Approximation Guarantees for the Minimum Linear Arrangement Problem by Higher Eigenvalues

Abstract

Given an undirected graph G = (V,E) and positive edge weights {w _e } _e ∈ E , a linear arrangement is a permutation π: V → [n]. The value of the arrangement is $\mathbf{val}(G,\pi):= \frac{1}{n}\sum_{e=\{u,v\}}w_e |\pi(u)-\pi(v)|$. In the minimum linear arrangement problem (MLA), the goal is to find a linear arrangement π ^* that achieves val(G,π ^*) = MLA(G): =  min _π val(G,π).

In this paper, we show that for any ε > 0 and positive integer r, there is an O(n ^r/ε)-time randomized algorithm which, given a graph G, returns a permutation π such that

with high probability. Here $\mathcal{L}$ is the normalized Laplacian of G and $\lambda_r({\mathcal{L}})$ is the r-th eigenvalue of $\mathcal{L}$. Our algorithm gives a constant factor approximation for regular graphs that are weak expanders.