### Abstract

We address problems of the general form: given a J-dimensional binary (0-or 1-valued) vector a, a system of E of linear equations which a satisfies and a domain script D sign ⊂ ℝ^{J} which contains a, when is a the unique solution of E in script D sign? More generally, we aim at finding conditions for the invariance of a particular position j, 1 ≤ j ≤ J (meaning that b_{j} = a_{j}, for all solutions b of E in script D sign). We investigate two particular choices for script D sign: the set of binary vectors of length J (integral invariance) and the set of vectors in ℝ^{J} whose components lie between 0 and 1 (fractional invariance). For each position j, a system of inequalities is produced, whose solvability in the appropriate space indicates variance of the position. A version of Farkas' Lemma is used to specify the alternative system of inequalities, giving rise to a vector using which one can tell for each position whether or not it is fractionally invariant. We show that if the matrix of E is totally unimodular, then integral invariance is equivalent to fractional invariance. Our findings are applied to the problem of reconstruction of two-dimensional binary pictures from their projections (equivalently, (0, 1)-matrices from their marginals) and lead to a "structure result" on the arrangement of the invariant positions in the set of all binary pictures which share the same row and column sums and whose values are possibly prescribed at some positions. The relationship of our approach to the problem of reconstruction of higher-dimensional binary pictures is also discussed.

Original language | English |
---|---|

Pages (from-to) | 1-16 |

Number of pages | 16 |

Journal | Discrete Mathematics |

Volume | 171 |

Issue number | 1-3 |

Publication status | Published - Jun 20 1997 |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Discrete Mathematics*,

*171*(1-3), 1-16.

**Binary vectors partially determined by linear equation systems.** / Aharoni, R.; Herman, G. T.; Kuba, A.

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 171, no. 1-3, pp. 1-16.

}

TY - JOUR

T1 - Binary vectors partially determined by linear equation systems

AU - Aharoni, R.

AU - Herman, G. T.

AU - Kuba, A.

PY - 1997/6/20

Y1 - 1997/6/20

N2 - We address problems of the general form: given a J-dimensional binary (0-or 1-valued) vector a, a system of E of linear equations which a satisfies and a domain script D sign ⊂ ℝJ which contains a, when is a the unique solution of E in script D sign? More generally, we aim at finding conditions for the invariance of a particular position j, 1 ≤ j ≤ J (meaning that bj = aj, for all solutions b of E in script D sign). We investigate two particular choices for script D sign: the set of binary vectors of length J (integral invariance) and the set of vectors in ℝJ whose components lie between 0 and 1 (fractional invariance). For each position j, a system of inequalities is produced, whose solvability in the appropriate space indicates variance of the position. A version of Farkas' Lemma is used to specify the alternative system of inequalities, giving rise to a vector using which one can tell for each position whether or not it is fractionally invariant. We show that if the matrix of E is totally unimodular, then integral invariance is equivalent to fractional invariance. Our findings are applied to the problem of reconstruction of two-dimensional binary pictures from their projections (equivalently, (0, 1)-matrices from their marginals) and lead to a "structure result" on the arrangement of the invariant positions in the set of all binary pictures which share the same row and column sums and whose values are possibly prescribed at some positions. The relationship of our approach to the problem of reconstruction of higher-dimensional binary pictures is also discussed.

AB - We address problems of the general form: given a J-dimensional binary (0-or 1-valued) vector a, a system of E of linear equations which a satisfies and a domain script D sign ⊂ ℝJ which contains a, when is a the unique solution of E in script D sign? More generally, we aim at finding conditions for the invariance of a particular position j, 1 ≤ j ≤ J (meaning that bj = aj, for all solutions b of E in script D sign). We investigate two particular choices for script D sign: the set of binary vectors of length J (integral invariance) and the set of vectors in ℝJ whose components lie between 0 and 1 (fractional invariance). For each position j, a system of inequalities is produced, whose solvability in the appropriate space indicates variance of the position. A version of Farkas' Lemma is used to specify the alternative system of inequalities, giving rise to a vector using which one can tell for each position whether or not it is fractionally invariant. We show that if the matrix of E is totally unimodular, then integral invariance is equivalent to fractional invariance. Our findings are applied to the problem of reconstruction of two-dimensional binary pictures from their projections (equivalently, (0, 1)-matrices from their marginals) and lead to a "structure result" on the arrangement of the invariant positions in the set of all binary pictures which share the same row and column sums and whose values are possibly prescribed at some positions. The relationship of our approach to the problem of reconstruction of higher-dimensional binary pictures is also discussed.

UR - http://www.scopus.com/inward/record.url?scp=0001595253&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0001595253&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0001595253

VL - 171

SP - 1

EP - 16

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 1-3

ER -