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In computer science, the largest differencing method is an algorithm for solving the partition problem and the multiway number partitioning. It is also called the Karmarkar–Karp algorithm after its inventors, Narendra Karmarkar and Richard M. Karp. [1] It is often abbreviated as LDM. [2] [3]
The simplest greedy partitioning algorithm is called list scheduling. It just processes the inputs in any order they arrive. It just processes the inputs in any order they arrive. It always returns a partition in which the largest sum is at most 2 − 1 k {\displaystyle 2-{\frac {1}{k}}} times the optimal (minimum) largest sum. [ 1 ]
For every partition of S # (d) with sums C i #, there is a partition of S with sums C i, where + # # +, and it can be found in time O(n). Given a desired approximation precision ε>0, let δ>0 be the constant corresponding to ε/3, whose existence is guaranteed by Condition F*.
The input to the algorithm is an undirected graph G = (V, E) with vertex set V, edge set E, and (optionally) numerical weights on the edges in E.The goal of the algorithm is to partition V into two disjoint subsets A and B of equal (or nearly equal) size, in a way that minimizes the sum T of the weights of the subset of edges that cross from A to B.
The graphical method was used by Paul Ehrenfest and Heike Kamerlingh Onnes—with symbol ε (quantum energy element) in place of a star and the symbol 0 in place of a bar—as a simple derivation of Max Planck's expression for the number of "complexions" for a system of "resonators" of a single frequency. [5] [6]
Denote the n objects to partition by the integers 1, 2, ..., n. Define the reduced Stirling numbers of the second kind, denoted (,), to be the number of ways to partition the integers 1, 2, ..., n into k nonempty subsets such that all elements in each subset have pairwise distance at least d.
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Equal-cardinality partition is a variant in which both parts should have an equal number of items, in addition to having an equal sum. This variant is NP-hard too. [5]: SP12 Proof. Given a standard Partition instance with some n numbers, construct an Equal-Cardinality-Partition instance by adding n zeros. Clearly, the new instance has an equal ...