Inverse booking problem: Inverse chromatic number problem in interval graphs

We consider inverse chromatic number problems in interval graphs having the following form: we are given an integer K and an interval graph G = (V,E), associated with n = |V| intervals I _i = ]a _i ,b _i [ (1 ≤ i ≤ n), each having a specified length s(I _i ) = b _i - a _i , a (preferred) starting time a _i and a completion time b _i . The intervals are to be newly positioned with the least possible discrepancies from the original positions in such a way that the related interval graph can be colorable with at most K colors. We propose a model involving this problem called inverse booking problem.We show that inverse booking problems are hard to approximate within O(n ^{1 - ε} ), ε > 0 in the general case with no constraints on lengths of intervals, even though a ratio of n can be achieved by using a result of [13]. This result answers a question recently formulated in [12] about the approximation behavior of the unweighted case of single machine just-in-time scheduling problem with earliness and tardiness costs. Moreover, this result holds for some restrictive cases.