Approximation Algorithms for NP-Hard Problems by Dorit Hochbaum

Approximation Algorithms for NP-Hard Problems



Download Approximation Algorithms for NP-Hard Problems




Approximation Algorithms for NP-Hard Problems Dorit Hochbaum ebook
Page: 620
ISBN: 0534949681, 9780534949686
Format: djvu
Publisher: Course Technology


Unsurprisingly, submodular maximization tends to be NP-hard for most natural choices of constraints, so we look for approximation algorithms. Yet most such problems are NP-hard. The problem is NP hard for all non-trivial values of k and d and there are various approximation algorithms for solving this problem. The Hitting Set problem is NP-hard [Karp' 72]. Because all of these problems are NP-hard, the primary goal of this research is to produce polynomial-time, approximation algorithms for each problem considered. We show both problems to be NP-hard and prove limits on approximation for both problems. I still maintain that someone could make a good movie with the premise "random guy finds easy algorithm to solve NP-complete problems now what?" in the vein of Primer (random guys . Linear programming has been a successful tool in combinatorial optimization to achieve polynomial time algorithms for problems in P and also to achieve good approximation algorithms for problems which are NP-hard. The theory of NP-completeness suggests that some problems in CS are inherently hard—that is, there is likely no possible algorithm that can efficiently solve them. (So to solve an instance of the Hitting Set Problem, it suffices to solve the instance of your problem with. Thus unless P =NP, there are no efficient algorithms to find optimal solutions to such problems. There are already arbitrarily good polynomial-time approximation algorithms for many NPO-complete problems like TSP, but TSP is actually APX-complete too, meaning you cannot even approximate answers beyond a certain factor unless P=NP. Many combinatorial optimization problems can be expressed as the minimization or maximization of a submodular function, including min- and max-cut, coverage problems, and welfare maximization in algorithmic game theory. We present integer programs for both GOPs that provide exact solutions.

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