Knapsack with SPECIFIC AMOUNT of items from different groups


So this is a variation of the Knapsack Problem I came with the other day.

It is like a 0-1 Knapsack Problem where there are multiple groups and each item belongs to only one group. The goal is to maximize the profits subject to the constraints. In this case, a fixed number of items from each group have to be chosen for each group.

It is similar to the Multiple Choice Knapsack Problem, but in that case you only pick 1 of item of each group, in this one you want to pick x amount of items of each group

So, each item has: value, weight and group

Each group has an item count (Ex: if group A (or 0) has 2, the final solution needs to have 2 items of group A, no more no less)

And and you also have a maximum capacity (not related to the groups)

This translates into:

  • values[i] = The value of the ith element
  • weights[i] = The weigth of the ith element
  • groups[i] = The group of the ith element
  • C = Capacity
  • n = Amount of elements
  • m = Amount of groups
  • count[j] = Amount of items of group j

I’m attempting a Recursive solution first and then I will try a Dynamic approach.

Any solution would be appreciated (preferably Python, but anything will do 🙂 ).

Usefull links I found:

Asked By: Pablo Roldán



Full code also in:

Explanation after the code.

This code is for an example where you have a Fantasy League with a playersDB where each player has price (weight), points (value) and position (group); there is a list of possible_formations (group variations); and a budget (W) you can’t go over.

Full code:


      from group_knapsack import best_full_teams
      playersDB = [
          Player(name="Keylor Navas", price=16, points=7.5, position="GK"),
          Player(name="Laporte", price=23, points=7.2, position="DEF"),
          Player(name="Modric", price=22, points=7.3, position="MID"),
          Player(name="Messi", price=51, points=8.2, position="ATT"),
      possible_formations = [
          [3, 4, 3],
          [3, 5, 2],
          [4, 3, 3],
          [4, 4, 2],
          [4, 5, 1],
          [5, 3, 2],
          [5, 4, 1],
      budget = 300
      best_full_teams(playersDB, possible_formations, budget)

      import itertools
      from MCKP import knapsack_multichoice_onepick
      def best_full_teams(players_list, formations, budget):
          formation_score_players = []
          for formation in formations:
              players_points, players_prices, players_comb_indexes = players_preproc(
                  players_list, formation)
              score, comb_result_indexes = knapsack_multichoice_onepick(
                  players_prices, players_points, budget)
              result_indexes = []
              for comb_index in comb_result_indexes:
                  for winning_i in players_comb_indexes[comb_index[0]][comb_index[1]]:
              result_players = []
              for res_index in result_indexes:
              formation_score_players.append((formation, score, result_players))
              print("With formation " + str(formation) + ": " + str(score))
              for best_player in result_players:
          formation_score_players_by_score = sorted(formation_score_players,
                                                    key=lambda tup: tup[1],
          for final_formation_score in formation_score_players_by_score:
              print((final_formation_score[0], final_formation_score[1]))
          return formation_score_players
      def players_preproc(players_list, formation):
          max_gk = 1
          max_def = formation[0]
          max_mid = formation[1]
          max_att = formation[2]
          gk_values, gk_weights, gk_indexes = generate_group(players_list, "GK")
          gk_comb_values, gk_comb_weights, gk_comb_indexes = group_preproc(gk_values,
          def_values, def_weights, def_indexes = generate_group(players_list, "DEF")
          def_comb_values, def_comb_weights, def_comb_indexes = group_preproc(
              def_values, def_weights, def_indexes, max_def)
          mid_values, mid_weights, mid_indexes = generate_group(players_list, "MID")
          mid_comb_values, mid_comb_weights, mid_comb_indexes = group_preproc(
              mid_values, mid_weights, mid_indexes, max_mid)
          att_values, att_weights, att_indexes = generate_group(players_list, "ATT")
          att_comb_values, att_comb_weights, att_comb_indexes = group_preproc(
              att_values, att_weights, att_indexes, max_att)
          result_comb_values = [gk_comb_values, def_comb_values, mid_comb_values,
          result_comb_weights = [gk_comb_weights, def_comb_weights, mid_comb_weights,
          result_comb_indexes = [gk_comb_indexes, def_comb_indexes, mid_comb_indexes,
          return result_comb_values, result_comb_weights, result_comb_indexes
      def generate_group(full_list, group):
          group_values = []
          group_weights = []
          group_indexes = []
          for i, item in enumerate(full_list):
              if item.position == group:
          return group_values, group_weights, group_indexes
      def group_preproc(group_values, group_weights, initial_indexes, r):
          comb_values = list(itertools.combinations(group_values, r))
          comb_weights = list(itertools.combinations(group_weights, r))
          comb_indexes = list(itertools.combinations(initial_indexes, r))
          group_comb_values = []
          for value_combinations in comb_values:
              values_added = sum(list(value_combinations))
          group_comb_weights = []
          for weight_combinations in comb_weights:
              weights_added = sum(list(weight_combinations))
          return group_comb_values, group_comb_weights, comb_indexes

      import copy
      def knapsack_multichoice_onepick(weights, values, max_weight):
          if len(weights) == 0:
              return 0
          last_array = [-1 for _ in range(max_weight + 1)]
          last_path = [[] for _ in range(max_weight + 1)]
          for i in range(len(weights[0])):
              if weights[0][i] < max_weight:
                  if last_array[weights[0][i]] < values[0][i]:
                      last_array[weights[0][i]] = values[0][i]
                      last_path[weights[0][i]] = [(0, i)]
          for i in range(1, len(weights)):
              current_array = [-1 for _ in range(max_weight + 1)]
              current_path = [[] for _ in range(max_weight + 1)]
              for j in range(len(weights[i])):
                  for k in range(weights[i][j], max_weight + 1):
                      if last_array[k - weights[i][j]] > 0:
                          if current_array[k] < last_array[k - weights[i][j]] + 
                              current_array[k] = last_array[k - weights[i][j]] + 
                              current_path[k] = copy.deepcopy(
                                  last_path[k - weights[i][j]])
                              current_path[k].append((i, j))
              last_array = current_array
              last_path = current_path
          solution, index_path = get_onepick_solution(last_array, last_path)
          return solution, index_path
      def get_onepick_solution(scores, paths):
          scores_paths = list(zip(scores, paths))
          scores_paths_by_score = sorted(scores_paths, key=lambda tup: tup[0],
          return scores_paths_by_score[0][0], scores_paths_by_score[0][1]

      class Player:
          def __init__(
                  name: str,
                  price: float,
                  points: float,
                  position: str
     = name
              self.price = price
              self.points = points
              self.position = position
          def __str__(self):
              return f"({}, {self.price}, {self.points}, {self.position})"
          def position(self):
              return self._position
          def position(self, pos):
              if pos not in ["GK", "DEF", "MID", "ATT"]:
                  raise ValueError("Sorry, that's not a valid position")
              self._position = pos
          def get_group(self):
              if self.position == "GK":
                  group = 0
              elif self.position == "DEF":
                  group = 1
              elif self.position == "MID":
                  group = 2
                  group = 3
              return group


Okay,so I managed to find a solution translating what was here: Solving the Multiple Choice Knapsack Problem from C++ to Python. My solution also gives the path that got you to that solution. It uses Dynamic Programming and it’s very fast.

The input data, instead of having groups[i], has the weights and the values as a list of lists, where every list inside represent the values of each group:

  • weights[i] = [weights_group_0, weights_group_1, ...]
  • values[i] = [values_group_0, values_group_1, ...]


  • weights_group_i[j] = The weigth of the jth element of the ith group
  • values_group_i[j] = The value of the jth element of the ith group

Those would be the inputs of knapsack_multichoice_onepick. Here is an example:

# Example
values = [[6, 10], [12, 2], [2, 3]]
weights = [[1, 2], [6, 2], [3, 2]]
W = 7

print(knapsack_multichoice_onepick(weights, values, W))  # (15, [(0, 1), (1, 1), (2, 1)])

After that I followed @user3386109 ‘s suggestion and did the combinations with the indexes. The group preprocesing methods are players_preproc, generate_group and group_preproc.

Again, this code is for an example where you have a Fantasy League with a playersDB where each player has price (weight), points (value) and position (group); there is a list of possible_formations (group variations); and a budget (W) you can’t go over.

The best_full_teams method prints everything and uses all the previous ones.

Answered By: Pablo Roldán