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2-a. Let be an instance constructed from by grouping items and rounding the size of items in each group to the highest item in the group. 3-a. Construct the configuration linear program for , without the integrality constraints. 4. Compute a (fractional) solution x for the relaxed linear program. 3-b.
The problem of fractional knapsack with penalties was introduced by Malaguti, Monaci, Paronuzzi and Pferschy. [44] They developed an FPTAS and a dynamic program for the problem, and they showed an extensive computational study comparing the performance of their models.
In mathematical optimization, fractional programming is a generalization of linear-fractional programming. The objective function in a fractional program is a ratio of two functions that are in general nonlinear.
In mathematical optimization, linear-fractional programming (LFP) is a generalization of linear programming (LP). Whereas the objective function in a linear program is a linear function, the objective function in a linear-fractional program is a ratio of two linear functions. A linear program can be regarded as a special case of a linear ...
The goal in finding these "hard" instances is for their use in public-key cryptography systems, such as the Merkle–Hellman knapsack cryptosystem. More generally, better understanding of the structure of the space of instances of an optimization problem helps to advance the study of the particular problem and can improve algorithm selection.
In applied statistics, fractional models are, to some extent, related to binary response models. However, instead of estimating the probability of being in one bin of a dichotomous variable , the fractional model typically deals with variables that take on all possible values in the unit interval .
In the fractional set cover problem, it is allowed to select fractions of sets, rather than entire sets. A fractional set cover is an assignment of a fraction (a number in [0,1]) to each set in S {\displaystyle {\mathcal {S}}} , such that for each element x in the universe, the sum of fractions of sets that contain x is at least 1.
In computer science, fractional cascading is a technique to speed up a sequence of binary searches for the same value in a sequence of related data structures. The first binary search in the sequence takes a logarithmic amount of time, as is standard for binary searches, but successive searches in the sequence are faster.