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Are there any known "hard" instances for NP-Complete Problems,

or are there no general hard instances. So for different algorithms different instances are hard?

guest
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    ... where "hard" means an NP-hard distribution or that there is no known polynomial-time algorithm for the distribution? $;$ –  Feb 18 '15 at 16:49
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    No single instance can be hard, since it is hard-coded. There could be an infinite sequence of instances which is hard in the sense that if some polytime algorithm works correctly on all of them, then P=NP. The set of all instances is such an example, but sometimes smaller sets also work. – Yuval Filmus Feb 18 '15 at 17:28
  • and many instance can be trivially solved, but then there has to be a minimum set of the "hard" instances! But is there anything known, where these instances lie, how do they look like? – guest Feb 18 '15 at 17:48
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    Why does there have "to be a minimum set of the 'hard' instances"? $;$ –  Feb 18 '15 at 18:30
  • Wouldn't otherwise all instance be easy? – guest Feb 18 '15 at 18:33
  • Any set of "hard" instances has to be infinite, because for any finite set there is a polynomial time algorithm deciding that set. However, take any set of "hard" instances and remove any finite subset of its elements. The set must remain "hard", since the finite subset we removed isn't "hard". Hence no set can be minimal (by any reasonable notion of minimal) since we can always remove any finite number of elements from the set and still end up with a "hard" set. – Tom van der Zanden Feb 18 '15 at 20:40
  • Yeah but this sounds to me like i can make this statement: All instances we can ever write down/try to calculate are finite, so the set of those instances is finite, so we have a polynomial time algorithm for all practical cases. – guest Feb 18 '15 at 20:46
  • guest, you've gotten some extremely helpful feedback that should help you improve question. Please edit your question based upon the feedback you've gotten, to ask a more precise, narrowly scoped, well-defined question. I also suggest you add more details on what your thoughts are, what research you've done, and what specifically you mean by a "hard instance". In general, one-sentence questions are rarely a good fit for this site -- we expect you to flesh out the question in more detail. – D.W. Feb 18 '15 at 22:51

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