Inspired by an article in the New Scientist I am returning to a favourite subject – whether P = NP and what the implications would be in the (unlikely) case that this were so.
Here’s a crude but quick explanation of P and NP: P problems are those that can be solve in a known time based on a polynomial (hence P) of the problem’s complexity – ie., we know in advance how to solve the problem. NP (N standing for non-deterministic) problems are those for which we can quickly (ie in P) verify that a solution is correct but for which we don’t have an algorithmic solution to hand – in other words we have to try all the possible algorithmic solutions in the hope of hitting the right one. Reversing one-way functions (used to encrypt internet commerce) is an NP problem – hence, it is thought/hoped that internet commerce is secure. On the other hand drawing up a school timetable is also an NP problem so solving that would be a bonus. There are a set of problems, known as NP-complete, which if any one was shown to be, in reality a P problem would mean that P = NP – in other words there would be no NP problems as such (we are ignoring NP-hard problems).
If it was shown we lived in a world where P=NP then we would inhabit ‘algorithmica’ – a land where computers could solve complex problems with, it is said, relative ease.
But what if, actually, we have polynomial solutions to P class problems but there were too complex to be of much use? The New Scientist article – which examines the theoretical problems faced by users of the ‘simplex algorithm’ points to just such a case.
The simplex algorithm aims to optimise a multiple variable problem using linear programming – as in an example they suggest, how do you get bananas from 5 distribution centres with varying numbers of supplies to 200 shops with varying levels of demand – a 1000 dimensional problem.
The simplex algorithm involves seeking the optimal vertex in the geometrical representation of this problem. This was thought to be rendered as a problem in P via the ‘Hirsch conjecture‘ – that the maximum number of edges we must traverse to get between any two corners on a polyhedron is never greater than the number of faces of the polyhedron minus the number of dimensions in the problem.
While this is true in the three dimensional world a paper presented in 2010 and published last month in the Annals of Mathematics – A counterexample to the Hirsch Conjecture by Francisco Santos has knocked down its universal applicability. Santos found a 43 dimensional shape with 86 faces. If the Hirsch conjecture was valid then the maximum distance between two corners would be 43 steps, but he found a pair at least 44 steps apart.
That leaves another limit – devised by Gil Kalai of the Hebrew University of Jerusalem and Daniel Kleitman of MIT, but this, says the New Scientist is “too big, in fact, to guarantee a reasonable running time for the simplex method“. Their two page paper can be read here. They suggest the diameter (maximal number of steps) is where is the number of faces and the dimensions. (The Hirsch conjecture is instead .)
So for Santos’s shape we would have a maximal diameter of (this is the upper limit, rather than the actual diameter). A much bigger figure even for a small dimensional problem, the paper also refers to a linear programming method that would require, in this case, a maximum of steps. Not a practical proposition if the dimension count starts to rise. (NB I am not suggesting these are the real limits for Santos’s shape, I am merely using the figures as an illustration of the many orders of magnitude difference they suggest might apply).
I think these figures suggest that proving P = NP might not be enough even if it were possible. We might have algorithms in P, but the time required would be such that quicker, if somewhat less accurate, approximations (as often used today) would still be preferred.
Caveat: Some/much of the above is outside my maths comfort zone, so if you spot an error shout it out.
First of all – a quick explanation of P and NP. The class of problems known as ‘P’ – for polynomial (as in they can be solved in a time which is dependent on a polynomial of their complexity) – are those for which a known algorithm – a sequence of mathematical steps – to solve them exists. For instance, solving (i.e., finding the value of x where the formula evaluates to zero) x – 2 is a P class problem. NP (not P) problems are much more interesting – these are problems for which an algorithm exists but which is unknown at the time the problem is posed. In the worst case the only way of solving the problem may be to try all the potential algorithms until we find the one that solves the problem. That said, once a potential solution is found it can be verified ‘simply’ (i.e. in polynomial time). It is not known if, in fact NP problems (such as drawing up school timetables or cracking internet public key encryption) are really P type problems after all and we just have not found the solution or are permanently excluded from ‘simple’ (P) solutions. A class of NP problems called ‘NP complete‘ are those that, if shown to really be P class problems, would indicate P=NP. Most, but not all, mathematicians think, in fact P!=NP.
So, here’s the problem. It sounds simple, but as it is NP, it’s not! (I got this from Making Embedded Systems: Design Patterns for Great Software)
You have a micro controller with a timer of fixed 4MHz frequency and two 8 bit registers that (a) counts tick and (b) a match register that triggers an interrupt which the count register matches the tick count and a 16 bit prescaler (that allows the scaling of the ticks e.g. – if set to 2 then twice as many ticks are required to trigger the interrupt).
So how can you set the match and prescaler to work for an arbitrary frequency? Sounds like it should be easily algorithmically managed, but it’s not.
- how do you prove that SAT is NP-complete? (cs.stackexchange.com)
- Are there NP problems, not in P and not NP Complete? (cs.stackexchange.com)
- Rule of thumb to know if a problem could be NP-complete (cs.stackexchange.com)
So, it seems internet commerce is safe for now. But Romanov is not throwing in the towel:
Thank you for your attention to my work. You’d better suspend your investigations.
A shortcoming possibly exists in the filtration procedure which requires an amendment.
A book to buy? P, NP, and NP-Completeness: The Basics of Computational Complexity – I admit to being tempted.
- What if P = NP? (cartesianproduct.wordpress.com)
- Polynomial Time Code For 3-SAT Released, P==NP (science.slashdot.org)
- The polynomial algorithm for 3-SAT problem (or P=NP) (romvf.wordpress.com)
- 2011 preview: Million-dollar mathematics problem (newscientist.com)
- P ≠ NP? It’s bad news for the power of computing [Math] (io9.com)