Is there a name for this modification to the subset sum problem?

Question

Let $S = \{\{x_{1},y_{1},z_{1}\},\{x_{2},y_{2},z_{2}\}, \ldots, \{x_{n},y_{n},z_{n}\}\}$ and a target $t$. Let $S_i$ be the subset list $\{x_{i},y_{i},z_{i}\}$. Find a subset sum that sums to $t$ such that one and only one element is chosen from each subset list $S_i$.

I can also imagine this is the same as finding the longest path from $s$ to some sink such that the path costs less than $t$, where the sum of the edges is the same as choosing from each subset list $S_i$.

Is there a name for this problem? Thanks.

Answer 1

I've seen this described as a "a variant of sum of subsets problem" by Arya et al.

Your formulation reminds me of some work I did on the problem of phase identification in electricity networks (e.g. see p118 of my Thesis, p135 of the pdf). Perhaps the references there (from 193 onwards) would be helpful.