Fast Deterministic Distributed Degree Splitting

2026-04-01

AI Summary

This work studies the problems of balanced orientation in directed graphs and degree splitting in undirected graphs within the LOCAL model, with the goal of minimizing load imbalance across vertices. By establishing a connection to sinkless orientations in hypergraphs, the authors propose a deterministic distributed algorithm that computes an orientation where each vertex’s discrepancy is at most ε·deg(v) in O(ε⁻¹·log n) rounds, and extend this approach to degree splitting. This significantly improves upon prior algorithms by eliminating polylogarithmic dependencies on ε in the round complexity. As an application, the method yields a (3/2+ε)Δ-edge-coloring algorithm running in O(ε⁻¹·log²Δ·log n + ε⁻²·log n) rounds, achieving state-of-the-art performance for certain parameter regimes.

Abstract

We obtain better algorithms for computing more balanced orientations and degree splits in LOCAL. Important to our result is a connection to the hypergraph sinkless orientation problem [BMNSU, SODA'25] We design an algorithm of complexity $\mathcal{O}(\varepsilon^{-1} \cdot \log n)$ for computing a balanced orientation with discrepancy at most $\varepsilon \cdot \mathrm{deg}(v)$ for every vertex $v \in V$. This improves upon a previous result by [GHKMSU, Distrib. Comput. 2020] of complexity $\mathcal{O}(\varepsilon^{-1} \cdot \log \varepsilon^{-1} \cdot (\log \log \varepsilon^{-1})^{1.71} \cdot \log n)$. Further, we show that this result can also be extended to compute undirected degree splits with the same discrepancy and in the same runtime. As as application we show that $(3 / 2 + \varepsilon)Δ$-edge coloring can now be solved in $\mathcal{O}(\varepsilon^{-1} \cdot \log^2 Δ\cdot \log n + \varepsilon^{-2} \cdot \log n)$ rounds in LOCAL. Note that for constant $\varepsilon$ and $Δ= \mathcal{O}(2^{\log^{1/3} n})$ this runtime matches the current state-of-the-art for $(2Δ- 1)$-edge coloring in [Ghaffari & Kuhn, FOCS'21].

Problem

• balanced orientation
• degree splitting
• distributed algorithm
• LOCAL model
• discrepancy

Innovation

• balanced orientation
• degree splitting
• LOCAL model
• distributed algorithm
• edge coloring