Upcoming Challenge: Hypergraph Partitioning

Challenge Design
1

We are excited to introduce our upcoming challenge for TIG:

\large{ \textbf{Hypergraph Partitioning}}

Impact: Practical and/or Scientific

Hypergraphs are a powerful tool for representing complex networks in which relationships may involve more than two elements simultaneously. Hypergraph partitioning refers to dividing such a network into a specified number of groups that are roughly equal in size while keeping as many related items together as possible. Although the problem is computationally challenging (NP-hard), it has broad applications across numerous fields:

  • Parallel Computing & Load Balancing: By intelligently distributing tasks across processors, hypergraph partitioning minimizes communication overhead and enhances overall computational efficiency [1][2][3][4][5].
  • Distributed Neural Network Training: It enables the partitioning of compute graphs across multiple GPUs or servers, significantly accelerating the training of deep learning models [6][7].
  • VLSI & Circuit Design: By effectively grouping circuit components, it optimizes chip layouts and reduces interconnect complexity, leading to faster and more efficient designs [8][9].
  • Social Networks & Community Detection: Capturing multi-way relationships, hypergraph partitioning reveals hidden community structures and provides deeper insights into group dynamics [10].
  • Bioinformatics & Computational Biology: It facilitates the clustering of proteins, genes, and genomic regions to identify functional modules, thereby aiding discovery in biological research [11].
  • Machine Learning & Data Mining: By effectively modeling higher-order interactions, it improves data clustering and feature selection, enhancing analytical outcomes [12].
  • Other Applications: From optimizing database sharding and segmenting GIS regions to modularizing software systems, hypergraph partitioning transforms large-scale challenges into more tractable problems [1:1][7:1][4:1].

In the rapidly evolving field of Decentralized Physical Infrastructure Networks (DePIN) — which leverage blockchain technology and distributed nodes to manage physical assets — hypergraph partitioning plays an especially important role. By accurately modeling complex interactions, it can effectively group related tasks and resources across scenarios such as decentralized compute/storage, blockchain data sharding, IoT networks, or supply chain logistics [13]. This grouping helps minimize cross-node communication and balances workloads, ultimately enhancing the scalability and performance of these decentralized systems [14].

Problem Description and Formulation

The Hypergraph Partitioning challenge at TIG can be formulated as follows:

Goal: Divide a hypergraph into a specified number of balanced partitions while minimizing the number of cut hyperedges.

Consider a hypergraph H = (V, N), with:

  • A set of vertices V, where each vertex v \in V has a weight w[v].
  • A set of hyperedges (nets) N, where each hyperedge n \in N is a subset of V and has a cost c[n].

Constraints:

  • Each vertex belongs to exactly one part, i.e., V_k \cap V_l = \emptyset for all 1 \leq k < l \leq K.
  • Every part must contain at least one vertex, i.e., V_k \neq \emptyset.
  • The total weight of each part, W_k, must not exceed the average part weight W_{\text{avg}} by more than an allowed imbalance tolerance:
W_k \leq W_{\text{avg}}(1+\epsilon),

where W_k = \sum_{v \in V_k} w[v], W_{\text{avg}} = \frac{\sum_{k=1}^{K} W_k}{K}, and \epsilon is the allowed imbalance tolerance.

Objective:

\text{Minimize:}\quad \sum_{n \in N} c[n]\bigl(\lambda_n - 1\bigr),

where the connectivity, \lambda_n, is the number of parts that hyperedge n spans.

Baseline Calculation

The baseline algorithm provides a reference performance metric for each instance. It is crucial that this baseline is both stable (e.g., consistently within 20% of the best solution) and efficient.

Our chosen baseline is a greedy bipartition algorithm. The algorithm repeatedly partitions the vertex set into two parts until reaching the desired number of partitions. Each bipartition step proceeds as follows:

  1. Determine target sizes. Given a current subset of vertices, calculate how many vertices should go to the left and right parts (e.g., if we aim for 2^d total parts, each subdivision targets two parts of prescribed sizes).
  2. Sort vertices by degree. For the vertices in the current subset, compute their degrees (the number of hyperedges to which each vertex belongs). Sort them in descending order so that higher-degree vertices are placed first.
  3. Place vertices greedily. Initialize two Boolean arrays (one per part) to track which hyperedges already have at least one vertex assigned to that part. For each vertex in sorted order:
  • Count how many of its hyperedges are “activated” in the left part and how many in the right.
  • If the left side has higher overlap, assign the vertex to the left part (unless it is at capacity); similarly, assign it to the right side if that overlap is higher. In case of a tie or if a part is filled, assign the vertex to the other part. Continue until one part reaches its target size, then assign any remaining vertices to the other part.
  1. Recursive Subdivision. After producing a bipartition, apply the same procedure to each newly formed part until the desired number of parts (e.g., 64) is reached.

Finally, the connectivity of the complete multiway partition is computed, giving the baseline_value. Although this local greedy strategy does not capture global interactions perfectly, it is fast, straightforward, and serves as a reasonable performance benchmark for more sophisticated methods.

Random Instance Generation

Importance

The instance generation process aims to produce scenarios that closely resemble established academic benchmark instances. By mirroring real-world conditions, we ensure that the challenge motivates algorithmic advances relevant to practical applications.

Generation

We plan to generate synthetic hypergraphs that mimic the characteristics of academic benchmark datasets. Although our approach is still under development (with expert consultation planned), a promising method involves the degree-corrected hypergraph stochastic block model (DCHSBM) by Chodrow et al. [15]. This model naturally produces realistic clustering, varied node degrees, and heavy-tailed edge sizes. Ultimately, we want hypergraph instances whose sizes and structures reflect those found in widely used sparse matrix collections, such as the SuiteSparse Matrix Collection [16], as referenced in [3:1].

Choice of Parameters

We are considering two approaches for setting the number of hyperedges. In one approach, the number of hyperedges equals the number of vertices, reflecting the sparse matrices in the SuiteSparse Matrix Collection [16:1]. Alternatively, we may draw the number of hyperedges from a Poisson distribution, whose mean is derived from a product of node degree parameters and a cluster-based affinity function, as in [15:1].

For the TIG challenge, we are currently considering fixing the number of parts at 64. The number of clusters will be decided as part of the generation. We will set all weights and costs to unity (i.e., w[v] = c[n] = 1), thereby reducing the problem to minimizing solely the connectivity \lambda_n.

Difficulty Parameters

We define the following difficulty parameters that influence the Pareto frontier for qualifying solutions:

  1. num_vertices: The number of vertices in the hypergraph instance. As num_vertices grows, the solution space expands significantly, making it harder to find or approximate an optimal partition. This tests an algorithm’s ability to scale efficiently while maintaining good solution quality.
  2. better_than_baseline: The required proportional improvement over the baseline solution cost (measured by partition connectivity). A stricter threshold demands higher-quality optimization, pushing algorithms to deliver more refined solutions.

Our Challenge

At TIG, the baseline connectivity is determined by the greedy bipartition approach. The challenge is to develop algorithms that improve upon this baseline by at least the specified factor (better_than_baseline).

  1. Devine, K.D., Boman, E.G., Heaphy, R.T., Bisseling, R.H., & Catalyurek, U.V. (2006). Parallel hypergraph partitioning for scientific computing. Proceedings 20th IEEE International Parallel & Distributed Processing Symposium. ↩︎ ↩︎

  2. Aykanat, C., Cambazoglu, B., & Uçar, B. (2008). Multi-level direct K-way hypergraph partitioning with multiple constraints and fixed vertices. Journal of Parallel and Distributed Computing, 68, 609–625. ↩︎

  3. Trifunovic, A., & Knottenbelt, W. (2008). Parallel multilevel algorithms for hypergraph partitioning. J. Parallel Distrib. Comput., 68, 563–581. ↩︎ ↩︎

  4. Gottesbüren, L., & Hamann, M. (2022). Deterministic Parallel Hypergraph Partitioning. In Euro-Par 2022: Parallel Processing (pp. 301–316). Springer International Publishing. ↩︎ ↩︎

  5. Schlag, S., Heuer, T., Gottesbüren, L., Akhremtsev, Y., Schulz, C., & Sanders, P. (2023). High-Quality Hypergraph Partitioning. ACM J. Exp. Algorithmics, 27(1.9), 39. ↩︎

  6. Zheng, D., Song, X., Yang, C., LaSalle, D., & Karypis, G. (2022). Distributed Hybrid CPU and GPU Training for Graph Neural Networks on Billion-Scale Heterogeneous Graphs. In Proceedings (pp. 4582–4591). :leftwards_arrow_with_hook: ↩︎

  7. Catalyurek, U., Devine, K., Fonseca Faraj, M., Gottesbüren, L., Heuer, T., Meyerhenke, H., Sanders, P., Schlag, S., Schulz, C., & Seemaier, D. (2022). More Recent Advances in (Hyper)Graph Partitioning. ↩︎ ↩︎

  8. Papa, D., & Markov, I. (2007). Hypergraph Partitioning and Clustering. In Handbook of Approximation Algorithms and Metaheuristics. ↩︎

  9. Karypis, G., Aggarwal, R., Kumar, V., & Shekhar, S. (1999). Multilevel hypergraph partitioning: applications in VLSI domain. IEEE Transactions on Very Large Scale Integration (VLSI) Systems, 7(1), 69–79. ↩︎

  10. Zhang, C., Cheng, W., Li, F., & Wang, X. (2024). Hypergraph-Based Influence Maximization in Online Social Networks. Mathematics, 12(17), 2769. ↩︎

  11. Wang, S., Cui, H., Qu, Y., & Yijia, Z. (2025). Multi-source biological knowledge-guided hypergraph spatiotemporal subnetwork embedding for protein complex identification. Briefings in Bioinformatics, 26. ↩︎

  12. Zhou, D., Huang, J., & Schölkopf, B. (2006). Learning with Hypergraphs: Clustering, Classification, and Embedding. In Advances in Neural Information Processing Systems 19 (2006), 1601–1608. ↩︎

  13. Qu C, Tao M, Yuan R. A Hypergraph-Based Blockchain Model and Application in Internet of Things-Enabled Smart Homes. Sensors (Basel). 2018 Aug 24;18(9):2784. doi: 10.3390/s18092784. PMID: 30149523; PMCID: PMC6164253. ↩︎

  14. K. Kumar et al. “SWORD: workload-aware data placement and replica selection for cloud data management systems”. In: The VLDB Journal 23 (Dec. 2014), pp. 845–870. doi: 10.1007/s00778-014-0362-1. ↩︎

  15. Chodrow, P.S., Veldt, N., & Benson, A.R. (2021). Generative hypergraph clustering: From blockmodels to modularity. Science Advances, 7. ↩︎ ↩︎

  16. Kolodziej, S., Mahmoudi Aznaveh, M., Bullock, M., David, J., Davis, T., Henderson, M., Hu, Y., & Sandstrom, R. (2019). The SuiteSparse Matrix Collection Website Interface. Journal of Open Source Software, 4, 1244. ↩︎ ↩︎

3 Likes
2

I’m personally looking forward to the new Hypergraph Partitioning Challenge, as hypergraphs are a crucial part of problem-solving in real-world applications. However, I would like the team to consider some suggestions regarding the baseline algorithm choice and its impact on the competitive nature of TIG.

Historically, many challenges in TIG have been dominated by greedy algorithms, which are often “optimized” for speed rather than solution quality. Given that solution quality and reliability are becoming central to the project, I believe it is important to establish a more complex and meaningful baseline algorithm, rather than relying on a greedy approach that may find solutions quickly but fails to capture the deeper structure of the problem. This is particularly concerning for large-scale and highly connected hypergraphs, where greedy methods tend to produce less reliable solutions.

To ensure that solution quality is a priority, I propose that the baseline algorithm should go beyond simple greedy bipartitioning. Instead, it could incorporate one of the following approaches:

  • Multi-Level Partitioning
  • Simulated Annealing (SA) or other refinement techniques
  • Spectral Partitioning

These are just initial suggestions, and I would be interested in hearing further input from the team.

With fuel limits increasing from 2 billion to 10 billion, we now have the ability to implement more complex and computationally feasible algorithms. This presents an opportunity to move beyond simple greedy approaches and encourage meaningful innovation rather than just incremental “tweaks” to existing heuristics.

Additionally, in real-world applications, purely greedy approaches are rarely used for hypergraph partitioning, as they tend to ignore global structure and fail to produce high-quality solutions. Aligning the baseline algorithm with more robust techniques would better reflect real-world problem-solving and support the project’s long-term goals.

I hope the team considers implementing a baseline solver that evaluates not just the number of solutions found, but also solution quality and reliability. This would create a more competitive and innovative challenge that pushes participants to develop truly novel approaches.

3 Likes
3

Hi Jake,

Thanks for contributing to our forum discussion! We’re also very excited about the Hypergraph Partitioning Challenge and want to design it as best as we can, so we really appreciate any and all feedback.

In case you weren’t aware, the team has been working on a new “Enhanced Block Reward Function” to counter the dominance of greedy algorithms across all challenges. We believe this issue stems from the current reward function prioritising the number of solutions found over the reliability of securing a solution for each instance attempted. You can find more details in our latest post here. We believe that once this mechanism is implemented, coupled with increased fuel limits, it will drive meaningful innovation by focusing more on solution quality and discouraging greedy approaches. This, in turn, will allow the better_than_baseline factor to more effectively incentivise higher solution quality.

The role of the baseline algorithm is to establish a performance target to be exceeded by the better_than_baseline factor for each randomly generated instance, creating a clear metric for what qualifies as a valid solution from one instance to the next. This is necessary because the randomness of our instance generation means we do not know the optimal solution in advance, so we need an alternative reference. We need the baseline to be efficient since instance generation is part of the solution verification process; if generating the instance took as long as solving it, we’d lose the asymmetry and compromise our Sybil-defense. Additionally, the baseline should offer a performance metric that is consistent and stable across instances—e.g., consistently about 20% below the best-known partition from a SOTA solver like KaHyPar. These two factors—efficiency and consistency—should be the only considerations for choosing our baseline solver.

Incentivising high-quality algorithms while ensuring that greedy approaches do not dominate is vital. To address these issues, we are implementing additional measures within the reward function and fuel limits. If our new updates or proposals don’t have the desired effect, we will continue to work on this issue until it is resolved.

Thanks again for your comments—we really appreciate them and hope this response clears up why the baseline isn’t a factor in incentivising higher solution quality and more innovative algorithms.

Thanks!

1 Like
4

Thank you for the response Aoibheann. I am glad to hear that you are actively working to address the issue with greedy algorithms. I did read that the reliability factor was going to be the base of the project soon, so hopefully all new challenges will adopt this.

I do understand where you are coming from about the baseline and it does make sense that it needs to be efficient and consistent, especially considering the instance verification and Sybil defense. I didn’t give that a thought in my response to be honest.

That said, I do still feel there’s an important balance to strike. Even with the new reward system and fuel limits encouraging better approaches, there’s always the risk that “innovators” focus on just beating a weak baseline rather than pushing toward truly high-quality solutions.

Would it make sense to benchmark the baseline against something stronger (like KaHyPar or a multi-level solver) to make sure the better_than_baseline factor actually challenges people to innovate? If the baseline is too weak, even with the reward function changes, we might not be making full use of the computational resources now available.

I look forward to seeing how this all plays out :slight_smile:

2 Likes