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FPT algorithms for a special block-structured integer program with applications in scheduling.

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Title: FPT algorithms for a special block-structured integer program with applications in scheduling.
Authors: Chen, Hua1 (AUTHOR), Chen, Lin2 (AUTHOR), Zhang, Guochuan1 (AUTHOR) zgc@zju.edu.cn
Source: Mathematical Programming. Nov2024, Vol. 208 Issue 1/2, p463-496. 34p.
Subjects: Computable functions, Absolute value, Integer programming, Integers, Logarithms
Abstract: In this paper, a special case of the generalized 4-block n-fold IPs is investigated, where B i = B and B has a rank at most 1. Such IPs, called almost combinatorial 4-block n-fold IPs, include the generalized n-fold IPs as a subcase. We are interested in fixed parameter tractable (FPT) algorithms by taking as parameters the dimensions of the blocks and the largest coefficient. For almost combinatorial 4-block n-fold IPs, we first show that there exists some λ ≤ g (γ) such that for any nonzero kernel element g , λ g can always be decomposed into kernel elements in the same orthant whose ℓ ∞ -norm is bounded by g (γ) (while g itself might not admit such a decomposition), where g is a computable function and γ is an upper bound on the dimensions of the blocks and the largest coefficient. Based on this, we are able to bound the ℓ ∞ -norm of Graver basis elements by O (g (γ) n) and develop an O (g (γ) n 3 + o (1) L ^ 2) -time algorithm (here L ^ denotes the logarithm of the largest absolute value occurring in the input). Additionally, we show that the ℓ ∞ -norm of Graver basis elements is Ω (n) . As applications, almost combinatorial 4-block n-fold IPs can be used to model generalizations of classical problems, including scheduling with rejection, bi-criteria scheduling, and a generalized delivery problem. Therefore, our FPT algorithm establishes a general framework to settle these problems. [ABSTRACT FROM AUTHOR]
Copyright of Mathematical Programming is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: FPT algorithms for a special block-structured integer program with applications in scheduling.
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  Data: <searchLink fieldCode="AR" term="%22Chen%2C+Hua%22">Chen, Hua</searchLink><relatesTo>1</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Chen%2C+Lin%22">Chen, Lin</searchLink><relatesTo>2</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Zhang%2C+Guochuan%22">Zhang, Guochuan</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> zgc@zju.edu.cn</i>
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  Data: <searchLink fieldCode="JN" term="%22Mathematical+Programming%22">Mathematical Programming</searchLink>. Nov2024, Vol. 208 Issue 1/2, p463-496. 34p.
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  Data: <searchLink fieldCode="DE" term="%22Computable+functions%22">Computable functions</searchLink><br /><searchLink fieldCode="DE" term="%22Absolute+value%22">Absolute value</searchLink><br /><searchLink fieldCode="DE" term="%22Integer+programming%22">Integer programming</searchLink><br /><searchLink fieldCode="DE" term="%22Integers%22">Integers</searchLink><br /><searchLink fieldCode="DE" term="%22Logarithms%22">Logarithms</searchLink>
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  Data: In this paper, a special case of the generalized 4-block n-fold IPs is investigated, where B i = B and B has a rank at most 1. Such IPs, called almost combinatorial 4-block n-fold IPs, include the generalized n-fold IPs as a subcase. We are interested in fixed parameter tractable (FPT) algorithms by taking as parameters the dimensions of the blocks and the largest coefficient. For almost combinatorial 4-block n-fold IPs, we first show that there exists some λ ≤ g (γ) such that for any nonzero kernel element g , λ g can always be decomposed into kernel elements in the same orthant whose ℓ ∞ -norm is bounded by g (γ) (while g itself might not admit such a decomposition), where g is a computable function and γ is an upper bound on the dimensions of the blocks and the largest coefficient. Based on this, we are able to bound the ℓ ∞ -norm of Graver basis elements by O (g (γ) n) and develop an O (g (γ) n 3 + o (1) L ^ 2) -time algorithm (here L ^ denotes the logarithm of the largest absolute value occurring in the input). Additionally, we show that the ℓ ∞ -norm of Graver basis elements is Ω (n) . As applications, almost combinatorial 4-block n-fold IPs can be used to model generalizations of classical problems, including scheduling with rejection, bi-criteria scheduling, and a generalized delivery problem. Therefore, our FPT algorithm establishes a general framework to settle these problems. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Mathematical Programming is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1007/s10107-023-02046-z
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      – Code: eng
        Text: English
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      – SubjectFull: Absolute value
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      – SubjectFull: Integer programming
        Type: general
      – SubjectFull: Integers
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      – SubjectFull: Logarithms
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      – TitleFull: FPT algorithms for a special block-structured integer program with applications in scheduling.
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            – D: 01
              M: 11
              Text: Nov2024
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              Y: 2024
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