The minimum linear arrangement problem on proper interval graphs

We present a linear time algorithm for the minimum linear arrangement problem on proper interval graphs. The obtained ordering is a 4-approximation for general interval graphs. 1 Preliminaries Let F be a family of nonempty sets. The intersection graph of F is obtained by representing each set in F by a vertex and connecting two vertices by an edge if and only if their corresponding sets intersenct. The intersection graph of a family of intervals on a linearly ordered set (like the real line) is called an interval graph. If these intervals are constructed such that no interval properly contains another then such graph is called a proper interval graph. The families of interval and proper interval graphs are widely studied and used in different fields. In this chapter we present an algorithm which produces an optimal solution of the MinLA on proper interval graphs. Let us construct graph G = (V,E) in a following way (algorithm A): • set n as number of vertices in a graph • drop n vertices on an axis with integer coordinates from 1 to n • take a subset of successive vertices and make a clique from them • return to the previous step t times As a result of this construction we obtain a graph with the representation like on Figure 1. If we have a situation with nested cliques, we can ignore the clique that is placed inside of some other clique. We solve the problem for a family of graphs obtained by applying the algorithm A and then show that there is an algorithm which produces such representation for proper interval graphs. In the following claims we will work with a graph G = (V,E) that is a chain of k cliques C1...Ck constructed using algorithm A. In all following orders we index the vertices from 1 to n, where |V | = n. The orders of the vertices that preserve the order of cliques C1, C2, ..., Ck and full rotation Ck, Ck−1, ..., C1 will be called ’natural orders’ (or N − order) of G. Denote by dv the degree of vertex v. ∗This work is done as a part of M.Sc. thesis [3]