On the Sum Coloring Problem on Interval Graphs

Abstract.

In this paper we study the following NP-complete problem: given an interval graph G = (V,E) , find a node p -coloring $\langle V_1, V_2, . . ., V_p\rangle$ such that the cost $\xi (\langle V_1, V_2, . . ., V_p \rangle ) = \sum^p_{i=1} i|V_i|$ is minimal, where $\langle V_1, V_2, . . ., V_p \rangle$ denotes a partition of V whose subsets are ordered by nonincreasing cardinality. We present an O(m χ (G) + n log n) time ε -approximate algorithm (ε < 2) to solve the problem, where n , m , and χ #(G) are the number of nodes of the interval graph, its number of cliques, and its chromatic number, respectively. The algorithm is shown to solve the problem exactly on some classes of interval graphs, namely, the proper and the containment interval graphs, and the intersection graphs of sets of ``short'' intervals. The problem of determining the minimum number of colors needed to achieve the minimum $\xi (\langle V_1, V_2, . . ., V_p \rangle )$ over all p -colorings of G is also addressed.