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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 (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 . [1] The problem is known to be NP-complete.
What is happening is that the problem becomes seemingly non- exponential when it is practical to count the entire solution space. There are two ways to count the solution space in the subset sum problem. One is to count the number of ways the variables can be combined. There are 2 N possible ways to combine the variables. However, with N = 10 ...
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]
The Stirling numbers of the second kind, S(n,k) count the number of partitions of a set of n elements into k non-empty subsets (indistinguishable boxes). An explicit formula for them can be obtained by applying the principle of inclusion–exclusion to a very closely related problem, namely, counting the number of partitions of an n -set into k ...
Maximum subarray problems arise in many fields, such as genomic sequence analysis and computer vision.. Genomic sequence analysis employs maximum subarray algorithms to identify important biological segments of protein sequences that have unusual properties, by assigning scores to points within the sequence that are positive when a motif to be recognized is present, and negative when it is not ...
The counting measure is a special case of a more general construction. With the notation as above, any function : [,) defines a measure on (,) via ():= (), where the possibly uncountable sum of real numbers is defined to be the supremum of the sums over all finite subsets, that is, := , | | < {}.
In 3-Partition the goal is to partition S into m = n/3 subsets, not just a fixed number of subsets, with equal sum. Partition is "easier" than 3-Partition: while 3-Partition is strongly NP-hard , Partition is only weakly NP-hard - it is hard only when the numbers are encoded in non-unary system, and have value exponential in n .