# Copyright 2021 Hakan Kjellerstrand hakank@bonetmail.com # # Licensed under the Apache License, Version 2.0 (the "License"); # you may not use this file except in compliance with the License. # You may obtain a copy of the License at # # http://www.apache.org/licenses/LICENSE-2.0 # # Unless required by applicable law or agreed to in writing, software # distributed under the License is distributed on an "AS IS" BASIS, # WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. # See the License for the specific language governing permissions and # limitations under the License. """ Ormat game in OR-tools CP-SAT Solver. From bit-player "The Ormat Game" http://bit-player.org/2010/the-ormat-game ''' I'm going to give you a square grid, with some of the cells colored and others possibly left blank. We'll call this a template. Perhaps the grid will be one of these 3x3 templates: [see pictures at the web page] You have a supply of transparent plastic overlays that match the grid in size and shape and that also bear patterns of black dots: [ibid.] Your task is to assemble a subset of the overlays and lay them on the template in such a way that dots cover all the colored squares but none of the blank squares. You are welcome to superimpose multiple dots on any colored square, but overall you want to use as few overlays as possible. To make things interesting, I'll suggest a wager. I'll pay you $3 for a correct covering of a 3x3 template, but you have to pay me $1 for each overlay you use. Is this a good bet? ''' This is a prototype which the following limitations: - the overlays is not generated dynamically for each n - it just shows which overlays that is used. It would be nice with a much more graphical output. - the questions asked by bit-player is not answered (it requires more analysis) That said, here is solutions of the three problems stated at the web page with minimum number of overlays. x is an indicator matrix which overlay to use overlay used is a set representation of the overlay used num_overlays is the number of overlays used This is a port of my old CP model ormat_game.py This model was created by Hakan Kjellerstrand (hakank@gmail.com) Also see my other OR-tools models: http://www.hakank.org/or_tools/ """ from __future__ import print_function from ortools.sat.python import cp_model as cp import math, sys # from cp_sat_utils import * import string, sys class SolutionCollector(cp.CpSolverSolutionCallback): """SolutionCollector for generating overlays""" def __init__(self, n, x, collection): cp.CpSolverSolutionCallback.__init__(self) self.__n = n self.__x = x self._collection = collection self.__solution_count = 0 def OnSolutionCallback(self): self.__solution_count += 1 n = self.__n ov = [[self.Value(self.__x[i][j]) for i in range(n)] for j in range(n)] self._collection.append(ov) def SolutionCount(self): return self.__solution_count # # Generate all the overlays for a specific size (n) # def get_overlays(n=3, debug=0): model = cp.CpModel() # # decision variables # x = [] for i in range(n): t = [] for j in range(n): t.append(model.NewIntVar(0,1, "x[%i][%i]"%(i,j))) x.append(t) # # constraints # for i in range(n): model.Add(sum([x[i][j] for j in range(n)]) == 1) model.Add(sum([x[j][i] for j in range(n)]) == 1) solver = cp.CpSolver() overlays = [] solution_collector = SolutionCollector(n, x,overlays) _status = solver.SearchForAllSolutions(model,solution_collector) # if status == cp.OPTIMAL: # True return overlays # # print a problem # def print_problem(problem, n): print("Problem:") for i in range(n): for j in range(n): print(problem[i][j], " ",end=" ") print('') print('') # # print a solution # def print_solution(x, overlays): f = len(x) n = len(overlays[0]) print("f:",f, " n: ", n) # print("x: ", x) print("num overlays:", sum(x)) for o in range(f): if x[o] == 1: print("Overlay", o) for i in range(n): for j in range(n): print(overlays[o][i][j], " ",end=" ") print('') print('') def main(problem, overlays, n, debug=0): model = cp.CpModel() # data f = len(overlays) # declare variables x = [model.NewIntVar(0, 1, "x[%i]" %i) for i in range(f)] num_overlays = model.NewIntVar(0, f, 'num_overlays') y = {} for i in range(n): for j in range(n): y[(i,j)] = model.NewIntVar(0,f,"y[%i,%i]"%(i,j)) # constraints for i in range(n): model.Add(sum([y[(i,j)] for j in range(n)]) > 0) model.Add(sum([y[(j,i)] for j in range(n)]) > 0) model.Add(sum(x) == num_overlays) for i in range(n): for j in range(n): model.Add(y[(i,j)] == sum([(x[o])*(overlays[o][i][j]) for o in range(f)])) if problem[i][j] == 1: model.Add(y[(i,j)] >= 1) if problem[i][j] == 0: model.Add(y[(i,j)] == 0) model.Minimize(num_overlays) # solution and search solver = cp.CpSolver() solver.parameters.search_branching = cp.PORTFOLIO_SEARCH solver.parameters.cp_model_presolve = False # solver.parameters.linearization_level = 0 # solver.parameters.cp_model_probing_level = 0 status = solver.Solve(model) the_solution = [] if status == cp.OPTIMAL: t = [solver.Value(x[i]) for i in range(f)] the_solution = t print("Num overlays: ", sum([solver.Value(x[i]) for i in range(f)])) print("\nOptimal solution:") print_solution(the_solution, overlays) print("Num overlays: ", sum(the_solution)) print() print("NumConflicts:", solver.NumConflicts()) print("NumBranches:", solver.NumBranches()) print("wall_time:", solver.WallTime()) print() # problem 1 problem1 = { "n": 3, "p": [ [1,0,0], [0,1,1], [0,1,1] ] } # # Problem grid 2 problem2 = { "n": 3, "p": [ [1,1,1], [1,1,1], [1,1,1] ] } # Problem grid 3 problem3 = { "n": 3, "p": [ [1,1,1], [1,1,1], [1,1,0] ] } # This rotation of the above works problem4 = { "n": 3, "p": [ [1,1,1], [1,1,1], [0,1,1] ] } # This is a _bad_ problem since all rows # and colutions must have at least one cell=1 problem5 = { "n": 3, "p": [ [0,0,0], [0,1,1], [0,1,1] ] } # # Problem grid 4 (n = 4) problem6 = { "n": 4, "p": [ [1,1,1,1], [1,1,1,1], [1,1,1,1], [1,1,0,0] ] } # variant problem7 = { "n": 4, "p": [ [1,1,1,1], [1,1,1,1], [1,1,1,1], [1,1,1,0] ] } # variant problem8 = { "n": 4, "p": [ [1,1,1,1], [1,1,1,1], [1,1,1,1], [1,1,1,1] ] } # Problem grid 5 (n = 5) # This is under the section "Out of bounds" problem9 = { "n": 5, "p": [ [1,1,1,1,1], [1,1,1,1,1], [1,1,1,1,1], [1,1,1,1,1], [1,1,0,0,0] ] } # Problem grid 6 (n = 6) # # This is under the section "Out of bounds"% problem10 = { "n": 6, "p": [ [1,1,1,1,1,1], [1,1,1,1,1,1], [1,1,1,1,1,1], [1,1,1,1,1,1], [1,1,1,1,1,1], [1,1,0,0,0,0] ] } # # Problem (n=7) # hard problem11 = { "n": 7, "p": [ [1,1,1,1,1,1,1], [1,1,1,1,1,1,1], [0,1,1,1,1,1,1], [0,0,1,1,1,1,1], [0,0,0,1,1,1,1], [0,0,0,0,1,1,1], [0,0,0,0,0,1,1] ] } # # Problem (n=7), "flags" # # quite hard # problem12 = { "n": 7, "p": [ [0,0,0,1,1,1,1], [0,0,0,1,1,1,1], [0,0,0,1,1,1,1], [1,1,1,1,1,1,1], [1,1,1,1,1,1,1], [1,1,1,1,1,1,1], [1,1,1,1,1,1,1] ] } # # Wrapper for solving a problem # def solve_problem(p, n): print_problem(p, n) overlays = get_overlays(n, debug) main(p, overlays, n, debug) # # Solve all problems # def solve_all(): ps = [problem1,problem2,problem3,problem4,problem5,problem6, problem7,problem8,problem9,problem10,problem11,problem12] for p in ps: solve_problem(p["p"],p["n"]) print() problem = problem12 if __name__ == '__main__': debug = 1 # solve_problem(problem["p"],problem["n"]) solve_all()