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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 ⟨jeffreyf@bu.edu⟩.
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.
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[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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