# Can an NP-complete problem be solved in quadratic or linear time?

Dear Intertwit,

As one of the world’s leading mathematicians, much like you, I ponder daily over whether the P versus NP problem is solvable.

Consider the subset sum problem that you’ll be well aware of.

One example of a problem that is easy to verify I present below – but let’s face it Intertwit, the answer may be difficult to compute even on your smart phone there.

Given a set of integers, does some nonempty subset of them sum to 0? I suspect you know the answer. But!…

For instance, does a subset of the set {−2, −3, 15, 14, 7, −10} add up to 0?

The answer “yes” of course Intertwit. Because the subset {−2, −3, −10, 15} adds up to zero. As you know this can be quickly verified with three additions. Simple stuff.

The key thing is: as far as we know here at the Institute, there is no known algorithm to find such a subset in polynomial time (there is one, however, in exponential time, which consists of 2n-n-1 tries), but such an algorithm exists if P = NP; hence this problem is in NP (quickly checkable on Facebook) but not necessarily in P (quickly solvable – if you see what I mean).

Discuss.

P = NP

Polynomial time = Nondeterministic polynomial time

Well, does it?

This whole mathematical can of formulae was first mentioned in a 1956 letter written by Kurt Gödel to John von Neumann – a couple of laugh a minute characters and proto-computer programmers. The rest of the letter contained a recipe for lemon drizzle cake and a long discourse about whether Bristol Rovers or City had a better chance of being promoted to the 1^{st} Division that season.

In the end no one still seems to have an answer for this equation and both Bristol clubs languish outside the top flight.

Gödel also developed a couple of theories on “incompleteness”. It seems to us old Kurt had a lot of questions and very few answers. Well that’s what Intertwit.com is here for isn’t it.

There’s a lot of people out there with a solution or theory on this subject and some algorithms for NP-complete problems exhibit exponential complexity.

Adam Hufangel a random chap with a marvellous surname reckoned thusly:

*As time approaches infinity P=NP, the problem is really solving a relative problem in a non-relative plain, in this case infinte time.*

*If Time reaches infinite amounts, it’s only logical to assume that every possible option to solving the problem has been exhausted, and eventually a solution, or in some cases the lack there-of would be discovered.*

*Also given an infinite amount of time and a finite set of options to derive the solution it’s only logical to believe that not only would the solution be discovered, but also the most easiest method at arriving at said solution…. thus P=NP*

*To be totally honest string theory pretty much makes this NP=P, but I don’t think we have enough comment space for that.*

That’s all very well but we at Intertwit like to look at things differently.

Consider the equation thusly:

P = NP

Pants = No Pants

That’s nonsense isn’t it? Pants are pants. No pants = arrest.

Therefore Pants cannot equal No Pants. It stands to reason.

However if you substitute “No” for “New” you have a completely different problem to evaluate.

Pants = New Pants

In this case the equation is correct. Pants can obviously be new as well as old so it’s a correct assertion. Add into the mix that New Pants aftershave (sponsored by Intertwit.com) is a gloriously and heady fragrance to allure and sedate all that come with 3 metres of its radius then P=NP is solvable polynomially or quadratically. And probably quantumly as well. Buy a bottle today. It even gives mathematicians a fair chance.

**New Pants aftershave – Buy a Bottle for the mathematician in your life.**

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