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

Abstract

For a fixed positive $\epsilon$, we show the existence of a constant $C_{\epsilon }$ with the following property: Given a $\pm 1$-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 $F^{\prime }$ of $F$ in $K_n$ with $|c(E(F^{\prime }))|\le \left(\frac{3}{4}+\epsilon \right)\Delta +C_{\epsilon }.$ Our approach is based on the method of conditional expectation.