Node-balancing by edge-increments

TL;DR

This paper investigates the problem of balancing vertex weights in graphs and hypergraphs through edge-increment steps, providing characterizations and efficient algorithms for when such balancing is possible.

Contribution

It offers a characterization of graphs that can be balanced via edge-increments and presents a strongly polynomial-time algorithm for finding a balancing sequence.

Findings

Characterization of graphs that can be balanced by edge-increments.

A strongly polynomial-time algorithm for computing a balancing sequence.

Connections established with fundamental matching theory results.

Abstract

Suppose you are given a graph $G=(V,E)$ with a weight assignment $w:V\rightarrow\mathbb{Z}$ and that your objective is to modify $w$ using legal steps such that all vertices will have the same weight, where in each legal step you are allowed to choose an edge and increment the weights of its end points by $1$. In this paper we study several variants of this problem for graphs and hypergraphs. On the combinatorial side we show connections with fundamental results from matching theory such as Hall's Theorem and Tutte's Theorem. On the algorithmic side we study the computational complexity of associated decision problems. Our main results are a characterization of the graphs for which any initial assignment can be balanced by edge-increments and a strongly polynomial-time algorithm that computes a balancing sequence of increments if one exists.