# Balanced subset sum problem

Suppose I have a set of $$2N$$ items with weights $$w_0, w_1, \ldots, w_{2N-1}$$. I want to identify the two most "balanced" sets of $$k$$ items each, where $$k$$ is not given but limited to the range $$k_\min \leq k \leq N$$.

More specifically: Identify the two disjunct subsets $$A = \{a_0, a_1, \ldots a_{k-1}\}$$ and $$B = \{b_0, b_1, \ldots a_{k-1}\}$$ subject to the following conditions:

• all the $$a_i, b_i \in [0,2N-1]$$
• $$A \cap B = \varnothing$$
• both sets have equal cardinality $$k$$ where $$k_\min \leq k \leq N$$
• the following quantity $$\epsilon$$ is minimized: \begin{align} \epsilon &= \frac{|W_A-W_B|}{W_A+W_B} \cr W_A &= \sum\limits_{i=0}^{k-1}w_{a_i} \cr W_B &= \sum\limits_{i=0}^{k-1}w_{b_i} \cr \end{align}

Is this a known problem? (I found Is there a "balanced knapsacks" problem with a known result? which is similar)

In my case the value of $$N$$ is not very large (under 50) so I don't really care about proving NP-hard, I just want to know a good strategy.