Abstract
It has long been known, since the classical work of (Arora, Karger, Karpinski, JCSS'99), that MAXCUT admits a PTAS on dense graphs, and more generally, MAXkCSP 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 (1epsilon)approximating any MAXkCSP problem in subexponential 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 MAXkCSP problems with Omega(n^{k1+delta}) constraints within a factor of (1epsilon) in time 2^{O(n^{1delta}*ln(n) / epsilon^3)}. The framework is quite general and includes classical optimization problems, such as MAXCUT, MAXDICUT, MAXkSAT, and (with a slight extension) kDENSEST SUBGRAPH, as special cases. For MAXCUT in particular (where k=2), it gives an approximation scheme that runs in time subexponential in n even for "almostsparse" 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, MAXkSAT instances with O(n^{k1}) clauses cannot be approximated within a ratio better than r in time 2^{O(n^{1delta})}. Second, the running time of our algorithm is almost tight for all densities. Even for MAXCUT there exists r<1 such that for all delta' > delta >0, MAXCUT instances with n^{1+delta} edges cannot be approximated within a ratio better than r in time 2^{n^{1delta'}}.
BibTeX  Entry
@InProceedings{fotakis_et_al:LIPIcs:2016:5738,
author = {Dimitris Fotakis and Michael Lampis and Vangelis Th. Paschos},
title = {{Subexponential Approximation Schemes for CSPs: From Dense to Almost Sparse}},
booktitle = {33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)},
pages = {37:137:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770019},
ISSN = {18688969},
year = {2016},
volume = {47},
editor = {Nicolas Ollinger and Heribert Vollmer},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2016/5738},
URN = {urn:nbn:de:0030drops57388},
doi = {10.4230/LIPIcs.STACS.2016.37},
annote = {Keywords: polynomial and subexponential approximation, sampling, randomized rounding}
}
Keywords: 

polynomial and subexponential approximation, sampling, randomized rounding 
Collection: 

33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016) 
Issue Date: 

2016 
Date of publication: 

16.02.2016 