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

TL;DR

This paper presents a polynomial-time method using conditional expectation to find spanning forests in complete graphs with zero-sum edge labelings, minimizing the sum of edge labels in the forest.

Contribution

It introduces a new polynomial-time algorithm for locating low-sum copies of spanning forests in zero-sum complete graphs, improving previous bounds.

Findings

Achieves low-sum spanning forests with bounds depending on maximum degree

Uses conditional expectation method for efficient algorithmic solution

Provides theoretical guarantees for the existence and construction of such forests

Abstract

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.