Efficiently finding low-sum copies of spanning forests in zero-sum complete graphs via conditional expectation

For a fixed positive ǫ, we show the existence of a constant Cǫ with the following property: Given a ±1-edge-labeling c : E(Kn) → {−1, 1} of the complete graph Kn with c(E(Kn)) = 0, and a spanning forest F of Kn of maximum degree ∆, one can determine in polynomial time an isomorphic copy F ′ of F in Kn with |c(E(F ))| ≤ ( 3 4 + ǫ ) ∆+ Cǫ. Our approach is based on the method of conditional expectation.