Postage stamp problem
{{more citations needed|date=June 2017}}
File:postage_stamp_problem_example.svg
The postage stamp problem (also called the Frobenius coin problem and the Chicken McNugget theorem) is a mathematical riddle that asks what is the smallest postage value which cannot be placed on an envelope, if the latter can hold only a limited number of stamps, and these may only have certain specified face values.Jeffrey Shallit (2001), [https://arxiv.org/abs/math.NT/0112257 The computational complexity of the local postage stamp problem]. SIGACT News 33 (1) (March 2002), 90-94. Accessed on 2009-12-30.
For example, suppose the envelope can hold only three stamps, and the available stamp values are 1 cent, 2 cents, 5 cents, and 20 cents. Then the solution is 13 cents; since any smaller value can be obtained with at most three stamps (e.g. 4 = 2 + 2, 8 = 5 + 2 + 1, etc.), but to get 13 cents one must use at least four stamps.
Mathematical definition
Mathematically, the problem can be formulated as follows:
: Given an integer m and a set V of positive integers, find the smallest integer z that cannot be written as the sum v1 + v2 + ··· + vk of some number k ≤ m of (not necessarily distinct) elements of V.
Complexity
This problem can be solved by brute force search or backtracking with maximum time proportional to |V |m, where |V | is the number of distinct stamp values allowed. Therefore, if the capacity of the envelope m is fixed, it is a polynomial time problem. If the capacity m is arbitrary, the problem is known to be NP-hard.
See also
References
{{reflist}}
External links
- {{cite journal| first1=W. F. | last1=Lunnon| title= A postage stamp problem
|journal = Comput. J. | number=4 | year= 1969| pages=377–380|volume=12|doi=10.1093/comjnl/12.4.377| doi-access=free}}
- {{cite journal| first1=R. | last1=Alter | first2=J. A. | last2=Barnett| title=A postage stamp problem
|journal = Amer. Math. Monthly | year=1980 | volume=87 | issue=3 | pages=206–210 | doi=10.2307/2321610| jstor=2321610 }}
- {{cite journal|first1=R. L. | last1=Graham | first2=N. J. A. | last2=Sloane
|title=On additive bases and harmonious graphs| journal=SIAM J. Algebr. Discrete Methods
|year =1980|volume=1 | issue=4 | pages=382–404|doi=10.1137/0601045|citeseerx=10.1.1.70.5521}}
- {{cite journal|first1=M. F. | last1=Challis| title=Two new techniques for computing extremal h-bases Ak
|journal=Comput. J. | volume=36|number=2 | pages=117–126| year=1993 |doi=10.1093/comjnl/36.2.117| doi-access=free}}
- {{cite arXiv| first1=J. | last1=Kohonen| first2=J. | last2=Corander | eprint=1310.7090
|title=Addition chains meet postage stamps: reducing the number of multiplications| year=2013| class=math.NT}}
- {{cite arXiv|first1=Jukka | last1=Kohonen| title=A meet-in-the-middle algorithm for finding extremal restricted additive 2-bases
|year=2014 | class=math.NT| eprint=1403.5945}}
- {{MathWorld | urlname=PostageStampProblem | title=Postage Stamp Problem}}
- {{OEIS el|A001212|Solution to the postage stamp problem with n denominations and 2 stamps}}
Category:Additive number theory