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 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(rmpk) library(dplyr) library(ROI.plugin.glpk) n <- 9 model <- MIPModel(ROI_solver("glpk")) # The number k stored in position i,j x <- model$add_variable(i = 1:n, j = 1:n, k = 1:9, type = "binary") # no objective model$set_objective(0) # only one number can be assigned per cell model$add_constraint(sum_expr(x[i, j, k], k = 1:9) == 1, i = 1:n, j = 1:n) # each number is exactly once in a row model$add_constraint(sum_expr(x[i, j, k], j = 1:n) == 1, i = 1:n, k = 1:9) # each number is exactly once in a column model$add_constraint(sum_expr(x[i, j, k], i = 1:n) == 1, j = 1:n, k = 1:9) # each 3x3 square must have all numbers model$add_constraint(sum_expr(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

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.

model$optimize() # the following dplyr statement plots a 9x9 matrix model$get_variable_value(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$add_constraint(x[1, 1, 1] == 1) model$add_constraint(x[1, 2, 2] == 1) model$add_constraint(x[1, 3, 3] == 1) model$add_constraint(x[2, 1, 4] == 1) model$add_constraint(x[2, 2, 5] == 1) model$add_constraint(x[2, 3, 6] == 1) model$add_constraint(x[3, 1, 7] == 1) model$add_constraint(x[3, 2, 8] == 1) model$add_constraint(x[3, 3, 9] == 1) model$optimize() model$get_variable_value(x[i,j,k]) %>% filter(value > 0) %>% select(i, j, k) %>% tidyr::spread(j, k) %>% select(-i)

Do you have any questions, ideas, comments? Or did you find a mistake? Let's discuss on Github.

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