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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.
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]
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 ...
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 ...
For example, given a binary tree of infinite depth, a depth-first search will go down one side (by convention the left side) of the tree, never visiting the rest, and indeed an in-order or post-order traversal will never visit any nodes, as it has not reached a leaf (and in fact never will). By contrast, a breadth-first (level-order) traversal ...
For example, the parent of 6 = 110 2 is 4 = 100 2. Implicit node 0 is the root. Implicit node 0 is the root. Each level k {\displaystyle k} of the tree contains nodes with indices corresponding to sums of k {\displaystyle k} distinct powers of 2 (with k = 0 {\displaystyle k=0} representing an empty sum 0).
A network formulation of the project selection problem with the optimal solution. In the project selection problem, there are n projects and m machines. Each project p i yields revenue r(p i) and each machine q j costs c(q j) to purchase. We want to select a subset of the project, and purchase a subset of the machines, to maximize the total ...