Search results
Results From The WOW.Com Content Network
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.
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.
In additive combinatorics, the sumset (also called the Minkowski sum) of two subsets and of an abelian group (written additively) is defined to be the set of all sums of an element from with an element from .
where A is a finite nonempty subset of a field F, and p(F) is a prime p if F is of characteristic p, and p(F) = ∞ if F is of characteristic 0. Various extensions of this result were given by Noga Alon , M. B. Nathanson and I. Ruzsa in 1996, [ 11 ] Q. H. Hou and Zhi-Wei Sun in 2002, [ 12 ] and G. Karolyi in 2004.
Every set can be the basis of a free abelian group, which is unique up to group isomorphisms. The free abelian group for a given basis set can be constructed in several different but equivalent ways: as a direct sum of copies of the integers, as a family of integer-valued functions, as a signed multiset, or by a presentation of a group.
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. There is also an optimization problem: find a partition of S into k subsets, such that the k sums are "as near as ...
For example, the set of odd numbers is a sum-free subset of the integers, and the set {N + 1, ..., 2N } forms a large sum-free subset of the set {1, ..., 2N }. Fermat's Last Theorem is the statement that, for a given integer n > 2, the set of all nonzero n th powers of the integers is a sum-free set.
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, := , | | < {}.