  Multiple interval graphs are variants of interval graphs where instead of a
single interval, each vertex is assigned a set of intervals on the real line.
We study the complexity of the MAXIMUM CLIQUE problem in several classes of
multiple interval graphs. The MAXIMUM CLIQUE problem, or the problem of finding
the size of the maximum clique, is known to be NP-complete for $t$-interval
graphs when $t\geq 3$ and polynomial-time solvable when $t=1$. The problem is
also known to be NP-complete in $t$-track graphs when $t\geq 4$ and
polynomial-time solvable when $t\leq 2$. We show that MAXIMUM CLIQUE is already
NP-complete for unit 2-interval graphs and unit 3-track graphs. Further, we
show that the problem is APX-complete for 2-interval graphs, 3-track graphs,
unit 3-interval graphs and unit 4-track graphs. We also introduce two new
classes of graphs called $t$-circular interval graphs and $t$-circular track
graphs and study the complexity of the MAXIMUM CLIQUE problem in them. On the
positive side, we present a polynomial time $t$-approximation algorithm for
WEIGHTED MAXIMUM CLIQUE on $t$-interval graphs, improving earlier work with
approximation ratio $4t$.
