  For any graph $G$, a separating path system of $G$ is a family of paths in
$G$ with the property that for any pair of edges in $E(G)$ there is at least
one path in the family that contains one edge but not the other. We investigate
the size of the smallest separating path system for $K_n$, denoted $f(K_n)$.
  Our first main result is a construction that shows $f(K_n) \leq
\left(\frac{21}{16}+o(1)\right)n$ for sufficiently large $n$. We also show that
$f(K_n) \leq n$ whenever $n=p,p+1$ for prime $p$. It is known by simple
argument that $f(K_n) \geq n-1$ for all $n \in \mathbb{N}$.
  A key idea in our construction is to reduce the problem to finding a single
path with some particular properties we call a Generator Path. These are
defined in such a way that the $n$ cyclic rotations of a generator path provide
a separating path system for $K_n$. Hence existence of a generator path for
some $K_n$ gives $f(K_n) \leq n$. We construct such paths for all $K_n$ with $n
\leq 20$, and show that generator paths exist whenever $n$ is prime.
