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Nicosia, Pacifici and Pferschy study the price of fairness, that is, the ratio between the maximum sum of utilities, and the maximum sum of utilities in a fair solution: For shared items: the price-of-fairness of max-min fairness is unbounded. For example, suppose there are four items with values 1, e, e, e, for some small e>0. The maximum sum ...
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]
Conversely, given a solution to the SubsetSumZero instance, it must contain the −T (since all integers in S are positive), so to get a sum of zero, it must also contain a subset of S with a sum of +T, which is a solution of the SubsetSumPositive instance. The input integers are positive, and T = sum(S)/2.
If the solution to any problem can be formulated recursively using the solution to its sub-problems, and if its sub-problems are overlapping, then one can easily memoize or store the solutions to the sub-problems in a table (often an array or hashtable in practice). Whenever we attempt to solve a new sub-problem, we first check the table to see ...
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.
A fractional set cover is an assignment of a fraction (a number in [0,1]) to each set in , such that for each element x in the universe, the sum of fractions of sets that contain x is at least 1. The goal is to find a fractional set cover in which the sum of fractions is as small as possible.
For example, for the array of values [−2, 1, −3, 4, −1, 2, 1, −5, 4], the contiguous subarray with the largest sum is [4, −1, 2, 1], with sum 6. Some properties of this problem are: If the array contains all non-negative numbers, then the problem is trivial; a maximum subarray is the entire array.
LeetCode: LeetCode has over 2,300 questions covering many different programming concepts and offers weekly and bi-weekly contests. The programming tasks are offered in English and Chinese. Project Euler [18] Large collection of computational math problems (i.e. not directly related to programming but often requiring programming skills for ...