A probabilistic algorithm for Optimal Linear Arrangements

The Optimal Linear Arrangement (OLA) problem seeks a vertex ordering of a graph that minimizes the sum of edge lengths, a fundamental challenge in graph layout, with applications in VLSI design, network optimization, and data visualization. We propose a basic randomized algorithm and enhance it using the method of conditional expectations to derive a deterministic algorithm with guaranteed performance bounds. Our theoretical analysis includes a concentration result showing that for random graphs, the optimal OLA value concentrates around the expected value of a random arrangement. Additionally, we extend the problem to weighted graphs and demonstrate the effectiveness of our algorithms through empirical evaluations.