Approximation Guarantees for the Minimum Linear Arrangement Problem by Higher Eigenvalues

Abstract

Given an n -vertex undirected graph G = ( V , E ) and positive edge weights { w e } 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 w e |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 n O ( r /ϵ) -time randomized algorithm that, given a graph G , returns a linear arrangement π, such that val ( G , π) ≤ (1 + 2/(1 − ε)λ r ( L )) MLA( G ) + O (√log n / n ∑ e ∈ E w e ) with high probability, where L is the normalized Laplacian of G and λ r ( L ) is the r th smallest eigenvalue of L . Our algorithm gives a constant factor approximation for regular graphs that are weak expanders.