\name{qpCliqueNumber}
\alias{qpCliqueNumber}

\title{
Clique number
}
\description{
Calculates the size of the largest maximal clique (the so-called clique number
or maximum clique size) in a given undirected graph.
}
\usage{
qpCliqueNumber(g, exact.calculation=TRUE, return.vertices=FALSE,
               approx.iter=100, verbose=TRUE)
}
\arguments{
  \item{g}{either a \code{graphNEL} object or an incidence matrix of the given
       undirected graph.}
  \item{exact.calculation}{logical; if TRUE then the exact clique number is
       calculated; if FALSE then a lower bound is given instead.}
  \item{return.vertices}{logical; if TRUE a set of vertices forming a maximal
       clique of maximum size is returned; if FALSE only the maximum clique
       size is returned.}
  \item{approx.iter}{number of iterations to be employed in the calculation of
       the lower bound (i.e., only applies when \code{exact.calculation=FALSE}.}
  \item{verbose}{show progress on calculations.}
}
\details{
The calculation of the clique number of an undirected graph is an NP-complete
problem which means that its computational cost is bounded by an exponential
running time (Pardalos and Xue, 1994). The current implementation uses C code
from the GNU GPL Cliquer library by Niskanen and Ostergard (2003) based on the,
probably the fastest to date, algorithm by Ostergard (2002).

The lower bound on the maximum clique size is calculated by ranking the
vertices by their connectivity degree, put the first vertex in a set and
go through the rest of the ranking adding those vertices to the set that
form a clique with the vertices currently within the set. Once the entire
ranking has been examined a large clique should have been built and eventually
one of the largests ones. This process is repeated a number of times
(\code{approximation.iterations}) each of which the ranking is altered with
increasing levels of randomness acyclically (altering 1 to $p$ vertices and
again). Larger values of \code{approximation.iterations} should provide
tighter lower bounds and eventually the exact maximum clique size.
}
\value{
the size of the largest maximal clique in the given graph, also known as its
clique number.
}
\references{
Castelo, R. and Roverato, A. A robust procedure for
Gaussian graphical model search from microarray data with p larger than n.
\emph{J. Mach. Learn. Res.}, 7:2621-2650, 2006.

Niskanen, S. Ostergard, P. Cliquer User's Guide, Version 1.0.
Communications Laboratory, Helsinki University of Technology, Espoo, Finland,
Tech. Rep. T48, 2003. (\url{http://users.tkk.fi/~pat/cliquer.html})

Ostergard, P. A fast algorithm for the maximum clique problem.
Discrete Appl. Math. 120:197-207, 2002.

Pardalos, P.M. and Xue, J. The maximum clique problem.
\emph{J. Global Optim.}, 4:301-328, 1994.
}
\author{R. Castelo}
\seealso{
  \code{\link{qpClique}}
}
\examples{
nVertices <- 50 # number of vertices
maxCon <- 5  # maximum connectivity per vertex

I <- qpRndGraph(n.vtx=nVertices, n.bd=maxCon)

qpCliqueNumber(I, verbose=FALSE)

maxCon <- 10  # maximum connectivity per vertex

I <- qpRndGraph(n.vtx=nVertices, n.bd=maxCon)

qpCliqueNumber(I, verbose=FALSE)

}
\keyword{models}
\keyword{multivariate}