A New Intersection Model and Improved Algorithms for Tolerance Graphs
George B. Mertzios, Ignasi Sau, Shmuel Zaks
Tolerance graphs model interval relations in such a way that intervals
can tolerate a certain degree of overlap without being in conflict. This
class of graphs, which generalizes in a natural way both interval and
permutation graphs, has attracted many research efforts since their
introduction in~\cite{GoMo82}, as it finds many important applications
in constraint-based temporal reasoning, resource allocation, and
scheduling problems, among others. In this article we propose the first
non-trivial intersection model for general tolerance graphs, given
by three-dimensional parallelepipeds, which extends the widely known
intersection model of parallelograms in the plane that characterizes the
class of bounded tolerance graphs. Apart from being important on its own,
this new representation also enables us to improve the time complexity
of three problems on tolerance graphs. Namely, we present optimal
$\mathcal{O}(n\log n)$ algorithms for computing a minimum coloring and
a maximum clique, and an $\mathcal{O}(n^{2})$ algorithm for computing a
maximum weight independent set in a tolerance graph with $n$ vertices,
thus improving the best known running times $\mathcal{O}(n^{2})$ and
$\mathcal{O}(n^{3})$ for these problems, respectively.