Using MindOpt to Solve Sudoku Problems

1. Problem Description and Mathematical Programming

Sudoku is a popular logic and number-based game. Players need to deduce the numbers for all the remaining empty spaces on a 9x9 grid based on the known numbers on the grid. The goal is to ensure that each row, column, and 3x3 sub-grid contains all the numbers from 1 to 9 without repetition. An example of a Sudoku puzzle is shown below:

The solution of the Sudoku problem can also be modeled as a mathematical programming problem.

To do so, we introduce the following decision variables and constraints:

Binary decision variables, x_ijk ∈{0,1}, where the index is i∈{0,⋯,8}，j∈{0,⋯,8} and k∈{1,⋯,9}. If x_ijk = 1 is, it means the index of the cell in the board is (i,j), and the cell value is k. If there is x_ijk = 0, it means (i,j), and the value is not k.
We introduce the constraints below to satisfy that the numerical value in each cell of the Sudoku board is between 1 and 9:

We introduce the constraints below to meet the requirement that numbers 1-9 appear only once in each row:

We introduce the constraints below to meet the requirement that numbers 1-9 appear only once in each column:

We introduce the constraints below to meet the requirement that numbers 1-9 appear only once in each 3x3 sub-grid:

Given the set ，we introduce the constraint x_ijk = 1,∀(i,j,k)∈Ω.
The essence of this problem is to find feasible solutions for the Sudoku puzzle. Therefore, we can omit the objective function.

Based on the discussion above, we can establish the following mathematical programming model for solving Sudoku problems:

2. Modeling with MAPL

MindOpt Algebraic Programming Language (MindOpt APL or MAPL) is an efficient algebraic modeling language for optimization. It supports a variety of solvers.

2.1. Model and Solve

Method 1：Write All Codes in the Cell

When solving a Sudoku problem, MAPL can model it as shown below:

Among them, use fixed.txt to give the known numbers and positions for the puzzle in Figure 1.

The * fixed.txt sudoku.mapl file can be viewed in the MindOpt Studio Sudoku case.

clear model;  #Used when running multiple times to clear the model
option modelname model/sudoku;  #link the model file

#--------------------------
# sudoku.mapl
# Declaring sets
set I := {0 .. 8};
set J := {0 .. 8};
set K := {1 .. 9};
set M := {0,1,2};
set N := {0,1,2};
set S := {0,1,2};
set F := { read "./model/sudoku_fixed.txt" as "<1n, 2n, 3n>" }; # Read in the given numbers and positions

# Declaring Variables
var x[I * J * K] binary;

# Declaring constraints
subto Value_Range:
   forall {(i,j) in I * J } sum {k in K} x[i,j,k] == 1; 

subto Rows:
   forall {(i,k) in I * K } sum {j in J} x[i,j,k] == 1; 

subto Cols:
   forall {(j,k) in J * K } sum {i in I} x[i,j,k] == 1; 

subto Cells:
   forall {m in M,n in N,k in K}sum{i in S,j in S} x[3*m+i,3*n+j,k] == 1; 

subto Fixed:
   forall {(i,j,k) in F } x[i,j,k] == 1; 
   
#------------------------------

print "Check the model, such as F:";
print F;


print "==================Solve using MindOpt Solver==================";
option solver mindopt;     # select the MindOpt solver
solve;  

# Change the solver
#print "==================Solve using Cbc Solver==================";
#option solver cbc;     # select the Cbc solver
#solve;

Method 2：Load the .mapl File

Users can also create a blank document using any text editor and save modeling codes as a sudoku.mapl file.

Then, run the following code to proceed with the modeling. Here, we can also change the solver to COIN-OR Cbc.

clear model;  #Used when running multiple times to clear the model

#------------------------------

model model/sudoku.mapl;  # modeling

#------------------------------

print "Check the model, such as F:";
print F;

#print "==================Solve using MindOpt Solver==================";
#option solver mindopt;     # select the MindOpt solver
#solve;  

# Change the solver
print "==================Solve using Cbc Solver==================";
option solver cbc;     # select the Cbc solver
solve;

3. Results

The results of Method 1 and Method 2 are consistent:

Check the model, such as F:
Multi: |3|{<0,1,2>,<0,4,8>,<0,7,7>,<1,1,7>,<1,0,4>,<1,5,9>,<2,7,2>,<2,5,3>,<2,6,5>,<3,1,9>,<3,6,1>,<3,2,2>,<3,3,3>,<4,1,1>,<4,4,7>,<4,7,3>,<4,8,5>,<5,5,5>,<5,6,6>,<5,2,7>,<5,3,9>,<6,0,7>,<6,6,2>,<6,2,4>,<6,8,6>,<7,4,3>,<7,5,4>,<7,3,6>,<8,4,9>,<8,7,5>,<8,0,6>,<8,8,3>}
==================Solve using MindOpt Solver==================
Running mindoptampl
wantsol=1
MindOpt Version 0.25.1 (Build date: 20230816)
Copyright (c) 2020-2023 Alibaba Cloud.

Start license validation (current time : 24-AUG-2023 16:16:38).
License validation terminated. Time : 0.005s

Model summary.
 - Num. variables     : 729
 - Num. constraints   : 324
 - Num. nonzeros      : 2916
 - Num. integer vars. : 697
 - Bound range        : [1.0e+00,1.0e+00]
 - Objective range    : [0.0e+00,0.0e+00]

Branch-and-cut method started.
Original model: nrow = 324 ncol = 729 nnz = 2916
Tolerance: primal = 1e-06 int = 1e-06 mipgap = 0.0001 mipgapAbs = 1e-06
Limit: time = 1.79769313486232e+308 node = -1 stalling = -1 solution = -1
presolver terminated; took 1 ms
presolver terminated; took 5 ms
Branch-and-cut method terminated. Time : 0.015s


OPTIMAL; objective 0.00

Completed.

-------------------------------------------------
Check the model, such as F:
Multi: |3|{<0,1,2>,<0,4,8>,<0,7,7>,<1,1,7>,<1,0,4>,<1,5,9>,<2,7,2>,<2,5,3>,<2,6,5>,<3,1,9>,<3,6,1>,<3,2,2>,<3,3,3>,<4,1,1>,<4,4,7>,<4,7,3>,<4,8,5>,<5,5,5>,<5,6,6>,<5,2,7>,<5,3,9>,<6,0,7>,<6,6,2>,<6,2,4>,<6,8,6>,<7,4,3>,<7,5,4>,<7,3,6>,<8,4,9>,<8,7,5>,<8,0,6>,<8,8,3>}
==================Solve using Cbc Solver==================
Running cbc
CBC 2.10.5Completed.

We can use display; to print the solution. In this case, it displays a lot of content, and multiple runs may make the Notebook kernel busy and stop. If this happens, you can click the restart arrow circle button (located near the play button) above to restart the kernel.

We can also use print and def to print a beautiful 9x9 grid.

print "-----------------Display---------------";
#display;      # this will print too much, if you need run, uncomment it


print "-----------Location and value of the non-zero variables-----------";
#forall {<i,j,k> in I*J*K with x[i,j,k] >= 1}  print 'x[{},{}] = {}' % i,j,k;   # this will print too much, if you need run, uncomment it


print "-----------------The 9*9 grid---------------";

# print the 9x9 grid with `def`

# get the location and value set of non-zero variables
def nonzero(m,n) = {<i,j,k> in I*J*K with m == i and n == j and x[i,j,k] >= 1};

# get the value of k
def get_num(m,n) = sum {<i, j, k> in nonzero(m,n)} k;

print "\\ | 0 1 2 | 3 4 5 | 6 7 8 ";
forall {i in 0..8} 
    if i mod 3 != 0 then print "{} | {} {} {} | {} {} {} | {} {} {}" % i, get_num(i,0), get_num(i,1), get_num(i,2), get_num(i,3), get_num(i,4), get_num(i,5), get_num(i,6), get_num(i,7), get_num(i,8)
    else print "--+-------+-------+-------
{} | {} {} {} | {} {} {} | {} {} {}" % i, get_num(i,0), get_num(i,1), get_num(i,2), get_num(i,3), get_num(i,4), get_num(i,5), get_num(i,6), get_num(i,7), get_num(i,8);

The running result is listed below:

-----------------Display---------------
-----------Location and value of the non-zero variables-----------
-----------------The 9*9 grid---------------
\ | 0 1 2 | 3 4 5 | 6 7 8 
--+-------+-------+-------
0 | 9 2 5 | 1 8 6 | 3 7 4
1 | 4 7 3 | 5 2 9 | 8 6 1
2 | 1 6 8 | 7 4 3 | 5 2 9
--+-------+-------+-------
3 | 5 9 2 | 3 6 8 | 1 4 7
4 | 8 1 6 | 4 7 2 | 9 3 5
5 | 3 4 7 | 9 1 5 | 6 8 2
--+-------+-------+-------
6 | 7 3 4 | 8 5 1 | 2 9 6
7 | 2 5 9 | 6 3 4 | 7 1 8
8 | 6 8 1 | 2 9 7 | 4 5 3

The answer to this Sudoku puzzle is shown below:

Community

Using MindOpt to Solve Sudoku Problems

1. Problem Description and Mathematical Programming

2. Modeling with MAPL

2.1. Model and Solve

Method 1：Write All Codes in the Cell

Method 2：Load the .mapl File

3. Results

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