Compendium of optimization problems admitting ... - Semantic Scholar
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Compendium of optimization problems admitting highly parallel approximations
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Jeffrey Finkelstein Computer Science Department, Boston University May 6, 2016
This is a listing of classes of optimization problems and known optimization problems which are members of those classes. See Section 2 of the corresponding paper for definitions of the complexity classes. Warning: some of these may not be correctly classified and need verification!
-p
• NNCO:
– Maximum Variable Weighted Satisfiability [4, Theorem 3.1] [1, Theorem 8.3] • ApxNCO:
in
– Maximum k-CNF Satisfiability [1, Theorem 8.6] – Maximum Acyclic Subgraph [2, Section 7.4] – Minimum k-Center [2, Section 7.4]
k-
– k-Switching Network [2, Section 7.4] – Maximum Bounded Weighted Satisfiability [5, Theorem 4]
• RNCAS:
W or
– Minimum Metric Traveling Salesperson [2, Theorem 7.1.1]
• NCAS:
– Maximum k-CSP [6, Corollary 13] – Maximum Independent Set for Planar Graphs [2, Theorem 6.4.1]
Copyright 2013, 2016 Jeffrey Finkelstein 〈[email protected]〉. This document is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License, which is available at https://creativecommons.org/licenses/by-sa/4.0/. The LATEX markup that generated this document can be downloaded from its website at https://github.com/jfinkels/ncapproximation. The markup is distributed under the same license.
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• FNCAS: – Subset Sum [2, Theorem 4.1.4] – Maximum Clause Weighted CNF Satisfiability [6, Theorem 8] – Minimum Weight Vertex Cover [2, Theorem 5.3.6] – 0-1 Knapsack [3, Theorem 2] – Bin Packing [3, Theorem 3] • PO ∩ NNCO: – Linear Programming • PO ∩ ApxNCO: – Induced Subgraph of High Weight for Linear Extremal Properties [2] • PO ∩ NCAS: – Maximum Matching [2, Theorem 5.2.1] – Maximum Weight Matching [2, Theorem 5.2.2] – Positive Linear Programming [2, Theorem 5.1.11] [7] – Maximum Flow [2, Theorem 5.2.2] • PO ∩ FRNCAS: – Maximum Flow [2, Theorem 4.5.2] – Maximum Weight Perfect Matching [2, Theorem 4.5.2] – Maximum Weight Matching [2, Theorem 4.5.2]
References [1] Giorgio Ausiello et al. Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties. Springer, 1999. isbn: 9783540654315. url: http : / / books . google . com / books ? id = Yxxw90d9AuMC. [2] J. Díaz et al. Paradigms for fast parallel approximability. Cambridge International Series on Parallel Computation. Cambridge University Press, 1997. isbn: 9780521117920. url: http://books.google.com/books?id= tC9gCQ2lmVcC. [3] Ernst W. Mayr. Parallel Approximation Algorithms. Tech. rep. Stanford, CA, USA: Stanford University, Department of Computer Science, 1988. [4] Pekka Orponen and Heikki Mannila. On approximation preserving reductions: Complete problems and robust measures. Tech. rep. Helsinki, Finland: Department of Computer Science, University of Helsinki, 1987. 2
[5] Maria Serna and Fatos Xhafa. “On Parallel versus Sequential Approximation”. In: Algorithms — ESA ’95. Ed. by Paul Spirakis. Vol. 979. Lecture Notes in Computer Science. Springer Berlin / Heidelberg, 1995, pp. 409– 419. isbn: 978-3-540-60313-9. doi: 10.1007/3- 540- 60313- 1_159. url: http://dx.doi.org/10.1007/3-540-60313-1_159. [6] Luca Trevisan. “Parallel Approximation Algorithms by Positive Linear Programming”. In: Algorithmica 21 (1 1998), pp. 72–88. issn: 0178-4617. doi: 10.1007/PL00009209. url: http://dx.doi.org/10.1007/PL00009209. [7] Luca Trevisan and Fatos Xhafa. “The parallel complexity of positive linear programming”. In: Parallel Processing Letters 8.04 (1998), pp. 527–533.
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Compendium of optimization problems admitting highly parallel approximations
ro gr
Jeffrey Finkelstein Computer Science Department, Boston University May 6, 2016
This is a listing of classes of optimization problems and known optimization problems which are members of those classes. See Section 2 of the corresponding paper for definitions of the complexity classes. Warning: some of these may not be correctly classified and need verification!
-p
• NNCO:
– Maximum Variable Weighted Satisfiability [4, Theorem 3.1] [1, Theorem 8.3] • ApxNCO:
in
– Maximum k-CNF Satisfiability [1, Theorem 8.6] – Maximum Acyclic Subgraph [2, Section 7.4] – Minimum k-Center [2, Section 7.4]
k-
– k-Switching Network [2, Section 7.4] – Maximum Bounded Weighted Satisfiability [5, Theorem 4]
• RNCAS:
W or
– Minimum Metric Traveling Salesperson [2, Theorem 7.1.1]
• NCAS:
– Maximum k-CSP [6, Corollary 13] – Maximum Independent Set for Planar Graphs [2, Theorem 6.4.1]
Copyright 2013, 2016 Jeffrey Finkelstein 〈[email protected]〉. This document is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License, which is available at https://creativecommons.org/licenses/by-sa/4.0/. The LATEX markup that generated this document can be downloaded from its website at https://github.com/jfinkels/ncapproximation. The markup is distributed under the same license.
1
• FNCAS: – Subset Sum [2, Theorem 4.1.4] – Maximum Clause Weighted CNF Satisfiability [6, Theorem 8] – Minimum Weight Vertex Cover [2, Theorem 5.3.6] – 0-1 Knapsack [3, Theorem 2] – Bin Packing [3, Theorem 3] • PO ∩ NNCO: – Linear Programming • PO ∩ ApxNCO: – Induced Subgraph of High Weight for Linear Extremal Properties [2] • PO ∩ NCAS: – Maximum Matching [2, Theorem 5.2.1] – Maximum Weight Matching [2, Theorem 5.2.2] – Positive Linear Programming [2, Theorem 5.1.11] [7] – Maximum Flow [2, Theorem 5.2.2] • PO ∩ FRNCAS: – Maximum Flow [2, Theorem 4.5.2] – Maximum Weight Perfect Matching [2, Theorem 4.5.2] – Maximum Weight Matching [2, Theorem 4.5.2]
References [1] Giorgio Ausiello et al. Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties. Springer, 1999. isbn: 9783540654315. url: http : / / books . google . com / books ? id = Yxxw90d9AuMC. [2] J. Díaz et al. Paradigms for fast parallel approximability. Cambridge International Series on Parallel Computation. Cambridge University Press, 1997. isbn: 9780521117920. url: http://books.google.com/books?id= tC9gCQ2lmVcC. [3] Ernst W. Mayr. Parallel Approximation Algorithms. Tech. rep. Stanford, CA, USA: Stanford University, Department of Computer Science, 1988. [4] Pekka Orponen and Heikki Mannila. On approximation preserving reductions: Complete problems and robust measures. Tech. rep. Helsinki, Finland: Department of Computer Science, University of Helsinki, 1987. 2
[5] Maria Serna and Fatos Xhafa. “On Parallel versus Sequential Approximation”. In: Algorithms — ESA ’95. Ed. by Paul Spirakis. Vol. 979. Lecture Notes in Computer Science. Springer Berlin / Heidelberg, 1995, pp. 409– 419. isbn: 978-3-540-60313-9. doi: 10.1007/3- 540- 60313- 1_159. url: http://dx.doi.org/10.1007/3-540-60313-1_159. [6] Luca Trevisan. “Parallel Approximation Algorithms by Positive Linear Programming”. In: Algorithmica 21 (1 1998), pp. 72–88. issn: 0178-4617. doi: 10.1007/PL00009209. url: http://dx.doi.org/10.1007/PL00009209. [7] Luca Trevisan and Fatos Xhafa. “The parallel complexity of positive linear programming”. In: Parallel Processing Letters 8.04 (1998), pp. 527–533.
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