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Gomory Integer Programs

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Serkan Hosten, Rekha R. Thomas

Optimization and Control · Submitted 5 Jun 2001

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Every TDI system gives a Gomory family, and we construct Gomory families from matrices whose columns form a Hilbert basis for the cone they generate.

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arXiv IDmath/0106031 (version 1)
Pith IDZEMVUW6B, Pith's permanent record of this paper and its review
CitationRekha R. Thomas, Serkan Hosten. "Gomory Integer Programs." arXiv:math/0106031 (2001). https://pith.science/paper/ZEMVUW6B
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arXiv listing abstract v1. Results and the main claim are underlined; .

The set of all group relaxations of an integer program contains certain special members called Gomory relaxations. A family of integer programs with a fixed coefficient matrix and cost vector but varying right hand sides is a Gomory family if every program in the family can be solved by one of its Gomory relaxations. In this paper, we characterize Gomory families. Every TDI system gives a Gomory family, and we construct Gomory families from matrices whose columns form a Hilbert basis for the cone they generate. The existence of Gomory families is related to the Hilbert covering problems that arose from the conjectures of Sebo. Connections to commutative algebra are outlined at the end.

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 The set of all group relaxations of an integer program contains certain special members called Gomory relaxations. background 1.00 0.08 0.03 asserted 1.00
s1 A family of integer programs with a fixed coefficient matrix and cost vector but varying right hand sides is a Gomory family if every program in the family can be solved by one of its Gomory relaxations. background 0.99 0.21 0.06 asserted 0.99
s2 In this paper, we characterize Gomory families. contribution 1.00 0.91 0.11 asserted 0.93
s3 Every TDI system gives a Gomory family, and we construct Gomory families from matrices whose columns form a Hilbert basis for the cone they generate. result 0.60 0.87 0.25 asserted 1.00
s4 The existence of Gomory families is related to the Hilbert covering problems that arose from the conjectures of Sebo. interpretation 0.50 0.34 0.07 asserted 0.99
s5 Connections to commutative algebra are outlined at the end. contribution 0.61 0.71 0.03 asserted 0.86

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Original abstract

The set of all group relaxations of an integer program contains certain special members called Gomory relaxations. A family of integer programs with a fixed coefficient matrix and cost vector but varying right hand sides is a Gomory family if every program in the family can be solved by one of its Gomory relaxations. In this paper, we characterize Gomory families. Every TDI system gives a Gomory family, and we construct Gomory families from matrices whose columns form a Hilbert basis for the cone they generate. The existence of Gomory families is related to the Hilbert covering problems that arose from the conjectures of Sebo. Connections to commutative algebra are outlined at the end.

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