There are two problems commonly known as the subset sum problem.
The first ("given sum problem") is the problem of finding what subset of a list of integers has a given sum, which is an integer relation problem where the relation coefficients are 0 or 1.
The ("same sum problem") is the problem of finding a set of distinct positive real numbers with as large a collection as possible of subsets with the same sum (Proctor 1982).
The same sum problem was solved by Stanley (1980) using the tools of algebraic geometry, with the answer given for numbers by the first positive integers: . Proctor (1982) gave the first elementary proof of this result. The maximal numbers of subsets of having the same sum for , 2, ... are 1, 1, 2, 2, 3, 5, 8, 14, 23, ... (OEIS A025591). Similarly, the numbers of different subset sums for , 2, ... are 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, ... (OEIS A000124). For example, for , the subsets of are
(1)
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(2)
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(3)
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(5)
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(6)
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(7)
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(8)
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so the sum appearing most often is 3, which occurs twice, and the number of different sums is 7.
The given sum problem is NP-complete. For small cases, it be solved using generating functions. Consider the number of ways to select out of given integers such that their sum equals , and define the generating function
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Upon expanding in powers of , this becomes
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But as a result of the exponent law , is precisely the desired generating function
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For example, consider the problem of picking objects from the set . The generating function is
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So, for example, selecting objects has the generating function
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so the number of ways of picking three of the integers 1 through 5 and having them sum to , 11, ..., 6 are the coefficients of , namely 1, 1, 2, 2, 2, 1, and 1. These solutions are summarized in the following table.
solutions | |
6 | (1, 2, 3) |
7 | (1, 2, 4) |
8 | (1, 2, 5), (1, 3, 4) |
9 | (1, 3, 5), (2, 3, 4) |
10 | (1, 4, 5), (2, 3, 5) |
11 | (2, 4, 5) |
12 | (3, 4, 5) |
A nice explicit example original proposed by Pólya (1956) asks for the number of ways to make change from an American dollar (using pennies, nickels, dimes, quarters, and half-dollars). The answer of 292 is provided as the coefficient of the term in the series
(15)
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(Borwein and Bailey 2003, p. 21).