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I am stuck at the complexity of the following problem: Given a multiset $S = \{x_1,..., x_n\}$ of $n$ integers and a natural number $k$. Can $S$ be partitioned into multisets $S_1,... S_j$ such that for all $S_i \subseteq S$ there is $\sum_{ s \in S_i} s = k?$

I have found out that this may be somehow related to the EqualSubetSum Problem or the partition problem. But is this problem really $NP$-complete or exists a polynomial algorithm depending on $k$ and $n$ to decide this problem?

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A set is a multiset (in which every element occurs once) and your partition problem can be used to find a subset of $S$ whose sum is $k$. [Note 1] So if there were a polynomial-time solution to your partition problem, it could easily be used to solve the subset sum problem in the same amount of time.

So the problem is NP-complete, but that doesn't stop it from having a pseudo-polynomial time solution. There are well-known DP algorithms to solve the subset sum problem which are polynomial in the magnitude of the inputs. Time complexity is always expressed as a function of the size of the inputs, as measured (for example) by the number of bits it takes to represent the input. The magnitude of a number is exponential in the size of the number (eg, a number represented in 100 bits could be as large as $2^{100}$). So time complexity polynomial in magnitude is still exponential in problem size.

In your problem, $n$ is a measure of problem size, whereas $k$ only measures numeric magnitude. So they are very different kinds of parameter.

Here's another longer explanation of pseudo-polynomial time in a StackOverflow answer.


Notes

  1. Let $S$ be a set for which we seek a subset whose sum is $k$. Without loss of generality, we can assume that $\Sigma S \le 2k$ (because a solution for $k' = \Sigma S - k$ leads directly to a solution for $k$ using the set difference from $S$.) If $\Sigma S = 2k$ then a solution to the multiset partition problem yields two solutions to the subset sum problem. Otherwise, we form the multiset $S' = S \cup \{2k - \Sigma S\}$. Obviously, $\Sigma S' = 2k$ and the element we added has a value less than $k$. We then compute the multiset partition of $\langle S', k\rangle$, which must be a partition into two pieces, at most one of which is a multiset (because there is at most one repeated element in $S'$, the one we added). If the first subset in the partition does not include the element we added, it is a solution to the original problem; otherwise, the second subset is.

    Computing $\Sigma S$ and examining the result of the multiset partition solution are both linear time. So if the multiset partition algorithm were polynomial time, so would be the derived subset sum algorithm.

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  • $\begingroup$ If $G=\{1,2\}$ and $k=2$, there is no solution to this question but there is a subset $\{2\}$ whose sum is 2. $\endgroup$ – Apass.Jack Dec 12 '18 at 6:42
  • $\begingroup$ @Apass.Jack: That's true; I should have mentioned the transformation from subset sum problem to multiset partition problem, and now I have. $\endgroup$ – rici Dec 12 '18 at 22:48
  • $\begingroup$ Nice explanation. $\endgroup$ – Apass.Jack Dec 13 '18 at 0:01
  • $\begingroup$ Have you answered the question, "a polynomial algorithm depending on k and n"? That means, I assume, time-complexity of $k^{O(1)}n^{O(1)}$. $\endgroup$ – Apass.Jack Dec 13 '18 at 0:02
  • $\begingroup$ @Apass.Jack: yes, that is what the expression means, I guess, and I think I did answer it: yes, there (probably) is a dynamic programming algorithm which is polynomial in $k$ but that's really pseudo-polynomial time because it is based on the magnitude of the input, not the size. In other words, the existence of a pseudo-polynomial time algorithm doesn't disprove the NP-completeness of the problem. I put the Wikipedia link in because it seems to be a complicated distinction to make and it didn't seem useful to dump it all into this answer. $\endgroup$ – rici Dec 13 '18 at 0:06

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