In dirkschumacher/ompr: Model and Solve Mixed Integer Linear Programs

Solve Sudokus using MILP

In this vignettes we will solve Sudoku puzzles using MILP. Sudoku in its most popular form is a constraint satisfaction problem and by setting the objective function to $0$ you transform the optimization problem into a pure constraint satistication problem. In this document we will consider Sudokus in a 9x9 grid with 3x3 sub-matrices.

Of course you can formulate an objective function as well that directs the solver towards solutions maximizing a certain linear function.

The model

The idea is to introduce a binary variable $x$ with three indexes $i, j, k$ that is $1$ if and only if the number $k$ is in cell $i, j$.

library(ompr)
library(dplyr)
n <- 9
model <- MIPModel() %>%

  # The number k stored in position i,j
  add_variable(x[i, j, k], i = 1:n, j = 1:n, k = 1:9, type = "binary") %>%

  # no objective
  set_objective(0) %>%

  # only one number can be assigned per cell
  add_constraint(sum_over(x[i, j, k], k = 1:9) == 1, i = 1:n, j = 1:n) %>%

  # each number is exactly once in a row
  add_constraint(sum_over(x[i, j, k], j = 1:n) == 1, i = 1:n, k = 1:9) %>%

  # each number is exactly once in a column
  add_constraint(sum_over(x[i, j, k], i = 1:n) == 1, j = 1:n, k = 1:9) %>% 

  # each 3x3 square must have all numbers
  add_constraint(sum_over(x[i, j, k], i = 1:3 + sx, j = 1:3 + sy) == 1, 
                 sx = seq(0, n - 3, 3), sy = seq(0, n - 3, 3), k = 1:9)
model

Solve the model

We will use glpk to solve the above model. Note that we haven't fixed any numbers to specific values. That means that the solver will find a valid sudoku without any prior hints.

library(ompr.roi)
library(ROI.plugin.glpk)
result <- solve_model(model, with_ROI(solver = "glpk", verbose = TRUE))

# the following dplyr statement plots a 9x9 matrix
result %>% 
  get_solution(x[i,j,k]) %>%
  filter(value > 0) %>%  
  select(i, j, k) %>% 
  tidyr::spread(j, k) %>% 
  select(-i)

If you want to solve a concrete sudoku you can fix certain cells to specific values. For example here we solve a sudoku that has the sequence from 1 to 9 in the first 3x3 matrix fixed.

model_fixed <- model %>% 
  add_constraint(x[1, 1, 1] == 1) %>% 
  add_constraint(x[1, 2, 2] == 1) %>% 
  add_constraint(x[1, 3, 3] == 1) %>% 
  add_constraint(x[2, 1, 4] == 1) %>% 
  add_constraint(x[2, 2, 5] == 1) %>% 
  add_constraint(x[2, 3, 6] == 1) %>% 
  add_constraint(x[3, 1, 7] == 1) %>% 
  add_constraint(x[3, 2, 8] == 1) %>% 
  add_constraint(x[3, 3, 9] == 1)
result <- solve_model(model_fixed, with_ROI(solver = "glpk", verbose = TRUE))

result %>% 
  get_solution(x[i,j,k]) %>%
  filter(value > 0) %>%  
  select(i, j, k) %>% 
  tidyr::spread(j, k) %>% 
  select(-i)