Abstract
Since no concrete lower bounds for problems in NP are known, we identify the “hardest” problems in this class. To express the “harder” (or rather “not simpler than”) relationship one uses the concept of reduction. In order to ensure that one does not get out of the class of polynomial time verification, the reduction function is required to be computable in polynomial time. One can then show that if P \(\ne \) NP is assumed, none of those “hardest” problem in NP can actually have a polynomial time solution. In other words, to show that P = NP it suffices to find a polynomial time solution for any of the “hardest” problems in NP.
How can we characterise and formalise that problems are harder than others in a class?
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Notes
- 1.
Richard Manning Karp (born January 3, 1935) is an American computer scientist, most known for his work in complexity and computability theory for which he received, among many other awards, the Turing Award in 1985.
- 2.
There are other variations of reduction where e.g. the question “is in B?” can be asked several times.
- 3.
Emil L. Post (February 11, 1897–April 21, 1954) was an American mathematician and logician, also famous for his “Post System” that can be used as a model of computation.
- 4.
This follows from the definition of time measures as the construction of large expressions is accounted for in the time measure. For each time unit consumed one can at most produce one “bit” of the output.
- 5.
“Another last minute terminology substitution happened when Aho, Hopcroft, and Ullman substituted “NP-complete” for “Polynomially-complete” in their text on Algorithms even though they had already gotten galley proofs using the original name. The name was changed at that late date as the result of a poll conducted throughout the Theoretical Computer Science community (suggested names were NP-Hard, Herculean Problem, and Augean Problem)” (see footnote 6) [6]. It was actually Knuth himself who triggered the discussion about the terminology in [5].
- 6.
In Greek mythology Heracles (Hercules) was known for his strength. As a penance for having killed his family, he had to perform ten labours, the fifth of which was to clean the stables of Augeas in one day. It should be noted that Augeas was known for having the most cattle in the country. Heracles solved this task by rerouting the local rivers into the stables (which was then considered cheating, but this is another story...).
- 7.
For those who know about the properties of order relations: relation \(\le _P\) is not totally ordered, it is not even quite an order. It is just a so-called preorder (or quasi-order). It is reflexive and transitive but not anti-symmetric, i.e. it does not hold that \(A\le B\) and \(B\le A\) implies \(A=B\). We can have, for instance two different “hardest” problems such that each must be reducible to the other but they are different problems still.
- 8.
See Proposition 2.1.
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Reus, B. (2016). How Hard Is a Problem?. In: Limits of Computation. Undergraduate Topics in Computer Science. Springer, Cham. https://doi.org/10.1007/978-3-319-27889-6_19
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