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Approximation Algorithms for NP-Hard Problems Dorit Hochbaum
Publisher: Course Technology

Approximation algorithms for the traveling salesman problem 443. My algorithms professor used to tell his students (including me) this story to motivate studying NP-complete problems and reductions. Many of the striking advances in theoretical computer science over the past two decades concern approximation algorithms, which compute provably near-optimal solutions to NP-hard optimization 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. As we know, NP-hard problems are nightmare for the computers. It further motivates the study of approximation algorithms and other techniques to cope with NP-Completeness. 12.3 approximation algorithms for np-hard problems 441. Here is an example to give a feeling. 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 . (So to solve an instance of the Hitting Set Problem, it suffices to solve the instance of your problem with. The Hitting Set problem is NP-hard [Karp' 72]. Approximation algorithms for the knapsack problem 453. The story goes something like this: say you're working as a software developer and your boss gives you this project so I give up,” you need to show your boss that it's NP-Hard and this motivates the studying of reductions. For these problems, approximation algorithms are good choices. Think too hard about approximation algorithms. They show roughly “if you have a problem for which each and every alpha approximation to the optimum is goodish, then here is a 1-line algorithm that will solve that problem perfectly.” Everyone is familiar with the notion of reduction in complexity theory … essentially a certificate that your problem instance might (in the worst case) be encoding an NP-complete problem.