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Max-Balancing Weighted Directed Graphs and Matrix Scaling
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Abstract
A weighted directed graph G is a triple (V, A, g) where (V, A) is a directed graph and g is an arbitrary real-valued function defined on the arc set A. Let G be a strongly-connected, simple weighted directed graph. We say that G is max-balanced if for every nontrivial subset of the vertices W, the maximum weight over arcs leaving W equals the maximum weight over arcs entering W. We show that there exists a (up to an additive constant) unique potential pv for v ∈ V such that (V, A, gp) is max-balanced where gap = pu + ga − pv for a = (u, v) ∈ A. We describe an O(|V|2|A|) algorithm for computing p using an algorithm for computing the maximum cycle-mean of G. Finally, we apply our principal result to the similarity scaling of nonnegative matrices.
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Cite it as
@article{schneider1991maxbalancing,
title = {Max-Balancing Weighted Directed Graphs and Matrix Scaling},
author = {Hans Schneider and Michael H. Schneider},
year = {1991},
journal = {Mathematics of Operations Research},
publisher = {Institute for Operations Research and the Management Sciences},
volume = {16},
number = {1},
pages = {208--222},
doi = {10.1287/moor.16.1.208},
}