Fast Deterministic Distributed Degree Splitting

Abstract

We obtain faster distributed algorithms for the directed and undirected degree splitting problems. In the directed degree splitting problem the goal is to orient the edges of a graph to minimize the discrepancy between in-degree and out-degree at every vertex. In the undirected degree splitting problem the goal is to color the edges red and blue to minimize the discrepancy between red-degree and blue-degree at every vertex.
In particular, we design a deterministic Local algorithm of complexity 𝒪(ε^{-1} ⋅ log n) for computing a directed degree split with discrepancy at most ε ⋅ deg(v) for every vertex v ∈ V. The dependence on n in our runtime is optimal due to the Ω(log n) lower bound by Chang, Kopelowitz and Pettie (STOC 2016), which already holds for ε = 1/3. We improve upon the previous result by Ghaffari, Hirvonen, Kuhn, Maus, Suomela and Uitto (DISC 2017) by eliminating all additional dependence on ε beyond the leading ε^{-1} factor. Furthermore, we extend our algorithm to compute undirected degree splits with the same discrepancy guarantee and in the same asymptotic runtime. Beyond our deterministic Local results, we also provide faster randomized algorithms for the same problems.
A key ingredient in our approach is a connection to the hypergraph sinkless orientation (HSO) problem introduced by Brandt, Maus, Narayanan, Schager and Uitto (SODA 2025). We extend their randomized HSO algorithm to apply to a larger class of instances, which in turn reduces the runtime of our randomized degree splitting algorithms. We also provide deterministic and randomized CONGEST algorithms for solving HSO.
Degree splittings are a natural tool to solve other graph problems via a recursive divide-and-conquer strategy. As a demonstration of this claim, we apply our degree splitting algorithm to show that (3/2 + ε)Δ-edge coloring can now be solved in 𝒪(ε^{-1} ⋅ log²Δ ⋅ log n + ε^{-2} ⋅ log n) rounds in Local. For constant ε and Δ = 𝒪(2^{log^{1/3} n}), this runtime matches the current state-of-the-art for (2Δ-1)-edge coloring in Ghaffari and Kuhn (FOCS 2021).