ComPart: Community-Guided Post-Coarsening for High-Quality Hypergraph Partitioning

Abstract: Hypergraph partitioning is a critical step in the design of complex embedded systems, essential for optimizing task mapping on heterogeneous MPSoCs and enabling multi-FPGA prototyping. Many existing methods rely on community detection to identify modules with dense internal and sparse external connections, typically utilizing them to constrain the coarsening phase--a widely adopted paradigm. In this work, we propose ComPart, a generalized framework that integrates diverse community detection methods to uncover high-quality clusterings throughout the post-coarsening stages (i.e., initial partitioning and uncoarsening). These discovered clusterings serve as distinct structural guides, enabling the refinement process to identify superior partitioning solutions. Our framework offers two key advantages: (1) it establishes a new paradigm that leverages community structures detected during uncoarsening to escape local optima and explore globally meaningful solution subspaces, transcending the limitations of standard local refinements; and (2) it flexibly accommodates both existing and future community detection methods. Furthermore, we theoretically generalize locally-dense decomposition--originally from graphs--to the hypergraph domain. We provide the formal extension and necessary proofs to apply this technique to hypergraphs, marking its first application in hypergraph partitioning. Specifically, we utilize this rigorously derived decomposition to guide the initial partitioning phase toward superior starting points. Experimental results on standard benchmarks demonstrate that our method consistently outperforms state-of-the-art methods in solution quality.

• The paper introduces ComPart, which applies community detection during initial partitioning and uncoarsening, enabling coordinated sub-community moves through ILP or KaHyPar’s V-cycle instead of relying only on vertex-level refinement.
• ComPart consistently outperforms KaHyPar and other baselines on Titan23 and ISPD98, reducing geometric-mean cut sizes versus KaHyPar by up to 5.74% and delivering its strongest gains for direct multi-way partitioning.
• The paper generalizes locally dense decomposition to weighted hypergraphs with a provably unique density hierarchy and globally convergent optimization, although ILP scalability and heuristic clique expansion remain practical limitations.

Overview

Hypergraph partitioning is a core algorithmic kernel in embedded system design, underpinning task mapping on heterogeneous MPSoCs and multi-FPGA logic emulation. The dominant approach is the multi-level framework—coarsening, initial partitioning, and uncoarsening with local refinement—embodied in solvers such as PaToH, hMETIS, and KaHyPar. Prior community-aware methods, including KaHyPar's Louvain-guided coarsening, SHyPar's flow-based community detection, and GenPart's GCN-generated initial solutions, almost exclusively exploit community structure during coarsening. ComPart departs from this convention by applying community detection to the post-coarsening phases—initial partitioning and uncoarsening—thereby establishing a complementary paradigm in which detected communities serve as structural guides for global solution adjustments rather than coarsening constraints.

Framework design

ComPart is built on KaHyPar's $n$-level architecture, in which each coarsening step merges one vertex pair. During uncoarsening, community detection is triggered at $\alpha$ distinct levels whose node counts follow a geometric progression between the coarsest node count $n_c$ and $n$. This spacing follows the "Kronecker product" principle: topological detail evolves smoothly as node counts grow proportionally, so structural signals detected at these intervals are more reliable.

At each trigger level, two stages execute. First, a community detection algorithm is applied to the hypergraph or, for graph-only methods, to a clique-model conversion in which each hyperedge $e$ becomes a clique with pairwise weights $w_e/(|e|-1)$. Hyperedges larger than a threshold ($S_2 = 200$) are ignored, and mid-sized hyperedges are held in a queue used to connect the graph if clique expansion alone leaves it disconnected. The paper uses four igraph methods—Label Propagation, Walktrap, Louvain, and Fast Greedy. Detected clusters are classified as intra-partition communities or edge communities spanning partitions; edge communities are intersected with partition boundaries to yield sub-communities.

Second, sub-community reassignment is formulated as an ILP over binary assignment variables $z_{u,i}$ and hyperedge containment variables $h_{e,i}$, maximizing the weight of uncut hyperedges subject to the balance constraint. Because ILP is expensive, the paper adopts a size-dependent strategy: small hypergraphs are solved to optimality (warm-started with the current partition), medium hypergraphs terminate at a 1% optimality gap, and large hypergraphs substitute KaHyPar's V-cycle for the ILP. The practical consequence is that entire community fragments move simultaneously, enabling large-scale adjustments that vertex-wise FM refinement cannot reach—directly addressing FM's tendency to stall in local optima.

Locally-dense decomposition on hypergraphs

The second contribution is a formal generalization of locally-dense decomposition (LDD) from graphs to hypergraphs with positive real vertex weights, applied for the first time in hypergraph partitioning. The decomposition produces a strictly nested chain of vertex sets $$B_0 \subsetneq B_1 \subsetneq \cdots \subsetneq B_\gamma = \mathcal{V}$$, where each $\alpha$0 is the maximal set strictly containing $\alpha$1 maximizing the ratio of internal hyperedge weight to the vertex weight of the added layer.

The paper establishes three results: (1) the decomposition is unique, proved by induction via an exchange argument on the union of differing sets; (2) in any optimal solution of an associated quadratic program, if $\alpha$2 within a hyperedge, then $\alpha$3—flow is only assigned to minimum-load vertices, proved by a shifting argument showing any other allocation admits a strictly objective-reducing perturbation; and (3) the optimal $\alpha$4 values coincide exactly with the per-layer density ratios $\alpha$5, which are strictly decreasing. Unlike heuristic clustering, this decomposition is a convex optimization problem, so convergence to the global optimum is guaranteed. The authors solve it with a block-coordinate descent scheme using water-filling per hyperedge; convergence follows from continuous differentiability and blockwise strict convexity of the objective (positive-definite diagonal Hessian).

The decomposition is used at the coarsest level, where coarsening has already collapsed local dense structures into single nodes, so the globally derived density hierarchy provides complementary guidance for constructing a better initial partition.

Experimental results

Experiments use the Titan23 and ISPD98 benchmarks with $\alpha$6 and equal wall-clock budgets for ComPart, hMETIS, and KaHyPar; SHyPar and SpecPart results are cited or executed where feasible. ComPart uses $\alpha$7 with Fast Greedy. The implementation is based on KaHyPar (cut_kKaHyPar_sea20.ini configuration).

Normalized cut sizes (geometric mean relative to KaHyPar) are:

Benchmark	$\alpha$8	hMETIS	K-SpecPart	KaHyPar	SHyPar	ComPart
Titan23	2	1.035	0.975	1	0.972	0.964
Titan23	3	1.102	1.016	1	—	0.943
Titan23	4	1.131	1.080	1	—	0.955
ISPD98	2	1.009	0.987	1	0.981	0.979
ISPD98	3	1.010	0.994	1	0.985	0.959
ISPD98	4	1.041	1.058	1	1.016	0.989

ComPart outperforms all baselines in every configuration. On Titan23 it improves over KaHyPar by 3.62% ($\alpha$9), 5.74% ($n_c$0), and 4.50% ($n_c$1), and over the best baseline (SHyPar) by 0.87% at $n_c$2. On ISPD98 the corresponding gains over KaHyPar are 2.10%, 4.11%, and 1.14%. Gains are consistently larger for $n_c$3 than for bipartitioning, indicating that existing methods are comparatively stronger at $n_c$4 and that community-guided post-coarsening is most valuable in direct multi-way partitioning. The larger improvement on Titan23 is attributed to the community detection strategy handling large hyperedges more effectively than the baselines.

The ablation isolates the two components: community detection during uncoarsening (U) alone yields a 3.6% improvement over KaHyPar, while LDD-guided initial partitioning (I) alone yields 0.7%. U is therefore the primary driver of the gains, with I providing a consistent but smaller contribution. Among the four detection strategies, Fast Greedy and Louvain deliver the strongest quality improvements; Label Propagation and Walktrap contribute modestly. A notable caveat is that SHyPar could not be evaluated on Titan23 for $n_c$5 due to excessive runtime (over 20 hours on medium-sized datasets), so the $n_c$6 comparisons on Titan23 rest on the cited $n_c$7 behavior and the other baselines.

Limitations and open questions

The paper concedes several constraints. The ILP stage does not scale, forcing a fallback to the V-cycle for large hypergraphs, which weakens the theoretical guarantee of globally optimal community reassignment precisely on the instances where partitioning is hardest. The clique-model conversion discards hyperedges above a size threshold and injects negligible-weight bridging edges when the graph remains disconnected; both are heuristic choices whose sensitivity to $n_c$8, $n_c$9, and bridging is not studied. The level-triggering schedule is motivated by the Kronecker principle but the choice of $n$0 and the geometric spacing are not ablated independently of the detection algorithm. The detailed uniqueness proof of Theorem 1 is omitted for brevity, and the runtime overhead of the water-filling BCD solver relative to heuristic clustering is not reported. Open questions include whether the LDD hierarchy can guide refinement at intermediate (rather than only coarsest) levels, and how the framework interacts with flow-based refinement within KaHyPar.

Conclusion

ComPart repositions community detection from coarsening guidance to post-coarsening optimization, combining periodic community-guided global moves (via ILP or V-cycle) with a provably convergent hypergraph locally-dense decomposition for initial partitioning. The theoretical extension of LDD to weighted hypergraphs is rigorous, and empirical results show consistent state-of-the-art cut quality on Titan23 and ISPD98, with the largest gains in direct $n$1-way partitioning and on hypergraphs with large hyperedges.