I am interested in the complexity of the following problem:
Input: A list $a_1\leq ⋯ \leq a_n$ of positive integers.
Question: Are there two vectors $x, x'\in\{−1,0,1\}^n$, with at least one $x_i$ and one $x'_i$ non-zero (the subsets must be non-empty), such that $$\sum_{i=1}^nx_ia_i=0, \sum_{i=1}^nx'_ia_i=0 \text{ and } x_ix'_i=0 \text{ }\forall i.$$ WEAK PARTITION is a variant of PARTITION where we are looking for a partition of a subset of the input, and not a partition of all the integers. This is the reason why we have $\{−1,0,1\}^n$ and not $\{−1,1\}^n$. Note that the subset must be non-empty, it means at least one $x_i$ must be non-zero.
But I am interested in the problem described above where we are looking for two distinct weak partitions. For example, with $A=(1,2,3,4,5,7,11,11)$, we have $x=(1,0,1,0,0,1,-1,0)$, $x'=(0,1,0,1,1,0,0,-1)$ and for each component $i$ we have $x_ix'_i=0$. It means we have two disjoint subsets of $A$ where there is a weak partition.
There is clearly a reduction from WEAK PARTITION but I suspect this problem to be NP-hard in the strong sense. Do you see any reduction from a problem which is NP-hard in the strong sense ?
Thank you very much.