049: Number Partitioning

# Copyright 2010 Hakan Kjellerstrand hakank@gmail.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.

"""

  Set partition problem in Google CP Solver.

  Problem formulation from
  http://www.koalog.com/resources/samples/PartitionProblem.java.html
  '''
   This is a partition problem.
   Given the set S = {1, 2, ..., n},
   it consists in finding two sets A and B such that:

     A U B = S,
     |A| = |B|,
     sum(A) = sum(B),
     sum_squares(A) = sum_squares(B)

  '''

  This model uses a binary matrix to represent the sets.


  Also, compare with other models which uses var sets:
  * MiniZinc: http://www.hakank.org/minizinc/set_partition.mzn
  * Gecode/R: http://www.hakank.org/gecode_r/set_partition.rb
  * Comet: http://hakank.org/comet/set_partition.co
  * Gecode: http://hakank.org/gecode/set_partition.cpp
  * ECLiPSe: http://hakank.org/eclipse/set_partition.ecl
  * SICStus: http://hakank.org/sicstus/set_partition.pl

  This model was created by Hakan Kjellerstrand (hakank@gmail.com)
  Also see my other Google CP Solver models: http://www.hakank.org/google_or_tools/
"""

import sys

from constraint_solver import pywrapcp


#
# Partition the sets (binary matrix representation).
#
def partition_sets(x, num_sets, n):
    solver = x.values()[0].solver()

    for i in range(num_sets):
        for j in range(num_sets):
            if i != j:
                b = solver.Sum([x[i,k]*x[j,k] for k in range(n)])
                solver.Add(b == 0)

    # ensure that all integers is in
    # (exactly) one partition
    b = [x[i,j] for i in range(num_sets) for j in range(n) ]
    solver.Add(solver.Sum(b) == n)


def main(n=16,num_sets=2):

    # Create the solver.
    solver = pywrapcp.Solver('Set partition')

    #
    # data
    #
    print "n:", n
    print "num_sets:", num_sets
    print

    # Check sizes
    assert n % num_sets == 0, "Equal sets is not possible."

    #
    # variables
    #

    # the set
    a = {}
    for i in range(num_sets):
        for j in range(n):
            a[i,j] = solver.IntVar(0, 1, 'a[%i,%i]' % (i,j))

    a_flat = [a[i,j] for i in range(num_sets) for j in range(n)]

    #
    # constraints
    #

    # partition set
    partition_sets(a, num_sets, n)

    for i in range(num_sets):
        for j in range(i, num_sets):

            # same cardinality
            solver.Add(solver.Sum([a[i,k] for k in range(n)])
                       ==
                       solver.Sum([a[j,k] for k in range(n)]))

            # same sum
            solver.Add(solver.Sum([k*a[i,k] for k in range(n)])
                       ==
                       solver.Sum([k*a[j,k] for k in range(n)]))


            # same sum squared
            solver.Add(solver.Sum([(k*a[i,k])*(k*a[i,k])
                                   for k in range(n)])
                       ==
                       solver.Sum([(k*a[j,k])*(k*a[j,k])
                                   for k in range(n)]))


    # symmetry breaking for num_sets == 2
    if num_sets == 2:
        solver.Add(a[0,0] == 1)


    #
    # search and result
    #
    db = solver.Phase(a_flat,
                 solver.INT_VAR_DEFAULT,
                 solver.INT_VALUE_DEFAULT)

    solver.NewSearch(db)


    num_solutions = 0
    while solver.NextSolution():
        a_val = {}
        for i in range(num_sets):
            for j in range(n):
                a_val[i,j] = a[i,j].Value()

        sq = sum([(j+1)*a_val[0,j] for j in range(n)])
        print "sums:", sq
        sq2 = sum([((j+1)*a_val[0,j])**2 for j in range(n)])
        print "sums squared:", sq2

        for i in range(num_sets):
            if sum([a_val[i,j] for j in range(n)]):
                print i+1, ":",
                for j in range(n):
                    if a_val[i,j] == 1:
                        print j+1,
                print

        print
        num_solutions += 1

    solver.EndSearch()

    print
    print "num_solutions:", num_solutions
    print "failures:", solver.Failures()
    print "branches:", solver.Branches()
    print "WallTime:", solver.WallTime()


n = 16
num_sets = 2
if __name__ == '__main__':
    if len(sys.argv) > 1:
        n = int(sys.argv[1])
    if len(sys.argv) > 2:
        num_sets = int(sys.argv[2])

    main(n, num_sets)