If the input for a subset sum problem is a multiset (with repetitions) instead of a set (without repetitions), e.g.Set $a = \{1282,1588,3590,5138,5505,5777,6198,7962,8827,9838,10137,12037,15296,16089\}$Multiset $a = \{1282,1282,3590,3590,5505,5505,6198,6198,8827,8827,10137,10137,15296,15296\}$(above sets are samples, the real sets have lengths up to 64 and each element is up to 64 bits, I'm working with density-1 instances).In both cases $N$ is the same (183 bits if we use $\sum_{i=1}^n \log_2 a_i$ or 224 bits if we use a more generic approach $\mathrm{Length}(a)\cdot \log_2 \max\,\{a_1, \dots, a_n\}$.Is there any chance the repetitions could affect the problem so bad that could move the multiset version out of NP?Everything I know and have read about subset sum for long time says both instances are equally hard, and subset sum and all its variant are NP-Complete and just dependent on the input size but I'm getting a weird different result. (I found this discrepancy when I was validating the solver works well with multisets.)
Is the multiset - subset sum problem variant not in NP?
complexity theory;np complete;np hard