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Abstract:
In this paper, we consider the mutual exclusion scheduling problem
for comparability graphs.
Given an undirected graph $G$ and a fixed constant $m$, the problem is to
find a minimum coloring of $G$ such that each color is used at most $m$
times. The complexity of this problem for comparability graphs was mentioned as an open problem
by M\"ohring (1985) and for permutation graphs (a
subclass of comparability graphs) as an open problem by Lonc (1991). We
prove that this problem is already NP-complete for permutation graphs and
for each fixed constant $m \ge 6$.
