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Recognizing Balanceable Matrices

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Abstract

A 0/±1 matrix is balanced if it does not contain a square submatrix with exactly two nonzero entries per row and per column in which the sum of all entries is 2 modulo 4. A 0/1 matrix is balanceable if its nonzero entries can be signed ±1 so that the resulting matrix is balanced. A signing algorithm due to Camion shows that the problems of recognizing balanced 0/±1 matrices and balanceable 0/1 matrices are equivalent. Conforti, Cornuéjols, Kapoor and Vušković gave an algorithm to test if a 0/±1 matrix is balanced. Truemper has characterized balanceable 0/1 matrices in terms of forbidden submatrices. In this paper we give an algorithm that explicitly finds one of these forbidden submatrices or shows that none exists.

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Authors and Affiliations

  1. Dipartimento di Matematica, Universitá degli Studi di Padova, Via Belzoni 7, 35131, Padova, Italy

    Michele Conforti

  2. Department of Combinatorics and Optimization, University of Waterloo, 200 University Ave, Waterloo, Ontario, Canada, N2J 3G1

    Giacomo Zambelli

Authors
  1. Michele Conforti
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  2. Giacomo Zambelli
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Correspondence to Michele Conforti.

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Received: October 2004

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Conforti, M., Zambelli, G. Recognizing Balanceable Matrices. Math. Program. 105, 161–179 (2006). https://doi.org/10.1007/s10107-005-0647-7

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  • DOI: https://doi.org/10.1007/s10107-005-0647-7

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