Abstract
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.
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Chung, Y., Culus, JF., Demange, M. (2008). Inverse Booking Problem: Inverse Chromatic Number Problem in Interval Graphs. In: Nakano, Si., Rahman, M.S. (eds) WALCOM: Algorithms and Computation. WALCOM 2008. Lecture Notes in Computer Science, vol 4921. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77891-2_17
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DOI: https://doi.org/10.1007/978-3-540-77891-2_17
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