i,j,k)) =l= sum((i,j,k),x.l(i,j,k))-1;

Indeed this solution is unique: adding the cut made the model infeasible.

Unique values in a cage

A different approach would be to add the cages to the amap set (and the set a) and then rewrite uniqueness constraint as:


I prefer the first approach, as this is a stronger formulation (it certainly makes life easier for the presolver: with the second approach the model was not completely eliminated by the presolver).

For our sample problem this constraint cunique does not change the solution. This is good news as we already established the solution was unique. Adding this constraint confirmed our solution already obeyed this rule.

References