Zielsumme

Zielsumme is a term used to denote a specific numeric value that a calculation, model or decision aims to achieve. In mathematics and computer science, the term is frequently applied to problems in which one tries to combine numbers to reach a given target sum.

Formal problem and complexity: Given integers a1, …, an and a target T, the question is whether there exists a subset S of {1, …, n} such that the sum of a_i over i in S equals T. This is the classic subset sum problem. If the numbers are all nonnegative, the problem remains NP-complete in general, though a dynamic programming approach yields a pseudo-polynomial time algorithm with complexity O(n · W), where W is the target sum. Alternative methods, such as meet-in-the-middle, can improve performance for larger instances in practice.

Example: Consider the set {3, 34, 4, 12, 5, 2} and the target T = 9. There exists a subset (4, 5) that sums to 9, so the answer is yes.

Variants and domains: Variations include allowing signs on the numbers (the Target Sum problem), or restricting to positive numbers. The concept of Zielsumme also appears in practical contexts outside pure theory, such as budgeting, financial planning, or sales targets, where a specific sum is the goal to be achieved through allocation of resources or aggregation of values.

In sum, Zielsumme captures the idea of pursuing a defined numeric goal in problems that involve selecting or arranging values to meet a precise total.