License: 
Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2016.37
URN: urn:nbn:de:0030-drops-57388
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2016/5738/
Fotakis, Dimitris ;
Lampis, Michael ;
Paschos, Vangelis Th.
Sub-exponential Approximation Schemes for CSPs: From Dense to Almost Sparse
Abstract
It has long been known, since the classical work of (Arora, Karger, Karpinski, JCSS'99), that MAX-CUT admits a PTAS on dense graphs, and more generally, MAX-k-CSP admits a PTAS on "dense" instances with Omega(n^k) constraints. In this paper we extend and generalize their exhaustive sampling approach, presenting a framework for (1-epsilon)-approximating any MAX-k-CSP problem in sub-exponential time while significantly relaxing the denseness requirement on the input instance.
Specifically, we prove that for any constants delta in (0, 1] and epsilon > 0, we can approximate MAX-k-CSP problems with Omega(n^{k-1+delta}) constraints within a factor of (1-epsilon) in time 2^{O(n^{1-delta}*ln(n) / epsilon^3)}. The framework is quite general and includes classical optimization problems, such as MAX-CUT, MAX-DICUT, MAX-k-SAT, and (with a slight extension) k-DENSEST SUBGRAPH, as special cases. For MAX-CUT in particular (where k=2), it gives an approximation scheme that runs in time sub-exponential in n even for "almost-sparse" instances (graphs with n^{1+delta} edges).
We prove that our results are essentially best possible, assuming the ETH. First, the density requirement cannot be relaxed further: there exists a constant r < 1 such that for all delta > 0, MAX-k-SAT instances with O(n^{k-1}) clauses cannot be approximated within a ratio better than r in time 2^{O(n^{1-delta})}. Second, the running time of our algorithm is almost tight for all densities. Even for MAX-CUT there exists r<1 such that for all delta' > delta >0, MAX-CUT instances with n^{1+delta} edges cannot be approximated within a ratio better than r in time 2^{n^{1-delta'}}.
BibTeX - Entry
@InProceedings{fotakis_et_al:LIPIcs:2016:5738,
author = {Dimitris Fotakis and Michael Lampis and Vangelis Th. Paschos},
title = {{Sub-exponential Approximation Schemes for CSPs: From Dense to Almost Sparse}},
booktitle = {33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)},
pages = {37:1--37:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-001-9},
ISSN = {1868-8969},
year = {2016},
volume = {47},
editor = {Nicolas Ollinger and Heribert Vollmer},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2016/5738},
URN = {urn:nbn:de:0030-drops-57388},
doi = {10.4230/LIPIcs.STACS.2016.37},
annote = {Keywords: polynomial and subexponential approximation, sampling, randomized rounding}
}
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Keywords: |
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polynomial and subexponential approximation, sampling, randomized rounding |
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Collection: |
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33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016) |
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Issue Date: |
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2016 |
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Date of publication: |
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16.02.2016 |
