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For example, there is a different approximation algorithm for minimum vertex cover that solves a linear programming relaxation to find a vertex cover that is at most twice the value of the relaxation. Since the value of the relaxation is never larger than the size of the optimal vertex cover, this yields another 2-approximation algorithm.
In 1972, an approximation algorithm for real convex minimization was studied by Arkadi Nemirovski and David B. Yudin (Judin). As an algorithm for solving linear programming problems with rational data, the ellipsoid algorithm was studied by Leonid Khachiyan; Khachiyan's achievement was to prove the polynomial-time solvability of linear programs.
This gives a TSP tour which is at most 1.5 times the optimal. It was one of the first approximation algorithms, and was in part responsible for drawing attention to approximation algorithms as a practical approach to intractable problems. As a matter of fact, the term "algorithm" was not commonly extended to approximation algorithms until later ...
The greedy algorithm for maximum coverage chooses sets according to one rule: at each stage, choose a set which contains the largest number of uncovered elements. It can be shown that this algorithm achieves an approximation ratio of .
A fully polynomial-time approximation scheme (FPTAS) is an algorithm for finding approximate solutions to function problems, especially optimization problems.An FPTAS takes as input an instance of the problem and a parameter ε > 0.
The following is an example of a possible implementation of Newton's method in the Python (version 3.x) programming language for finding a root of a function f which has derivative f_prime. The initial guess will be x 0 = 1 and the function will be f ( x ) = x 2 − 2 so that f ′ ( x ) = 2 x .
Covering and packing LPs commonly arise as a linear programming relaxation of a combinatorial problem and are important in the study of approximation algorithms. [12] For example, the LP relaxations of the set packing problem, the independent set problem, and the matching problem are packing LPs.
Furthermore, there can be no approximation algorithm with absolute approximation ratio smaller than unless =. This can be proven by a reduction from the partition problem : [ 10 ] given an instance of Partition where the sum of all input numbers is 2 T {\displaystyle 2T} , construct an instance of bin-packing in which the bin size is T .