The postage stamp problem 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), 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 Coin problem Knapsack problem Subset sum problem
References
External links