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The subset sum problem (SSP) is a decision problem in computer science. ... If the state is found, then by backtracking we can find a subset with a sum of exactly T.
The multiple subset sum problem is an optimization problem in computer science and operations research. It is a generalization of the subset sum problem . The input to the problem is a multiset S {\displaystyle S} of n integers and a positive integer m representing the number of subsets.
The subset sum problem is a special case of the decision and 0-1 problems where each kind of item, the weight equals the value: =. In the field of cryptography, the term knapsack problem is often used to refer specifically to the subset sum problem. The subset sum problem is one of Karp's 21 NP-complete problems. [2]
For this class of problems, the instance data P would be the integers m and n, and the predicate F. In a typical backtracking solution to this problem, one could define a partial candidate as a list of integers c = (c[1], c[2], …, c[k]), for any k between 0 and n, that are to be assigned to the first k variables x[1], x[2], …, x[k]. The ...
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. For this reason, many special cases and generalizations have been examined.
[1]: sec.5 The problem is parametrized by a positive integer k, and called k-way number partitioning. [2] The input to the problem is a multiset S of numbers (usually integers), whose sum is k*T. The associated decision problem is to decide whether S can be partitioned into k subsets such that the sum of each subset is exactly T.
The exact cover problem is NP-complete [3] and is one of Karp's 21 NP-complete problems. [4] It is NP-complete even when each subset in S contains exactly three elements; this restricted problem is known as exact cover by 3-sets, often abbreviated X3C. [3] Knuth's Algorithm X is an algorithm that finds all solutions to an exact cover problem.
The statement in the generalizations section, "[the subset sum problem] can actually be defined using any group", is not exactly accurate. For example Z_2 (integers modulo 2) under addition is a group, but finding the answer to the subset sum problem for a set of integers in Z_2 alone is trivial - the power set of Z_2