Interval Graph Representation with Given Interval and Intersection Lengths

We consider the problem of finding interval representations of graphs that additionally respect given interval lengths and/or pairwise intersection lengths, which are represented as weight functions on the vertices and edges, respectively. Pe’er and Shamir proved that the problem is NP-complete if only the former are given [SIAM J. Discr. Math. 10.4, 1997]. We give both a linear-time and a logspace algorithm for the case when both are given, and both an O(n ·m) time and a logspace algorithm when only the latter are given. We also show that the resulting interval systems are unique up to isomorphism. Complementing their hardness result, Pe’er and Shamir give a polynomialtime algorithm for the case that the input graph has a unique interval ordering of its maxcliques. For such graphs, their algorithm computes an interval representation that respects a given set of distance inequalities between the interval endpoints (if it exists). We observe that deciding if such a representation exists is NL-complete.