Why Some NP Problems Are Hard To Solve
Because of the many important problems in this class, there have been extensive efforts to find polynomial-time algorithms for problems in NP. However, there remain a large number of problems in NP that defy such attempts, seeming to require super-polynomial time. Whether these problems really aren't decidable in polynomial time is one of the greatest open questions in computer science (see P=NP problem for an in-depth discussion).
An important notion in this context is the set of NP-complete decision problems, which is a subset of NP and might be informally described as the "hardest" problems in NP. If there is a polynomial-time algorithm for even one of them, then there is a polynomial-time algorithm for all the problems in NP. Because of this, and because dedicated research has failed to find a polynomial algorithm for any NP-complete problem, once a problem has been proven to be NP-complete this is widely regarded as a sign that a polynomial algorithm for this problem is unlikely to exist.
However, in practical uses, instead of spending computational resources looking for an optimal solution, a good enough (but potentially suboptimal) solution may often be found in polynomial time. Also, the real life applications of some problems are easier than their theoretical equivalents. For example, inputs to the general Travelling salesman problem need not obey the triangle inequality, unlike real road networks.
Read more about this topic: NP (complexity)
Famous quotes containing the words problems, hard and/or solve:
“While the onset of puberty can vary by as much as six years, every adolescent wants to be right on the 50-yard line, right in the middle of the field. One is always too tall, too short, too thin, too fat, too hairy, too clear-skinned, too early, too late. Understandably, problems of self-image are rampant.”
—Joan Lipsitz (20th century)
“Monday’s child is fair in face,
Tuesday’s child is full of grace,
Wednesday’s child is full of woe,
Thursday’s child has far to go,
Friday’s child is loving and giving,
Saturday’s child works hard for its living;
And a child that is born on a Christmas day,
Is fair and wise, good and gay.”
—Anonymous. Quoted in Traditions, Legends, Superstitions, and Sketches of Devonshire, vol. 2, ed. Anna E.K.S. Bray (1838)
“To a poet the mere making of a poem can seem to solve the problem of truth ... but only a problem of art is solved in poetry.”
—Laura Riding (1901–1991)