Engineering Edge Orientation Algorithms

TL;DR

The paper tackles the edge orientation problem, which seeks an orientation of an undirected graph to minimize the maximum vertex out-degree, a quantity linked to the graph's pseudoarboricity. It introduces a novel path-based framework that extends Venkateswaran's improving-path approach, augmented with practical engineering techniques such as two-approximation data reduction, fast initialization, and a variety of path-search strategies, all interpreted through a bipartite flow perspective. The authors also provide an alternate correctness proof and demonstrate substantial empirical gains: on average, their RapidPathOrientation is $6.59$ times faster than Kowalik's exact method and $36.27$ times faster than Georgakopoulos–Politopoulos, while handling graphs with billions of edges. This work significantly advances the scalability of edge-orientation solvers for massive networks and points toward parallelism as a promising direction for future improvements.

Abstract

Given an undirected graph G, the edge orientation problem asks for assigning a direction to each edge to convert G into a directed graph. The aim is to minimize the maximum out degree of a vertex in the resulting directed graph. This problem, which is solvable in polynomial time, arises in many applications. An ongoing challenge in edge orientation algorithms is their scalability, particularly in handling large-scale networks with millions or billions of edges efficiently. We propose a novel algorithmic framework based on finding and manipulating simple paths to face this challenge. Our framework is based on an existing algorithm and allows many algorithmic choices. By carefully exploring these choices and engineering the underlying algorithms, we obtain an implementation which is more efficient and scalable than the current state-of-the-art. Our experiments demonstrate significant performance improvements compared to state-of-the-art solvers. On average our algorithm is 6.59 times faster when compared to the state-of-the-art.

Figures (7)

• Figure 1: Example bipartite repr. transformation.
• Figure 2: Left: Layered example orientation in a graph with nine edges and vertices. Right: Visualization of the onion-like structure of the orientation problem.
• Figure 3:
• Figure 4: Running time performance profile on 58 instances for state-of-the-art algorithms TwoA+Kowalik, TwoA+G&P and our RapidPathOrientation (RPO) approach. On the right hand side we present a detailed zoom to values up to 6.5.
• Figure : Orientation by Exhaustive Search.