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

The past few years have seen a flurry of results, including surprises such as a subexponential-time approximation algorithm, as well as algorithms for all “natural” families of instances we can think of. Today is for its application to the field of hardness of approximation algorithms: It turns out that the PCP theorem is equivalent to saying that there are problems where computing even an approximate solution is NP-hard. Thus unless P = NP, there are no efficient algorithms to find optimal solutions to such problems. Approximation algorithm: identifies approximate solutions to problems (mostly often NP-complete and NP-hard problems) to a certain bound. He helped create new approximation algorithms for fundamental optimization problems such as the Sparsest Cuts problem and the Euclidean Travelling Salesman problem, and contributed to the development of semi-definite programming as a practical algorithmic tool. 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. 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. An infinitesimal advance in the traveling salesman problem breathes new life into the search for improved approximate solutions. It is known that the decisional subset-sum is NP-complete (I believe this result is essentially due to Karp). It further motivates the study of approximation algorithms and other techniques to cope with NP-Completeness. Have you ever wondered if a specific NP-hard problem has an approximation algorithm or not? If yes, you may like to visit this site: A Compendium of NP optimization problems. Problem classes P, NP, NP-hard and NP-complete, deterministic and non deterministic polynomial time algorithms., Approximation algorithms for some NP-complete problems. The study of approximation algorithms for NP-hard problems has blossomed into a rich field, especially as a result of intense work over the last two decades. Approximation Algorithm vs Heuristic. My algorithms professor used to tell his students (including me) this story to motivate studying NP-complete problems and reductions. Khot's Unique Games Conjecture (UGC) —which asserts the NP-hardness of approximating a very simple constraint satisfaction problem— has assumed a central role in the effort to understand the optimal approximation ratios achievable for various NP-hard problems. The computer scientist Richard Karp, of the University of California at Berkeley, showed that the traveling salesman problem is “NP-hard,” which means that it has no efficient algorithm (unless a famous conjecture called P=NP is true — but the majority of computer scientists now suspect that it is false). Yet most such problems are NP-hard. Sanjeev Arora is one of the architects of the Probabilistically Checkable Proofs (PCP) theorem, which revolutionized our understanding of complexity and the approximability of NP-hard problems.