Matrix Scaling by Network Flow

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Matrix Scaling by Network Flow (2007)

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by Günter Rote , Martin Zachariasen

Venue:	IN PROCEEDINGS OF THE ACMSIAM SYMPOSIUM ON DISCRETE ALGORITHMS
Citations:	7 - 1 self

BibTeX

@INPROCEEDINGS{Rote07matrixscaling,
    author = {Günter Rote and Martin Zachariasen},
    title = {Matrix Scaling by Network Flow},
    booktitle = {IN PROCEEDINGS OF THE ACMSIAM SYMPOSIUM ON DISCRETE ALGORITHMS},
    year = {2007},
    pages = {848--854},
    publisher = {}
}

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Abstract

A given nonnegative n × n matrix A =(aij) is to be scaled, by multiplying its rows and columns by unknown positive multipliers λi and µj, such that the resulting matrix (aijλiµj) has specified row and column sums ri and sj. We give an algorithm that achieves the desired row and column sums with a maximum absolute error ε in O(n4 (log n +logh ε)) steps, where h is the overall total of the result matrix. Our algorithm is a scaling algorithm. It solves a sequence of more and more refined discretizations. The discretizations are minimum-cost network flow problems with convex piecewise linear costs. These discretizations are interesting in their own right because they arise in proportional elections.

Keyphrases

matrix scaling network flow desired row network flow problem column sum ri convex piecewise linear cost column sum proportional election refined discretizations unknown positive multiplier maximum absolute error overall total result matrix scaling algorithm

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