Subset sum for exactly k integers?

Question

Following from these question Subset sum problem and Sum-subset with a fixed subset size I was wondering what the general algorithm for solving a subset sum problem, where we are forced to use EXACTLY k integers, k <= n.

Evgeny Kluev mentioned that he would go for using optimal for k = 4 and after that use brute force approach for k- 4 and optimal for the rest. Anyone could enlight what he means by a brute force approach here combined with optimal k=4 algo?

Perhaps someone knows a better, general solution?

Answer 1

The original dynamic programming algorithm applies, with a slight extension - in addition to remembering partial sums, you also need to remember number of ints used to get the sums.

In the original algorithm, assuming the target sum is M and there are n integers, you fill a boolean n x M array A, where A[i,m] is true iff sum m can be achieved by picking (any number of) from first i+1 ints (assuming indexing from 0).

You can extend it to a three dimensional array nxMxk, which has a similar property - A[i,m,l] is true iff, sum m can be achieved by picking exactly l from first i+1 ints.

Assuming the ints are in array j[0..n-1]:

The recursive relation is pretty similar - the field A[0,j[0],1] is true (you pick j[0], getting sum j[0] with 1 int (duh)), other fields in A[0,*,*] are false and deriving fields in A[i+1,*,*] from A[i,*,*] is also similar to the original algorithm: A[i+1,m,l] is true if A[i,m,l]is true (if you can pick m from first i ints, then obviously you can pick m from first i+1 ints) or if A[i, m - j[i+1], l-1] is true (if you pick j[i+1] then you increase the sum by j[i+1] and the number of ints by 1).

If k is small then obviously it makes sense to skip all of the above part and just iterate over all combinations of k ints and checking their sums. k<=4 indeed seems like a sensible threshold.