Approximation Guarantees for the Minimum Linear Arrangement Problem by Higher Eigenvalues

Given an n-vertex undirected graph G = (V,E) and positive edge weights {we}e∈E, a linear arrangement is a permutation π : V → {1, 2, …, n}. The value of the arrangement is val(G, π) := 1/n∑ e ={u, v} ∈ E we|π(u) − π (v)|. In the minimum linear arrangement problem, the goal is to find a linear arrangement π * that achieves val(G, π*) = MLA(G) := min π val(G, π). In this article, we show that for any ε > 0 and positive integer r, there is an nO(r/ϵ)-time randomized algorithm that, given a graph G, returns a linear arrangement π, such that val(G, π) ≤ (1 + 2/(1 − ε)λr(L)) MLA(G) + O(√log n/n ∑ e ∈ E we) with high probability, where L is the normalized Laplacian of G and λr(L) is the rth smallest eigenvalue of L. Our algorithm gives a constant factor approximation for regular graphs that are weak expanders.