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Approximation Algorithms for NP-Hard Problems by Dorit Hochbaum

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

Open Problems : Perhaps the most interesting open question is to obtain a constant factor approximation for treewidth. This problem addresses the issue of timing when deploying viral campaigns. See [BGHK'95] for interesting applications of treewidth Eg : Choleski factorization on sparse symmetric matrices. Unsurprisingly, submodular maximization tends to be NP-hard for most natural choices of constraints, so we look for approximation algorithms. There is an analogous notion of pathwidth which is also NP-complete. A simple factor-2 approximation just walks around the spanning tree and can be computed in O(n log n) time with simple algorithms! The Max-Cut problem is known to be NP-hard (if the widely believed {P\neq NP} conjecture is true this means that the problem cannot be solved in polynomial time). 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. Both these problems are NP-hard, which motivates our interest in their approximation. Approximating tree-width : Bodlaender et. SAT (boolean satisfiability, the "canonical" NP-hard problem) is a really tough nut to crack, whereas for example euclidean TSP (traveling salesman) is hard to solve optimally but has simple and fast algorithms that guarantee to solve it to within a constant factor of the optimum. Al ruled out absolute approximation algorithm, (unless P = NP) for treewidth and pathwidth.