List of knapsack problems

The knapsack problem is one of the most studied problems in combinatorial optimization, with many real-life applications. For this reason, many special cases and generalizations have been examined.^[1][2]

Common to all versions are a set of n items, with each item ${\displaystyle 1\leq j\leq n}$ having an associated profit p_j and weight w_j. The binary decision variable x_j is used to select the item. The objective is to pick some of the items, with maximal total profit, while obeying that the maximum total weight of the chosen items must not exceed W. Generally, these coefficients are scaled to become integers, and they are almost always assumed to be positive.

The knapsack problem in its most basic form:

maximize ${\displaystyle \sum _{j=1}^{n}p_{j}x_{j}}$
subject to	${\displaystyle \sum _{j=1}^{n}w_{j}x_{j}\leq W,}$
	${\displaystyle x_{j}\in \{0,1\}}$	${\displaystyle \forall j\in \{1,\ldots ,n\}}$