tig-monorepopool/docs/challenges/knapsack.md

Knapsack Problem

The quadratic knapsack problem is one of the most popular variants of the single knapsack problem with applications in many optimization problems. The aim is to maximise the value of individual items placed in the knapsack while satisfying a weight constraint. However, pairs of items also have interaction values which may be negative or positive that are added to the total value within the knapsack.

Example

For our challenge, we use a version of the quadratic knapsack problem with configurable difficulty, where the following two parameters can be adjusted in order to vary the difficulty of the challenge:

• Parameter 1: num\textunderscore{ }items is the number of items from which you need to select a subset to put in the knapsack.
• Parameter 2: better\textunderscore{ }than\textunderscore{ }baseline \geq 1 is the factor by which a solution must be better than the baseline value [link TIG challenges for explanation of baseline value].

The larger the num\textunderscore{ }items, the more number of possible S_{knapsack}, making the challenge more difficult. Also, the higher better\textunderscore{ }than\textunderscore{ }baseline, the less likely a given S_{knapsack} will be a solution, making the challenge more difficult.

The weight w_i of each of the num\textunderscore{ }items is an integer, chosen independently, uniformly at random, and such that each of the item weights 1 <= w_i <= 50, for i=1,2,...,num\textunderscore{ }items. The individual values of the items v_i are selected by random from the range 50 <= v_i <= 100, and the interaction values of pairs of items V_{ij} are selected by random from the range -50 <= V_{ij} <= 50.

The total value of a knapsack is determined by summing up the individual values of items in the knapsack, as well as the interaction values of every pair of items (i,j) where i > j in the knapsack:

V_{knapsack} = \sum_{i \in knapsack}{v_i} + \sum_{(i,j)\in knapsack}{V_{ij}}

We impose a weight constraint W(S_{knapsack}) <= 0.5 \cdot W(S_{all}), where the knapsack can hold at most half the total weight of all items.

Consider an example of a challenge instance with num_items=4 and better_than_baseline = 1.10. Let the baseline value be 150:

weights = [26, 20, 39, 13]
individual_values = [63, 87, 52, 97]
interaction_values = [  0,  23, -18, -37
                       23,   0,  42, -28
                      -18,  42,   0,  32
                      -37, -28,  32,   0]
max_weight = 60
min_value = baseline*better_than_baseline = 165

The objective is to find a set of items where the total weight is at most 60 but has a total value of at least 165.

Now consider the following selection:

selected_items =  [0, 1, 3]

When evaluating this selection, we can confirm that the total weight is less than 60, and the total value is more than 165, thereby this selection of items is a solution:

• Total weight = 26 + 20 + 13 = 59
• Total value = 63 + 52 + 97 + 23 - 37 - 28 = 170

Our Challenge

In TIG, the baseline value is determined by a greedy algorithm that simply iterates through items sorted by potential value to weight ratio, adding them if knapsack is still below the weight constraint.