Search results
Results from the WOW.Com Content Network
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.
Despite its worst-case hardness, optimal solutions to very large instances of the problem can be produced with sophisticated algorithms. In addition, many approximation algorithms exist. For example, the first fit algorithm provides a fast but often non-optimal solution, involving placing each item into the first bin in which it will fit.
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 ...
For the set cover problem, Lovász proved that the integrality gap for an instance with n elements is H n, the nth harmonic number. One can turn the linear programming relaxation for this problem into an approximate solution of the original unrelaxed set cover instance via the technique of randomized rounding. [2]
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.
The SciPy scientific library, for instance, uses HiGHS as its LP solver [13] from release 1.6.0 [14] and the HiGHS MIP solver for discrete optimization from release 1.9.0. [15] As well as offering an interface to HiGHS, the JuMP modelling language for Julia [ 16 ] also describes the specific use of HiGHS in its user documentation. [ 17 ]
The "decimal" data type of the C# and Python programming languages, and the decimal formats of the IEEE 754-2008 standard, are designed to avoid the problems of binary floating-point representations when applied to human-entered exact decimal values, and make the arithmetic always behave as expected when numbers are printed in decimal.
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.