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<H1>NP-hard</H1>
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(definition)

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<strong>Definition:</strong>
A <a href="complexitycl.html" tppabs="http://hissa.nist.gov/dads/HTML/complexitycl.html"><em>complexity class</em></a> of problems that are intrinsically harder than those that can be solved by a <a href="turingmachin.html" tppabs="http://hissa.nist.gov/dads/HTML/turingmachin.html"><em>Turing machine</em></a> in <a href="nondetrmnstc.html" tppabs="http://hissa.nist.gov/dads/HTML/nondetrmnstc.html"><em>nondeterministic</em></a> <a href="polynomialtm.html" tppabs="http://hissa.nist.gov/dads/HTML/polynomialtm.html"><em>polynomial time</em></a>.  When a decision version of a combinatorial <a href="optimization.html" tppabs="http://hissa.nist.gov/dads/HTML/optimization.html"><em>optimization problem</em></a> is proven to belong to the class of <a href="npcomplete.html" tppabs="http://hissa.nist.gov/dads/HTML/npcomplete.html"><em>NP-complete</em></a> problems, which includes well-known problems such as satisfiability, <a href="travelsales.html" tppabs="http://hissa.nist.gov/dads/HTML/travelsales.html"><em>traveling salesman</em></a>, etc., an optimization version is <a href="nphard.html" tppabs="http://hissa.nist.gov/dads/HTML/nphard.html"><em>NP-hard</em></a>.
<P><strong>See also</strong>
<a href="stronglyNP.html" tppabs="http://hissa.nist.gov/dads/HTML/stronglyNP.html"><em>strongly NP-hard</em></a>, <a href="ap.html" tppabs="http://hissa.nist.gov/dads/HTML/ap.html"><em>AP</em></a>.
<P><em>Note:
For example, to decide if there exist k <a href="starshaped.html" tppabs="http://hissa.nist.gov/dads/HTML/starshaped.html"><em>star-shaped polygons</em></a> whose union is equal to a given simple polygon, for some parameter k, is NP-complete.  The optimization problem, i.e., finding a minimum of star-shaped polygons whose union is equal to a given simple polygon, is NP-hard. <P> From Algorithms and Theory of Computation Handbook, page 19-26, Copyright &copy; 1999 by CRC Press LLC.</em>
<P>Author: <a href="terms.html#authorCRC-A" tppabs="http://hissa.nist.gov/dads/terms.html#authorCRC-A">CRC-A</a>
<H2>More information</H2>
History, definitions, examples, etc. given in <A HREF="javascript:if(confirm('http://www.nist.gov/cgi-bin/exit_nist.cgi?url=http://www.cs.unb.ca/%7Ealopez-o/comp-faq/faq.html  \n\nThis file was not retrieved by Teleport Pro, because it is addressed on a domain or path outside the boundaries set for its Starting Address.  \n\nDo you want to open it from the server?'))window.location='http://www.nist.gov/cgi-bin/exit_nist.cgi?url=http://www.cs.unb.ca/%7Ealopez-o/comp-faq/faq.html'" tppabs="http://www.nist.gov/cgi-bin/exit_nist.cgi?url=http://www.cs.unb.ca/%7Ealopez-o/comp-faq/faq.html">Comp.Theory FAQ</A>, scroll down to <strong>P vs. NP</strong>.

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Entry modified Thu Oct  7 12:25:48 1999.<BR>
HTML page formatted Wed Dec 22 09:35:58 1999.

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