Inverse Booking Problem: Inverse Chromatic Number Problem in Interval Graphs

On Inverse Chromatic Number problems (Extended abstract)

“…In [6] we addressed the following problem dealing with a booking system: given a set of intervals, each representing a reservation of a resource for a certain duration, it is assumed that there are more than K reservations overlapping at the same time while only K identical resources are available. The task is to rearrange the given intervals by translating some of them so that (i) all intervals can be legally assigned to K identical parallel lines with no intersection between intervals on a same line and (ii) the total discrepancy between the original and the new interval position vectors (the translation cost) is minimum under the L 1 norm.…”

“…Some related real life models also motivate this version of inverse combinatorial optimization. Inverse chromatic number problems have already been discussed in [5,6]. It is NP-hard in general graphs: it is indeed shown in [5] that for any combinatorial optimization problem, its decision problem is polynomially Karp-reducible to the decision problem of the related inverse number problem.…”

“…([6]) IBP K=1 and IBP K=1,→ are both n-approximable and cannot be approximated in polynomial time within the ratio of O(n 1− ) for any > 0.…”