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. ↩︎ ↩︎

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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.

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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
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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:

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Hypergraph Partitioning Challenge Design Update

Random Instance Generation

Motivation

To ensure that the challenge drives progress on practically relevant problems,
our synthetic instances must resemble the hypergraphs most frequently encountered in the literature that resemble real world workloads. We therefore aim to match key structures found in hypergraphs from the SuiteSparse Matrix Collection [1] (used as a data source for evaluating hypergraph partitioners by Knottenbelt et. al. [2]).

Methodology

We generate each instance with HyperLap [3], a parallelizable, hierarchical hypergraph extension of the Fast Chung-Lu (FCL) model [4], that preserves an input hypergraph’s degree distribution by sampling hyperedges according to expected node degrees. HyperLap extends this model by employing a hierarchical multilevel partitioning scheme designed to reproduce the heavy-tailed, community-centric overlap patterns observed in real-world hypergraphs. HyperLap also maps naturally onto GPUs, which was a major motivation for this choice of hypergraph generator.

Fixed Parameters

  • Nodes vs. hyperedges. The number of hyperedges is one of the difficulty parameters, num_hyperedges. To reflect the sparsity pattern observed in SuiteSparse matrices, the number of nodes is chosen to be approximately equal to num_hyperedges. In practice, the generation method produces a node count around 92% of the hyperedge count.
  • Uniform weights and costs. All node weights and hyperedge costs are one, w[n]=c[n]=1, so the objective simplifies to minimising the connectivity metric now defined as \sum_n (\lambda_n-1).
  • Node degree and hyperedge size distribution. Node degrees and hyperedge sizes are sampled from truncated power-law distributions. This method effectively captures the heavy-tailed distributions commonly observed in real-world hypergraphs.

Given:

  • an array of node degrees;
  • an array of desired hyperedge sizes; and
  • a level‑weight vector \mathbf{p}=(p_1,\dots,p_L) satisfying \sum_{\ell}p_\ell=1,

the generator proceeds in two stages:

  1. Hierarchical layout:
    For num_hyperedges, we create L=\lfloor\log_2\text{num_hyperedges}\rfloor levels. Each level \ell contains exactly 2^{\ell-1} nested groups of equal size: if two nodes share a group at level \ell, they also share a group at every lower level.

  2. Edge construction:
    For every desired hyperedge size s we
    i) sample a level \ell with probability proportional to p_\ell;
    ii) choose one of the 2^{\ell-1} groups at that level uniformly at random; and
    iii) repeatedly sample nodes within the chosen group proportional to their node degree until s distinct nodes are selected.

This method ensures the preservation of both the input degree sequence and hyperedge size distribution while introducing realistic community structures and overlaps.

Interpreting the Level-Weight Vector

The level-weight vector \mathbf{p} dictates the resolution at which hyperedges form: low levels span large node sets, high levels cover tiny groups. Shifting probability mass therefore tunes how strongly hyperedges overlap and how “visible’’ the community structure is:

  1. Low-level weight dominance (p_1{+}p_2 \approx 1). Edges sampled from the coarsest groups, so overlaps are largely accidental and community structure is weak, these instances would be hard for partitioners to solve due to their lack of structure.
  2. High-level weight dominance (p_{L,L-1,...}\gg p_{1,2, ...}). Edges stay inside very small groups, yielding tight micro-communities; coarse partitioning would be relatively easy.
  3. Multi-dominant low levels (p_2{+}p_3{+}p_4{+}p_5\approx 1). Weight spread over the first few refinements produces a realistic, multiscale hierarchy with heavy-tailed overlaps.
  4. One-level dominant. Nearly all mass on one level k makes block-diagonal structure at a single, known scale - easy to partition if k is exploited.

Using HyperLap+ (which automatically fits optimal level-weight vectors to observed hypergraph data) we observed two dominant regimes in SuiteSparse hypergraphs: a single-level dominant pattern and a multi-dominant low-level pattern. We adopt the latter pattern due to its richer overlap structure, posing greater challenges for partitioning algorithms.

Baseline Calculation

Our chosen baseline is a greedy bipartition algorithm. The algorithm recursively partitions the node set into two subsets until the desired number of partitions is reached. Before applying any greedy refinements, the algorithm begins with an initial bipartition seeded by the two level-1 groups produced by the hypergraph generation method. Utilizing this initial partition capitalizes on the inherent structure of the node set, providing an effective starting point that results in a more stable baseline partition. Each subsequent bipartitioning step proceeds as follows:

  1. Determine target sizes. Given a current subset of nodes, calculate how many nodes 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 nodes by degree. For the nodes in the current subset, compute their degrees (the number of hyperedges to which each node belongs). Sort them in descending order so that higher-degree nodes are placed first.
  3. Place nodes greedily. Initialize two arrays (one per part) to track hyperedges already “activated” (those with at least one node assigned) within each part. For each node in sorted order:
  • Count how many of its hyperedges are activated in the left part and how many in the right.
  • If one side has a strictly higher overlap, assign the node to that side (provided it has not reached its target size). If overlaps are equal, assign to the part with fewer nodes. If one part has already reached capacity, the node is assigned to the other part by default.
  • Continue assigning nodes until one part reaches its target size, then assign any remaining nodes to the other part.
  1. Recursive Subdivision. After producing each bipartition, recursively apply the same procedure to each newly formed part until the desired number of parts (e.g., 64) is reached.

Finally, the connectivity metric of the complete multiway partition is computed, giving the baseline_value. Although this local greedy strategy does not capture global interactions perfectly, it remains computationally efficient, intuitive, and serves as a solid performance benchmark for more sophisticated methods.

Expert Review

We are collaborating with a leading expert in combinatorial optimisation, machine learning, submodular optimisation, and graph and hypergraph algorithms to review and refine the design of our upcoming Hypergraph Partitioning Challenge. This consultation is ongoing, and any adjustments or enhancements resulting from this expert review will be shared and implemented once finalized.

  1. Scott Kolodziej et al. The SuiteSparse Matrix Collection Website Interface. In: Journal of Open Source Software 4 (Mar. 2019), p. 1244. doi: 10.21105/joss.01244. ↩︎

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

  3. Geon Lee, Minyoung Choe, and Kijung Shin. How Do Hyperedges Overlap in Real-World Hypergraphs? – Patterns, Measures, and Generators. 2021. arXiv: 2101.07480 [cs.SI]. url: [2101.07480] How Do Hyperedges Overlap in Real-World Hypergraphs? -- Patterns, Measures, and Generators. ↩︎

  4. Fan Chung and Linyuan Lu. 2002. The average distances in random graphs with
    given expected degrees    . PNAS 99, 25 (2002), 15879–15882. ↩︎