Abstract
This chapter presents another set of useful and famous problems. This time, however, nobody knows yet whether they are in P. No algorithms have been discovered yet that solve these problems in polynomial time. For simplicity, all problems in this chapter will be formulated as decision problems. The problems include the Travelling Salesman Problem, the Graph Colouring Problem, the Max-Cut Problem, which are all graph problems, the 0-1 Knapsack Problem and the Integer Programming Problem. The consequence for a problem “being decidable in polynomial time” is also discussed.
For which common and useful problems are polynomial time algorithms (yet) unknown?
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Notes
- 1.
An advantage is that we do not have to care how long it takes to construct the solution.
- 2.
Sir William Rowan Hamilton (3/4 August 1805–2 September 1865) was an Irish physicist, astronomer, and mathematician who made significant contributions to all those fields.
- 3.
Thomas Penyngton Kirkman (31 March 1806–3 February 1895) was a British mathematician.
- 4.
Karl Menger (January 13, 1902–October 5, 1985) was an Austrian-American mathematician and son of the famous Viennese economist Carl Menger. He is probably most renowned for the three-dimensional generalisation of the Cantor set and Sierpinski carpet (fractal curves) called Menger sponge.
- 5.
Hassler Whitney (23 March 1907–10 May 1989) was an American award-winning mathematician.
- 6.
In 2012, there has even been a movie released with this title [13] that won the “Best Movie” award at the Silicon Valley Film Festival.
- 7.
This is the so-called symmetric version of the TSP, there is also an asymmetric one where the cost of travelling from u to v does not equal the cost of travelling from v to u.
- 8.
Apparently bees can learn quickly to find the shortest tour from their bee hive to all the food sources (i.e. flowers) in a meadow. Researchers of Royal Holloway University, London, have published some results in 2010. However, it is questionable how big the route can be that they manage to solve.
- 9.
RaaS in the context of Network Operating Systems means “Routing-as-a-Service”.
- 10.
One usually calculates between \(10^{78}\) and \(10^{80}\) atoms in the known universe. In the “J” series of the BBC panel game quiz show “QI” (Quite Interesting), host Stephen Fry mentioned a fact about 52!, the number of possibilities you can shuffle a pack of (poker) cards. The number is so big (around \(8\times 10^{67}\)) that it is unlikely that anybody who shuffled a deck of cards before produced the same order of cards in the entire history of humanity! This requires, of course, that the deck is shuffled properly, but apparently 4 decent shuffles suffice. And of course, it just expresses an extremely low probability. So when Fry introduced his shuffling of a deck with the words “I’m going to do something that has never been done by any human being since the beginning of time”, he is not quite right actually. He should have said “that is very unlikely (with virtually zero probability) to have been done by any human being before”.
- 11.
In other words: countries that share a border need to be drawn in different colours.
- 12.
Kenneth Ira Appel (October 8, 1932–April 19, 2013) was an American mathematician. With Wolfgang Haken he solved the Four-Colour theorem , one of the most famous problems in mathematics. He was awarded the Fulkerson Prize.
- 13.
Wolfgang Haken (born June 21, 1928) is a German-American mathematician who works in topology. With Appel he proved the Four-Colour-Thereom and was awarded the Fulkerson Prize in 1979.
- 14.
The proof by Appel and Haken was the first proof of a major theorem that used computers to “compute” parts of the proof in order to deal with the many resulting different cases and combinatorial problems. It was therefore initially not trusted by everyone. In 2004 a fully machine-checked proof of the Four-Colour theorem has been carried out [7].
- 15.
When do you want to colour a graph in real life?
- 16.
Very Large Scale Integration.
- 17.
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Reus, B. (2016). Common Problems Not Known to Be in P . In: Limits of Computation. Undergraduate Topics in Computer Science. Springer, Cham. https://doi.org/10.1007/978-3-319-27889-6_17
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