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The golden-section search is a technique for finding an extremum (minimum or maximum) of a function inside a specified interval. For a strictly unimodal function with an extremum inside the interval, it will find that extremum, while for an interval containing multiple extrema (possibly including the interval boundaries), it will converge to one of them.
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 ]
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.
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*.
For a partition of V into subsets U and W, an edge xy is balanced if either s(xy) = + and x and y are in the same subset, or s(xy) = – and x and y are different subsets. BSP aims at finding a partition with the maximum number b(G) of balanced edges in G. The Edwards-Erdős gives a lower bound on b(G) for every connected signed graph G.
The goal is to partition the items among the people in as fair way as possible. The natural generalization of the greedy number partitioning algorithm is the envy-graph algorithm . It guarantees that the allocation is envy-free up to at most one item (EF1).
The output is a partition of the items into m subsets, such that the number of items in each subset is at most k. Subject to this, it is required that the sums of sizes in the m subsets are as similar as possible. An example application is identical-machines scheduling where each machine has a job-queue that can hold at most k jobs. [1]
The following is a dynamic programming implementation (with Python 3) which uses a matrix to keep track of the optimal solutions to sub-problems, and returns the minimum number of coins, or "Infinity" if there is no way to make change with the coins given. A second matrix may be used to obtain the set of coins for the optimal solution.