Knapsack Problem

Overview

The area of Knapsack problems is one of the most active research areas of combinatorial optimization.

The problem is to maximise the value of items placed in a knapsack given the constraint that the total weight of items cannot exceed some limit.

Example

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

The larger the $num\_items$, the more number of possible $S_{knapsack}$, making the challenge more difficult. Also, the higher $better\_than\_baseline$, the less likely a given $S_{knapsack}$ will be a solution, making the challenge more difficult.

The weight $w_j$ of each of the $num\_items$ is an integer, chosen independently, uniformly at random, and such that each of the item weights $1 \leq w_j \leq 50$, for $j=1,2,...,num\_items$. The values of the items $v_j$ are similarly selected at random from the same distribution.

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

Consider an example of a challenge instance with $num\_items=6$ and $better\_than\_baseline = 1.09$. Let the baseline value be 100:

weights = [48, 20, 39, 13, 25, 16]
values = [24, 42, 27, 31, 44, 31]
max_weight = 80
min_value = 109

The objective is to find a set of items where the total weight is at most 80 but has a total value of at least 109. Now consider the following selection:

selected_items = [1, 3, 4, 5]

When evaluating this selection, we can confirm that the total weight is less than 80, and the total value is more than 109, thereby this selection of items is a solution:

Our Challenge

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

Applications

The Knapsack problems have a wide variety of practical applications. The use of knapsack in integer programming led to breakthroughs in several disciplines, including energy management and cellular network frequency planning.

Although originally studied in the context of logistics, Knapsack problems appear regularly in diverse areas of science and technology. For example, in gene expression data, there are usually thousands of genes, but only a subset of them are informative for a specific problem. The Knapsack Problem can be used to select a subset of genes (items) that maximizes the total information (value) without exceeding the limit of the number of genes that can be included in the analysis (weight limit).

Figure 1: Microarray clustering of differentially expressed genes in blood. Genes are clustered in rows, with red indicating high expression, yellow intermediate expression and blue low expression. The Knapsack problem is used to analyse gene expression clustering.