Hard P
Hard P
NP-hardness Euler diagram for P, NP, NP-complete, and NP-hard set of problems. The left side is valid under the assumption that P≠NP, while the right side is valid under the assumption that P=NP (except that the empty language and its complement are never NP-complete).
NP-hard class An NP-hard problem is at least as hard as the hardest problem in NP and it is a class of problems such that every problem in NP reduces to NP-hard.
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3.4. NP-Hard Algorithms Our last set of problems contains the hardest, most complex problems in computer science. They are not only hard to solve but are hard to verify as well. In fact, some of these problems aren't even decidable. Among the hardest computer science problems are: K-means Clustering Traveling Salesman Problem, and Graph Coloring
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NP-hard Intuitively, these are the problems that are at least as hard as the NP-complete problems. Note that NP-hard problems do not have to be in NP, and they do not have to be decision problems. The precise definition here is that a problem X is NP-hard, if there is an NP-complete problem Y, such that Y is reducible to X in polynomial time.
Conclusion P, NP, NP-Complete, and NP-Hard are fundamental classifications in computational theory. They define the limits of what can be solved or verified efficiently by computers.
After reading this chapter you will… understand how computer scientists classify problems. be able to define some of the most common classes of problems in computer science. be able to explain the relationship between P and NP problem classes. understand some key properties of NP-complete and NP-hard problems. understand the role of approximate solutions and heuristics in battling hard problems.
Finding the shortest TSP path, TSP-shortest, is NP-Hard and is harder than NP because verifying the shortest path is harder than a polynomial time check. #P (sharp-P) is the problem of finding the number of solutions to a problem in NP.
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