  For a fixed positive $\epsilon$, we show the existence of a constant
$C_\epsilon$ with the following property: Given a $\pm1$-edge-labeling
$c:E(K_n)\to \{ -1,1\}$ of the complete graph $K_n$ with $c(E(K_n))=0$, and a
spanning forest $F$ of $K_n$ of maximum degree $\Delta$, one can determine in
polynomial time an isomorphic copy $F'$ of $F$ in $K_n$ with $|c(E(F'))|\leq
\left(\frac{3}{4}+\epsilon\right)\Delta+C_\epsilon.$ Our approach is based on
the method of conditional expectation.
