Suppose we have a set $S = \{(a_1,b_1),...,(a_n,b_n)\}$ where $a_i < m$, $b_i = m-a_i$, $m \in \mathbb{Z}^{+}$, $m>2$ and $n$ is an even number greater than $3$. What is the most efficient algorithm to determine if it is possible to partition $S$ into two distinct subsets, $C$ and $D$, of equal size such that
$\sum_{a \in C} a > \sum_{b \in C} b$ and $\sum_{a \in D} a > \sum_{b \in D} b$
or $\sum_{b \in C} b > \sum_{a \in C} a$ and $\sum_{b \in D} b > \sum_{a \in D} a$ ?
For example, if $S = \{(56,44),(48,52),(43,57),(60,40)\}$, $C = \{(56,44),(48,52)\}$, and $D = \{(43,57),(60,40)\}$.
I am considering iteratively matching a pair with the best value of $a$ with a pair with the worst value of $a$. Is there another algorithm?