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'}}.

@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}
}