Computing the Bandwidth of Interval Graphs

Tractabilities and Intractabilities on Geometric Intersection Graphs

“…In [26], Kleitman and Vohra developed an algorithm for determining whether an interval graph G = (V, E) has a bandwidth less than or equal to a given integer k. Their algorithm plays an important role in the proof of Lemma 5. To be self-contained, we give the details of their algorithm below: …”

“…Only a few graph classes have been known for which the bandwidth problem can be solved in polynomial time. These include chain graphs [24], cographs and related classes (see [25] for the details), interval graphs [26][27][28], and bipartite permutation graphs [6,29] (see [25] for a comprehensive survey). One of the interesting graph classes for which the bandwidth problem can be solved efficiently is the class of interval graphs.…”

“…Unfortunately, the algorithm has a flaw, which has been fixed by Mahesh et al [27]. Kleitman and Vohra also show a polynomial time algorithm [26], and Sprague improves the time complexity to O(n log n) by sophisticated implementation of the algorithm [28]. All the algorithms above solve the decision problem that asks if an interval graph G has bandwidth at most k for given G and k. Thus, using binary search for k, we can compute the bandwidth bw(G) of an interval graph G in O(M (n) · log bw(G)) time, where M (n) = O(n log n) is the time complexity to solve the decision problem [28].…”

Optimal Linear Arrangement of Interval Graphs

“…For an interval graph with n vertices given by an interval model, Kleitman and Vohra's algorithm solves the decision problem "Is bw(G) ≤ k?" in O(nk) time, and it can be used to produce a layout with the minimum bandwidth in O(n 2 log n) time [13]. Furthermore, Sprague has shown how to implement Kleitman and Vohra's algorithm to answer the decision problem in O(n log n) time, and thus produce a minimum bandwidth layout in O(n log 2 n) time [19].…”

Mixed search number and linear‐width of interval and split graphs

“…For example, the bandwidth minimization problem is solved in polynomial time on interval graphs [14] and is NP hard on split graphs [16]. Profile (or SumCut) problem is trivially solvable in polynomial time on interval graphs and is NP hard on split graphs [21], while Optimal Linear Arrangement is NP hard on interval graphs [5].…”