Inverse booking problem: inverse chromatic number problem in interval graphs

  • Authors:

  • Yerim Chung;Jean-François Culus;Marc Demange

  • Affiliations:

  • Paris School of Economics, Paris I University, France;LIPN, Paris XIII University, France;ESSEC Business School, DIS department, France

  • Venue:
  • WALCOM'08 Proceedings of the 2nd international conference on Algorithms and computation
  • Year:
  • 2008

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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 Ii =[ai, bi] (1 ≤ i ≤ n), each having a specified length s(Ii) = bi - ai, a (preferred) starting time ai and a completion time bi. 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(n1-Ɛ), Ɛ 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.