algorithm - Subset-sum prob. (congruence variation) -


i'm wondering np-completeness of variation of subset-sub problem:

subset-sum problem: given set of integers , integer s, non-empty subset sum s?

this problem known in np , np-complete. consider variation:

subset-sum problem (congruence variation): given 2 integers s , m , set of integers modulo m, non-empty subset sum s mod m?

(i.e., numbers in set modulo m , expected sum s in mod m).

i'm wondering if problem has been studied before? (would know if np-complete or not). know if there paper or similar variation of subset-sum problem? thank you!

yes, problem np-complete. since normal subset sum np complete, there reduction of other np-complete problem subset sum.

the same reduction work prove modular subset sum np complete, if can additionally generate sufficiently large modulus size polynomial in input size. modulus has bigger largest number used in subset sum solution, , difference between subset sum , modular subset sum irrelevant.

for reduction can think of, easy generate such modulus. remember size of modulus has polynomial in input size, so, 100^(n^2) works fine -- it's 2*(n^2) digits long.


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