Dissertation/ Thesis
Exact and Parameterized Algorithms for Subset Sum Problems
| Τίτλος: | Exact and Parameterized Algorithms for Subset Sum Problems |
|---|---|
| Συγγραφείς: | Randolph, Timothy William |
| Έτος έκδοσης: | 2024 |
| Συλλογή: | Columbia University: Academic Commons |
| Θεματικοί όροι: | Computer science, Computer algorithms, Problem solving--Computer programs |
| Περιγραφή: | We present a variety of exact and parameterized algorithms for Subset Sum and otherrelated problems. The major contributions of this thesis include: 1. Average-case algorithms for Generalized Subset Sum, the problem that generalizes both Subset Sum and Equal Subset Sum, as well as structural results describing the parameter regime in which solutions exist. These results extend the application of the “Representation Method” and are the fastest known in this setting. 2. A proof that Either-Or Subset Sum, the problem of solving either Subset Sum or Equal Subset Sum, can be solved exponentially faster than time 2⁰‧⁵ⁿ in the worst case. In our view, this result illustrates the potential of the “structure vs. randomness” approach for Subset Sum. 3. Algorithms that solve worst-case Subset Sum faster than time 2⁰‧⁵ⁿ by a polynomial factor. These improvements on the best known exact algorithms for Subset Sum represent the successful application of “log shaving” techniques to the problem. 4. Algorithms for Subset Sum and k-SUM with constant doubling. When considered in terms of a novel parameterization in the doubling constant, Subset Sum admits an XP-algorithm, while k-SUM is Fixed-Parameter Tractable. We also show that Subset Sum is FPT in the doubling constant if and only if an instance I of Hyperplane-Constrained Integer Linear Programming with n variables, m constraints, and constraint matrix entries bounded by ∆ can be solved in time ∆^{O(m)} · poly(|I|). |
| Τύπος εγγράφου: | thesis |
| Γλώσσα: | English |
| DOI: | 10.7916/baym-5m55 |
| Διαθεσιμότητα: | https://doi.org/10.7916/baym-5m55 |
| Αριθμός Καταχώρησης: | edsbas.89872E4F |
| Βάση Δεδομένων: | BASE |
| FullText | Text: Availability: 0 CustomLinks: – Url: https://doi.org/10.7916/baym-5m55# Name: EDS - BASE (ns324271) Category: fullText Text: View record from BASE |
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| Items | – Name: Title Label: Title Group: Ti Data: Exact and Parameterized Algorithms for Subset Sum Problems – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Randolph%2C+Timothy+William%22">Randolph, Timothy William</searchLink> – Name: DatePubCY Label: Publication Year Group: Date Data: 2024 – Name: Subset Label: Collection Group: HoldingsInfo Data: Columbia University: Academic Commons – Name: Subject Label: Subject Terms Group: Su Data: <searchLink fieldCode="DE" term="%22Computer+science%22">Computer science</searchLink><br /><searchLink fieldCode="DE" term="%22Computer+algorithms%22">Computer algorithms</searchLink><br /><searchLink fieldCode="DE" term="%22Problem+solving--Computer+programs%22">Problem solving--Computer programs</searchLink> – Name: Abstract Label: Description Group: Ab Data: We present a variety of exact and parameterized algorithms for Subset Sum and otherrelated problems. The major contributions of this thesis include: 1. Average-case algorithms for Generalized Subset Sum, the problem that generalizes both Subset Sum and Equal Subset Sum, as well as structural results describing the parameter regime in which solutions exist. These results extend the application of the “Representation Method” and are the fastest known in this setting. 2. A proof that Either-Or Subset Sum, the problem of solving either Subset Sum or Equal Subset Sum, can be solved exponentially faster than time 2⁰‧⁵ⁿ in the worst case. In our view, this result illustrates the potential of the “structure vs. randomness” approach for Subset Sum. 3. Algorithms that solve worst-case Subset Sum faster than time 2⁰‧⁵ⁿ by a polynomial factor. These improvements on the best known exact algorithms for Subset Sum represent the successful application of “log shaving” techniques to the problem. 4. Algorithms for Subset Sum and k-SUM with constant doubling. When considered in terms of a novel parameterization in the doubling constant, Subset Sum admits an XP-algorithm, while k-SUM is Fixed-Parameter Tractable. We also show that Subset Sum is FPT in the doubling constant if and only if an instance I of Hyperplane-Constrained Integer Linear Programming with n variables, m constraints, and constraint matrix entries bounded by ∆ can be solved in time ∆^{O(m)} · poly(|I|). – Name: TypeDocument Label: Document Type Group: TypDoc Data: thesis – Name: Language Label: Language Group: Lang Data: English – Name: DOI Label: DOI Group: ID Data: 10.7916/baym-5m55 – Name: URL Label: Availability Group: URL Data: https://doi.org/10.7916/baym-5m55 – Name: AN Label: Accession Number Group: ID Data: edsbas.89872E4F |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.7916/baym-5m55 Languages: – Text: English Subjects: – SubjectFull: Computer science Type: general – SubjectFull: Computer algorithms Type: general – SubjectFull: Problem solving--Computer programs Type: general Titles: – TitleFull: Exact and Parameterized Algorithms for Subset Sum Problems Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Randolph, Timothy William IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 01 Type: published Y: 2024 Identifiers: – Type: issn-locals Value: edsbas – Type: issn-locals Value: edsbas.oa |
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