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Knapsack problems appear in real-world decision-making processes in a wide variety of fields, such as finding the least wasteful way to cut raw materials, [3] selection of investments and portfolios, [4] selection of assets for asset-backed securitization, [5] and generating keys for the Merkle–Hellman [6] and other knapsack cryptosystems.
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 having an associated profit p j and weight w j.
When the number of bins is restricted to 1 and each item is characterized by both a volume and a value, the problem of maximizing the value of items that can fit in the bin is known as the knapsack problem. A variant of bin packing that occurs in practice is when items can share space when packed into a bin.
The variant where variables are required to be 0 or 1, called zero-one linear programming, and several other variants are also NP-complete [2] [3]: MP1 Some problems related to Job-shop scheduling; Knapsack problem, quadratic knapsack problem, and several variants [2] [3]: MP9 Some problems related to Multiprocessor scheduling
The following is a dynamic programming implementation (with Python 3) which uses a matrix to keep track of the optimal solutions to sub-problems, and returns the minimum number of coins, or "Infinity" if there is no way to make change with the coins given. A second matrix may be used to obtain the set of coins for the optimal solution.
A minimum spanning tree of a weighted planar graph.Finding a minimum spanning tree is a common problem involving combinatorial optimization. Combinatorial optimization is a subfield of mathematical optimization that consists of finding an optimal object from a finite set of objects, [1] where the set of feasible solutions is discrete or can be reduced to a discrete set.
Given a solution to the SubsetSumPositive instance, adding the −T yields a solution to the SubsetSumZero instance. Conversely, given a solution to the SubsetSumZero instance, it must contain the − T (since all integers in S are positive), so to get a sum of zero, it must also contain a subset of S with a sum of + T , which is a solution of ...
Specific applications of search algorithms include: Problems in combinatorial optimization, such as: . The vehicle routing problem, a form of shortest path problem; The knapsack problem: Given a set of items, each with a weight and a value, determine the number of each item to include in a collection so that the total weight is less than or equal to a given limit and the total value is as ...