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Fixed-Threshold Peeling in Sublinear MPC: Round-Approximation Tradeoffs and Applications

Authors Slobodan Mitrović , Theodore Pan , Wen-Horng Sheu

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LIPIcs.DISC.2026.46.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Distributed algorithms
Keywords
  • Distributed graph algorithms
  • Massively Parallel Computation
  • Densest subgraph
  • Graph orientation
  • k-core decomposition
  • Graph coloring

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Abstract

A number of fundamental graph problems admit simple algorithms based on iterative peeling: repeatedly remove all vertices whose current degree is below a fixed threshold. This paradigm underlies algorithms for density-dependent edge orientation, density-dependent coloring, densest subgraph, and k-core decomposition. In this paper, we study these problems in the sub-linear MPC model.
We achieve the following round-approximation tradeoffs for this family of problems. For any constant δ ∈ (0,1), our algorithms use O(n^δ) local memory and Õ(m+n^{1+δ}) global memory.
- Density-dependent edge orientation. For any integer t > 0, we compute an orientation with maximum out-degree at most (2+ε)(t+1)α(G) in O(lg^{1/(t+2)} n ⋅ poly(lg lg n)) rounds, where α(G) denotes the minimum possible maximum out-degree of an orientation of G. In the poly(lg lg n)-round regime, obtained by setting t = ⌈lg lg n/lg lg lg n⌉, this gives an O(lg lg n/lg lg lg n)-approximation, improving the approximation factor of the recent work by Ghaffari and Grunau [PODC 2025].
- Densest subgraph. We obtain a (4+ε)-approximation in Õ(lg^{1/3} n) MPC rounds and a (6+ε)-approximation in Õ(lg^{1/4} n) MPC rounds. This improves the Õ(√{lg n}) round complexity of Ghaffari, Lattanzi, and Mitrović [ICML 2019] with a slightly larger approximation factor. This is the first O(1)-approximate algorithm for densest subgraph to break the Θ(√{lg n}) round-complexity barrier in the sub-linear MPC model.
- k-core decomposition. For any integer t > 0, we compute approximate coreness values within a factor of (2+ε)(t+1) in O(lg^{1/(t+2)} n ⋅ poly(lg lg n)) MPC rounds. This improves the Õ(√{lg n}) round complexity of Ghaffari, Lattanzi, and Mitrović [ICML 2019], again giving a round-approximation tradeoff.
- Density-dependent coloring. By combining our orientation algorithm with standard coloring reductions, we obtain a coloring with (2+ε)(t+1)α(G)+1 colors in O(lg^{1/(t+2)} n ⋅ poly(lg lg n)) MPC rounds. In particular, this improves the approximation obtainable in poly(lg lg n) rounds over Ghaffari and Grunau [PODC 2025].

Cite As Get BibTex

Slobodan Mitrović, Theodore Pan, and Wen-Horng Sheu. Fixed-Threshold Peeling in Sublinear MPC: Round-Approximation Tradeoffs and Applications. In 40th International Symposium on Distributed Computing (DISC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 397, pp. 46:1-46:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.DISC.2026.46

Author Details

Slobodan Mitrović
  • University of California, Davis, CA, USA
Theodore Pan
  • University of California, Davis, CA, USA
Wen-Horng Sheu
  • University of California, Davis, CA, USA

Funding

The authors were supported by the NSF Faculty Early Career Development Program No. 2340048.

Acknowledgements

AI Disclosure: The authors used ChatGPT 5.5 to prove an upper bound for a recurrence in round complexity analysis. The upper bound was first conjectured by the authors based on computations of small cases. The tool generated a proof, which can be verified by a straightforward induction argument. More specifically, the tool materially affected the proof of 7.17 in the full version. The authors verified the correctness and originality of all content including references.

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