Abstract
Mathematicians are always interested in problems and their solutions. In this chapter we will be interested specifically in problems that can be solved using an algorithm, that is, by a step-by-step procedure.
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- 1.
Often the invention of the computer is attributed to Charles Babbage (1791–1871), who designed several mechanical computers in the 1800s. While Babbage’s machines were partially built during his lifetime, none was completed. A functioning model of Babbage’s Difference Engine No. 2 was completed in 1991. It was built using machining tolerances achievable in the nineteenth century, providing final validation of Babbage’s work. A fairly comprehensive history of computing machines can be found in [Kra 11, Chap. 6].
- 2.
For instance, Herman Hollerith’s machine to automate analysis of the census, a machine that used punch cards, was invented in 1890. Hollerith went on to found the company IBM.
- 3.
Here we use roman letters “P” and “NP” instead of the caligraphic letters “\(\mathcal{P}\)” and “\(\mathcal{N}\mathcal{P}\),” because we have used the caligraphic letters for the restricted problem of language decidability.
- 4.
The linear programming problem is not solvable in polynomial time using the simplex method, but it is solvable in polynomial time by other methods, as was first shown by Leonid Khachiyan in 1979.
References and Further Reading
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Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, San Francisco (1991)
Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum Press, New York (1972)
Krantz, S.G.: The Proof Is in the Pudding. Birkhäuser, Boston (2011)
Lawler, E.L.: Combinatorial Optimization: Networks and Matroids. Holt, Rinehart, and Winston, New York (1976)
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Krantz, S.G., Parks, H.R. (2014). The P/NP Problem. In: A Mathematical Odyssey. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-8939-9_10
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