Dynamic programming | Weakly NP-complete problems

Subset sum problem

The subset sum problem (SSP) is a decision problem in computer science. In its most general formulation, there is a multiset of integers and a target-sum , and the question is to decide whether any subset of the integers sum to precisely . The problem is known to be NP. Moreover, some restricted variants of it are NP-complete too, for example: * The variant in which all inputs are positive. * The variant in which inputs may be positive or negative, and . For example, given the set , the answer is yes because the subset sums to zero. * The variant in which all inputs are positive, and the target sum is exactly half the sum of all inputs, i.e., . This special case of SSP is known as the partition problem. SSP can also be regarded as an optimization problem: find a subset whose sum is at most T, and subject to that, as close as possible to T. It is NP-hard, but there are several algorithms that can solve it reasonably quickly in practice. SSP is a special case of the knapsack problem and of the multiple subset sum problem. (Wikipedia).

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Related pages

Approximation algorithm | Depth-first search | David S. Johnson | Knapsack problem | Multiset | Pseudo-polynomial time | Dynamic programming | Merkle–Hellman knapsack cryptosystem | Exponential time | 3SUM | 3-dimensional matching | Optimization problem | Decision problem | Partition problem | NP-completeness | Breadth-first search | Multiple subset sum